# References — Chapter ᚂ(L): Garden variety (part 1) *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ref-ch-4, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [The chapter](ch04.md) *Each entry: one plain sentence saying exactly what it is cited for, its status, then every source with a note saying what that source actually carries. "Own computation" / "Own reading" entries are the author's own work, not outside sources.* ### 4.1 {#r-4.1} The chapter’s elementary identities and classifications of integers, rationals, and algebraic numbers are correct, and whether any factorial other than 4!, 5!, and 7! is one less than a square is Brocard’s problem, which is open. 42 is the Catalan number C₅, which counts the ways to bracket six factors. 0.999… and 1 are exactly the same number: the 10x − x argument shows it informally, proofs by series and limits show it rigorously, and the misconception is old. φ is defined as the positive root of x² − x − 1, so φ² − φ − 1 = 0. The silver ratio 1 + √2 ≈ 2.414 is the positive root of x² − 2x − 1 and the limit of ratios of consecutive Pell numbers; it is constructible. Sixty divides evenly by 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30, and base sixty survives in the 60 minutes of an hour, the 60 seconds of a minute, and the 360 degrees of a circle. 4 = 2² and 9 = 3² are the first two square numbers after 1. The decimal expansion of 1/3 repeats the digit 3 forever. 0! = 1 is the empty product and the one arrangement of nothing; it keeps n! = n·(n−1)! valid at n = 1, makes the binomial coefficients and the power series of e^x come out right, and matches Γ(1) = 1. *Status: confirmed* - [DLMF, §24.2 Bernoulli and Euler numbers](https://dlmf.nist.gov/24.2) — confirms the Bernoulli-number table and generating-function conventions. - [Wolfram MathWorld, Algebraic Number](https://mathworld.wolfram.com/AlgebraicNumber.html) — confirms the algebraic-number definition and the listed algebraic constants. - [Wikipedia, Brocard’s problem](https://en.wikipedia.org/wiki/Brocard%27s_problem) — n! + 1 = m² is known only for n = 4, 5, and 7; whether there are others is open, no further solution has been found below n = 10¹⁵, and Erdős conjectured there are none. - [Catalan number, Wikipedia](https://en.wikipedia.org/wiki/Catalan_number) — “The first Catalan numbers for n = 0, 1, 2, 3, ... are 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, ...”; C_n is “the number of different ways n + 1 factors can be completely parenthesized” - [0.999..., Wikipedia](https://en.wikipedia.org/wiki/0.999...) — “Despite common misconceptions, 0.999... is not “almost exactly 1” or “very, very nearly but not quite 1”; rather, “0.999...” and “1” represent exactly the same number.”; Byers’s argument displayed: x = 0.999…, 10x = 9.999…, 10x = 9 + x, 9x = 9, x = 1; “Other proofs generally involve basic properties of real numbers and methods of calculus, such as series and limits.”; “the 1895 Arithmetic for Schools says, “when a large number of 9s is taken, the difference between 1 and .99999... becomes inconceivably small.”” - [Golden ratio, Wikipedia](https://en.wikipedia.org/wiki/Golden_ratio) — the defining polynomial: “Its minimal polynomial, the polynomial of lowest degree with integer coefficients that has the golden ratio as a root, is x^2 − x − 1.”; “φ is defined as the positive solution.” - [Silver ratio, Wikipedia](https://en.wikipedia.org/wiki/Silver_ratio) — polynomial, value, and Pell limit: “It follows that the silver ratio is the positive solution of quadratic equation σ^2 − 2σ − 1 = 0. The quadratic formula gives the two solutions 1 ± √2, the decimal expansion of the positive root begins with 2.414213562373095...”; of the Pell numbers: “The first few terms are 0, 1, 2, 5, 12, 29, 70, 169,...”; “The limit ratio of consecutive terms is the silver mean.” - [Constructible number, Wikipedia](https://en.wikipedia.org/wiki/Constructible_number) — constructibility of numbers built from square roots: “Equivalently, r is constructible if and only if there is a closed-form expression for r using only integers and the operations for addition, subtraction, multiplication, division, and square roots.” - [Sexagesimal, Wikipedia](https://en.wikipedia.org/wiki/Sexagesimal) — carries the divisors, the hour, and the degree: “The number 60, a superior highly composite number, has twelve divisors, namely 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60, of which 2, 3, and 5 are prime numbers. With so many factors, many fractions involving sexagesimal numbers are simplified.”; “One hour of time is divided into 60 minutes, and one minute is divided into 60 seconds.”; “Similarly, the practical unit of angular measure is the degree, of which there are 360 (six sixties) in a circle.” - [Square number, Wikipedia](https://en.wikipedia.org/wiki/Square_number) — lists the small squares: “The squares (sequence A000290 in the OEIS) smaller than 60² = 3600 are:” … “2² = 4” … “3² = 9” - [Repeating decimal, Wikipedia](https://en.wikipedia.org/wiki/Repeating_decimal) — “the decimal representation of 1/3 becomes periodic just after the decimal point, repeating the single digit “3” forever, i.e. 0.333....” - [Factorial, Wikipedia](https://en.wikipedia.org/wiki/Factorial) — § Factorial of zero: “the definition of n! as a product involves the product of no numbers at all, and so is an example of the broader convention that the empty product, a product of no factors, is equal to the multiplicative identity.”; “There is exactly one permutation of zero objects: with nothing to permute, the only rearrangement is to do nothing.”; “the number of ways to choose all n elements from a set of n is (n choose n) = n!/(n!0!) = 1, a binomial coefficient identity that would only be valid with 0!=1.”; “With 0!=1, the recurrence relation for the factorial remains valid at n=1.”; “Setting 0!=1 allows for the compact expression of many formulae, such as the exponential function, as a power series”; “This choice matches the gamma function 0! = Γ(0+1) = 1”; and “there are n! different ways of arranging n distinct objects into a sequence.” ### 4.2 {#r-4.2} 1729 is the smallest number expressible as two positive cubes in two distinct ways, and subtracting the two representations gives the displayed zero. *Status: confirmed* - [Wolfram MathWorld, Taxicab Number](https://mathworld.wolfram.com/TaxicabNumber.html) — lists Taxicab(2)=1729 and its two cube representations. ### 4.3 {#r-4.3} The named integer curiosities, perfect and amicable numbers, Kaprekar’s constant, the cyclic number 142857, the Answer of Douglas Adams’s Deep Thought, and the displayed arithmetic values have the stated definitions. Belphegor’s prime, 1000000000000066600000000000001, is a 31-digit palindromic prime named by Clifford A. Pickover after the demon Belphegor, for its 666 and two runs of thirteen zeros. Deep Thought’s words are quoted from the novel, from “The Answer to the Great Question . . .” to “Forty-two,” said Deep Thought, with infinite majesty and calm. Keeping leading zeros, any four-digit number with at least two distinct digits reaches 6174 within seven iterations; if they are dropped, some reach 0 and up to eight are needed. *Status: confirmed* - [Wolfram MathWorld, Kaprekar Routine](https://mathworld.wolfram.com/KaprekarRoutine.html) — confirms 6174 and the four-digit routine. - [Wolfram MathWorld, Perfect Number](https://mathworld.wolfram.com/PerfectNumber.html) — confirms the proper-divisor definition and the first even perfect numbers; the Euclid form is perfect “as stated in Proposition IX.36 of Euclid’s Elements,” and “In a posthumous 1849 paper, Euler provided the first proof that Euclid’s construction gives all possible even perfect numbers.” - [Wikipedia, Amicable Numbers](https://en.wikipedia.org/wiki/Amicable_numbers) — confirms 220 and 284 and their divisor sums. - [Wolfram MathWorld, Cyclic Number](https://mathworld.wolfram.com/CyclicNumber.html) — 142857 is the cyclic number generated by 1/7, a cyclic number being an (n−1)-digit integer whose products with 1, 2, …, n−1 are its own digits reordered; 7 is the first full reptend prime, and multiplying a cyclic number by its generator gives a string of nines. - [Wikipedia, Phrases from The Hitchhiker’s Guide to the Galaxy](https://en.wikipedia.org/wiki/Phrases_from_The_Hitchhiker%27s_Guide_to_the_Galaxy) — Deep Thought’s Answer is 42, first broadcast in Fit the Fourth on 29 March 1978 and published in the novel of 1979; Adams wrote on 3 November 1993 that “It was a joke. It had to be a number, an ordinary, smallish number, and I chose that one.” - [Belphegor’s prime, Wikipedia](https://en.wikipedia.org/wiki/Belphegor%27s_prime) — “Belphegor’s prime is the palindromic prime number 1000000000000066600000000000001”; “The name “Belphegor’s prime” was coined by author Clifford A. Pickover in 2012.”; “Belphegor is one of the Seven Princes of Hell”; “This number is surrounded on either side by thirteen zeroes and is 31 digits in length (thirteen reversed)” - [Quote Origin: The Answer to Life, the Universe and Everything Is Forty-Two, Quote Investigator](https://quoteinvestigator.com/2022/10/29/answer-universe/) — quotes the novel’s passage: ““All right,” said Deep Thought. “The Answer to the Great Question . . .”” … ““Of Life, the Universe and Everything . . .” said Deep Thought.” … ““Is . . .” said Deep Thought, and paused.” … ““Forty-two,” said Deep Thought, with infinite majesty and calm.” - [Kaprekar’s routine, Wikipedia](https://en.wikipedia.org/wiki/Kaprekar%27s_routine) — “Any four-digit number (in base 10) with at least two distinct digits will reach 6174 within seven iterations.”; “When leading zeroes are retained, only repdigits lead to the trivial Kaprekar’s constant.” ### 4.4 {#r-4.4} The large-number values and historical attributions in the integer brace, including Eddington’s number, Shannon’s chess-game estimate, googol/googolplex, and Skewes’ bound, are as stated. Leonhard Euler proved 2³¹ − 1 prime in 1772 and it held the record for the largest known prime until 1867, when Fortuné Landry found (2⁵⁹ − 1)/179951 = 3,203,431,780,337. 2³¹ − 1 = 2,147,483,647; Euler’s proof used trial division; it is the largest value of a signed 32-bit integer, hexadecimal 7FFFFFFF, and as seconds from 1970 it ends at 19 January 2038. Moser’s number is 2 in a megagon in Steinhaus–Moser notation, Moser’s extension of Steinhaus’s polygons; it exceeds Skewes’ number and is far below Graham’s. Archimedes’ Sand Reckoner, third century BC, used Aristarchus’ universe and his orders and periods to bound the grains of sand filling it by 10^63. *Status: confirmed* - [Wikipedia, Large Numbers](https://en.wikipedia.org/wiki/Large_numbers) — gives the values of googol, googolplex, and the first Skewes’s number (approximately 10^10^10^34), and the Shannon number for chess, estimated at 10^120; the historical attributions are on this entry’s other lines. - [Wikipedia, Shannon Number](https://en.wikipedia.org/wiki/Shannon_number) — confirms Shannon’s 10^120 lower-bound estimate. - [Wikipedia, Skewes’ Number](https://en.wikipedia.org/wiki/Skewes%27_number) — confirms the historical bound and the modern upper bound. - [Largest known prime number, Wikipedia](https://en.wikipedia.org/wiki/Largest_known_prime_number) — read as raw HTML. Its “Largest known prime by year” table gives, consecutively, “M₁₇ | 6 | 1588 | Pietro Cataldi,” “M₁₉ | 6 | 1588 | Pietro Cataldi,” “M₃₁ | 10 | 1772 | Leonhard Euler,” “M₅₉/179951 | 13 | 1867 | Fortuné Landry” and “M₁₂₇ | 39 | 1876 | Édouard Lucas,” which is the 95-year gap the #m31 card names. - [2,147,483,647, Wikipedia](https://en.wikipedia.org/wiki/2,147,483,647) — “By 1772, Leonhard Euler had proven that 2,147,483,647 is a prime.”; “Euler used trial division, improving on Pietro Cataldi’s method, so that at most 372 divisions were needed.”; “In 1867, the number 3,203,431,780,337 was proven to be prime.”; “In computing, this number is the largest value that a signed 32-bit integer field can hold.”; “hexadecimal 7FFFFFFF”; “The latest time that can be represented in this form is 03:14:07 UTC on Tuesday, 19 January 2038 (corresponding to 2,147,483,647 seconds since the start of the epoch).”; “This means that systems using a 32-bit time_t type are susceptible to the Year 2038 problem.”; “On 1 January 2022, a bug was reported for Microsoft Exchange systems where email delivery would fail. An internal malware scanner (enabled by default since 2013) used the date and time as a signed 32-bit integer.” - [Steinhaus–Moser notation, Wikipedia](https://en.wikipedia.org/wiki/Steinhaus%E2%80%93Moser_notation) — “It is an extension (devised by Leo Moser) of Hugo Steinhaus’s polygon notation.”; “a number n in a triangle means n^n”; “a number n in a square is equivalent to “the number n inside n triangles, which are all nested.””; “Moser’s number is the number represented by “2 in a megagon.””; “Therefore, Moser’s number, although incomprehensibly large, is vanishingly small compared to Graham’s number” - [big numbers, Susan Stepney (University of York)](https://www-users.york.ac.uk/~ss44/cyc/b/big.htm) — gives Skewes’ number as e^e^e^79, approximately 10^10^10^34, “< 4^^5, occurs in theories on the distribution of primes”; with Wikipedia’s mega ≈ (256↑)^256 257 this puts Moser’s number far above it - [The Sand Reckoner, Wikipedia](https://en.wikipedia.org/wiki/The_Sand_Reckoner) — “The Sand Reckoner (Greek: Ψαμμίτης, Psammites) is a work by Archimedes, an Ancient Greek mathematician of the 3rd century BC, in which he set out to determine an upper bound for the number of grains of sand that fit into the universe.”; “To do this, he used the heliocentric model of Aristarchus of Samos.”; “He then constructed the orders of the second period by taking multiples of this unit”; “Multiplying 10^21 by 10^42 yields 10^63, the number of grains of sand in the Aristarchian Universe.” ### 4.5 {#r-4.5} The notation and comparisons involving Knuth arrows, Conway chains, hyperoperations, the fast-growing hierarchy, TREE(3), SSCG(3), Graham’s number, and Loader’s number are standard as qualified in the cards. Loader’s number is said on a community wiki (the Googology Wiki line below) to be far larger than TREE(3), a comparison this book does not rest on, and the Busy Beaver function grows faster than any computable function. The last digits of Graham’s number are known; its leading digits are not. 3↑↑4 has about 3.6 trillion digits: more than the 41,024,320 digits of the largest known prime (September 2026), fewer than the googolplex, 10^(10^100). Skewes’ 1933 bound is below 10^(10^(10^34)), far below a tower of 7,625,597,484,987 threes. SSCG(k) is Harvey Friedman’s function: the longest sequence of simple subcubic graphs, the i-th on at most i + k vertices, with no earlier one homeomorphically embeddable in a later one; the Robertson–Seymour theorem makes every such sequence finite; SSCG(0) = 2, SSCG(1) = 5, SSCG(2) = 3·2^(3·2^95) − 8, and SSCG(3) is greater than TREE(3). *Status: confirmed* - [Wikipedia, Knuth’s up-arrow notation](https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation) — confirms the recursive arrow definitions and 3↑↑3 value. - [Wikipedia, Conway chained arrow notation](https://en.wikipedia.org/wiki/Conway_chained_arrow_notation) — confirms the chain notation and its growth comparisons. - [Wikipedia, Fast-growing hierarchy](https://en.wikipedia.org/wiki/Fast-growing_hierarchy) — confirms the ordinal-indexed hierarchy and its use for growth comparison; it gives “f_{ω+1}(64) = f_{ω}^{64}(64) > Graham’s number” (sub- and superscripts as typeset on the page). - [Wikipedia, Graham’s number](https://en.wikipedia.org/wiki/Graham%27s_number) — the number Gardner published in 1977 is not the bound in Graham and Rothschild’s 1971 paper, which is smaller; the best known bounds on the original problem are 13 (Barkley, 2008) below and, above, the 2014 bound 2↑↑(2↑↑(3 + 2↑↑8)) of Lavrov, Lee and Mackey, sharpened again in 2019; the last ten digits are …2464195387; Martin Gardner described it in Scientific American “in November 1977,” writing that Graham had established “a bound so vast that it holds the record for the largest number ever used in a serious mathematical proof.”; “At the time of its introduction, it was the largest specific positive integer ever to have been used in a published mathematical proof.” - [Loader’s number, Googology Wiki](https://googology.miraheze.org/wiki/Loader's_number) — community wiki, not primary: “The final output of D^5(99) is much larger than TREE(3), SCG(13), and, say, BH(100).” - [Busy beaver, Wikipedia](https://en.wikipedia.org/wiki/Busy_beaver) — “This is a noncomputable function, because it grows asymptotically faster than any computable function.” - [Largest known prime number, Wikipedia](https://en.wikipedia.org/wiki/Largest_known_prime_number) — perishable: “The largest known prime number as of September 2026 is 2^136,279,841 − 1, a number that has 41,024,320 digits when written in the decimal system.” - [Googolplex, Wikipedia](https://en.wikipedia.org/wiki/Googolplex) — “A googolplex is the large number 10^(10^100), that is, 10 raised to the power of a googol.” - [Skewes’s number, Wikipedia](https://en.wikipedia.org/wiki/Skewes%27s_number) — “Skewes (1933) proved that, assuming that the Riemann hypothesis is true, there exists a number x violating π(x) < li(x), below e^e^e^79 < 10^10^10^34.” The comparison with 3↑↑↑3 is arithmetic. - [Friedman’s SSCG function, Wikipedia](https://en.wikipedia.org/wiki/Friedman%27s_SSCG_function) — “Friedman’s SSCG function is a mathematical function defined by Harvey Friedman.”; definition (displayed): “There is a sequence G1,…,Gn of simple subcubic graphs such that each Gi has at most i+k vertices and for no i < j is Gi homeomorphically embeddable into Gj.”; “The Robertson–Seymour theorem proves that subcubic graphs (simple or not) are well-quasi-ordered by homeomorphic embeddability, implying such a sequence cannot be infinite.”; “The first few terms of the sequence are SSCG(0) = 2, SSCG(1) = 5, and SSCG(2) = 3·2^(3·2^95) − 8” (displayed formulas); “It has been shown that the next term, SSCG(3), is greater than TREE(3).” - [Marco Ripà, Graham’s number stable digits: An exact solution (Notes on Number Theory and Discrete Mathematics 31(3), 2025, 607–616; arXiv 2024)](https://arxiv.org/abs/2411.00015) — read in the arXiv HTML full text: the most significant digit of G is 1 in binary and in ternary, but “the problem of expressing the most significant digit of G in the decimal number system is still open.” ### 4.6 {#r-4.6} 65537 is the fifth known Fermat prime and the largest known; the regular 65537-gon can be drawn with compass and straightedge, by the sufficiency Gauss proved and the necessity Wantzel proved in 1837; that no sixth Fermat prime exists is a heuristic expectation and not a theorem. 65537 is commonly used as the public exponent in RSA. *Status: judgement* - [Wolfram MathWorld, 65537-gon](https://mathworld.wolfram.com/65537-gon.html) — confirms constructibility from 65537 being a Fermat prime and attributes the result to Gauss. - [Wikipedia, Fermat number](https://en.wikipedia.org/wiki/Fermat_number) — confirms 65537 is the largest known Fermat prime. - [Boklan and Conway, Expect at most one billionth of a new Fermat prime!](https://arxiv.org/abs/1605.01371) — gives the heuristic evidence for no further Fermat primes. - [Wikipedia, Constructible polygon](https://en.wikipedia.org/wiki/Constructible_polygon) — Gauss proved the Fermat-prime condition sufficient and stated its necessity without proof; Wantzel gave the full proof of necessity in 1837, which is why the result is the Gauss–Wantzel theorem. - [65,537, Wikipedia](https://en.wikipedia.org/wiki/65,537) — “65537 is commonly used as a public exponent in the RSA cryptosystem.” Editorial note: The page hedges to “almost certainly the last” rather than asserting it. What is proved is that 65537 is the largest Fermat prime anyone has found; Boklan and Conway put the chance of another at under one in a billion, which is evidence and not a proof. On the constructibility, Gauss’s celebrated result of 1796 is the 17-gon; the general criterion is his Disquisitiones Arithmeticae of 1801, and the converse half is Wantzel’s. ### 4.7 {#r-4.7} The rational-number identities, continued-fraction facts, Bernoulli values, and harmonic-number statement are correct. The first harmonic numbers are 1, 3/2, 11/6, and 25/12, so H₄ is the first whose denominator (12) is not a factorial (1, 2, and 6 are). Hₙ − ln n decreases towards Euler’s γ ≈ 0.5772, so Hₙ grows like ln n + γ. 355/113, the Milü found by Zu Chongzhi in the fifth century, is accurate to six decimal places, and the next better rational approximation of π, by size of denominator, is 52163/16604.Counting the initial 3 as the first convergent, 22/7 and 355/113 are the second and fourth convergents of π = [3;7, 15, 1,…]; both π and 355/113 round to 3.141593 at six decimal places. *Status: confirmed* - **Own computation** (the author's own work) — direct high-precision evaluation of π gives initial partial quotients 3, 7, 15, 1. Their exact truncations are 3, 22/7, 333/106, and 355/113. π ≈ 3.141592653589793 and 355/113 ≈ 3.141592920353982, both rounding to 3.141593 at six places. - [Wolfram MathWorld, Continued Fraction](https://mathworld.wolfram.com/ContinuedFraction.html) — the simple-continued-fraction notation (convergents are only linked from it, under See also; the π approximations are on the Milü line). - [DLMF, §24.2 Bernoulli and Euler numbers](https://dlmf.nist.gov/24.2) — confirms B0, B1 convention, B2, and B4. - [Wikipedia, Harmonic Number](https://en.wikipedia.org/wiki/Harmonic_number) — confirms H4=25/12 and the nonintegrality theorem for Hn, n>1; on growth: “The values of the sequence H_n − ln n decrease monotonically towards the limit” … “where γ ≈ 0.5772156649 is the Euler–Mascheroni constant.” - [Milü, Wikipedia](https://en.wikipedia.org/wiki/Mil%C3%BC) — “355/113 is the best rational approximation of π with a denominator of four digits or fewer, being accurate to six decimal places.”; “The next rational number (ordered by size of denominator) that is a better rational approximation of π is 52163/16604, though it is still only correct to six decimal places.”; the Milü “is the name given to an approximation of π ( pi ) found by the Chinese mathematician and astronomer Zu Chongzhi during the 5th century.” ### 4.8 {#r-4.8} The constructibility and algebraic descriptions of φ, √2, the silver ratio, cos(2π/17), ∛2, cos(2π/7), the plastic, tribonacci, Lehmer, and Conway constants are correct as stated. cos(2π/17) ≈ 0.93247 has degree 8; Gauss’s diary records the 17-gon’s constructibility on 30 March 1796, the first progress in regular polygon construction since the Greeks; Gauss, born on 30 April 1777, was eighteen at that diary entry; that he asked for it on his gravestone is a story, probably apocryphal. Lehmer’s number ≈ 1.176280818, of degree 10, is the smallest known Salem number; it is the spectral radius of the Coxeter element of E₁₀ and the Mahler measure of the Alexander polynomial of the (−2, 3, 7)-pretzel knot. *Status: confirmed* - [Wolfram MathWorld, Constructible Polygon](https://mathworld.wolfram.com/ConstructiblePolygon.html) — confirms the Fermat-prime criterion and the 17-gon/heptagon distinction. - [Wolfram MathWorld, Plastic Constant](https://mathworld.wolfram.com/PlasticConstant.html) — confirms x^3=x+1 and the value 1.32471795… . - [Wolfram MathWorld, Conway’s Constant](https://mathworld.wolfram.com/ConwaysConstant.html) — confirms the look-and-say growth rate, value, and degree-71 polynomial. - [Constructible number, Wikipedia](https://en.wikipedia.org/wiki/Constructible_number) — the power-of-two criterion and the √2 example: “For instance, the square root of 2 is constructible, because it can be described by the formulas √2 or √(1+1).”; “It follows from this decomposition that the degree of the field extension [Q(γ):Q] is 2^r, where r counts the number of quadratic extension steps.”; “Consequently, if a complex number γ is constructible, then the above characterization implies that [Q(γ):Q] is a power of two.”; Gauss’s argument “was generalized in his 1801 book Disquisitiones Arithmeticae giving the sufficient condition for the construction of a regular n-gon.” - [Trigonometry Angles--Pi/17, Wolfram MathWorld](https://mathworld.wolfram.com/TrigonometryAnglesPi17.html) — the value, equation (12): “cos((2pi)/(17)) = 1/(16)[delta+sqrt(2)(alpha+epsilon^*)] approx 0.93247” - [Minimal polynomial of 2cos(2pi/n), Wikipedia](https://en.wikipedia.org/wiki/Minimal_polynomial_of_2cos(2pi/n)) — the degree, 8 for n = 17: “it follows that the degree of Ψn(x) is 1 for n=1,2 and φ(n)/2 for n≥3.” - [Gauss’s diary, Wikipedia](https://en.wikipedia.org/wiki/Gauss%27s_diary) — the date: “Entry 1, dated 1796, March 30, states “Principia quibus innititur sectio circuli, ac divisibilitas eiusdem geometrica in septemdecim partes etc.,” which records Gauss’s discovery of the construction of a heptadecagon by ruler and compass.” - [Heptadecagon, Wikipedia](https://en.wikipedia.org/wiki/Heptadecagon) — the first since the Greeks: “This proof represented the first progress in regular polygon construction in over 2000 years.” - [Constructing the Heptadecagon, MathPages](https://www.mathpages.com/home/kmath487.htm) — the gravestone as a story: “Gauss was clearly fond of this discovery, and there’s a story that he asked to have a heptadecagon carved on his tombstone, like the sphere incribed in a cylinder on Archimedes’ tombstone. The story is probably apochryphal” - [Heptagon, Wikipedia](https://en.wikipedia.org/wiki/Heptagon) — gives 2cos(2π/7) ≈ 1.247 and: “The impossibility of straightedge and compass construction follows from the observation that 2cos(2π/7) ≈ 1.247 is a zero of the irreducible cubic x³ + x² − 2x − 1. Consequently, this polynomial is the minimal polynomial of 2cos(2π/7), whereas the degree of the minimal polynomial for a constructible number must be a power of 2.” - [Lehmer’s conjecture, Wikipedia](https://en.wikipedia.org/wiki/Lehmer%27s_conjecture) — the value to nine places: “The smallest known Mahler measure (greater than 1) is for “Lehmer’s polynomial,”” x^10 + x^9 − x^7 − x^6 − x^5 − x^4 − x^3 + x + 1, “for which the Mahler measure is the Salem number” 1.176280818… - [Salem number, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Salem_number) — degree and polynomial: “The smallest known Salem number is the number σ1 = 1.1762808… of degree 10 known as Lehmer’s number. The minimum polynomial of σ1 is Lehmer’s polynomial: x^10 + x^9 − x^7 − x^6 − x^5 − x^4 − x^3 + x + 1.” - [Eriko Hironaka, What is... Lehmer’s number?, Notices of the AMS 56 (2009), author’s copy](https://www.math.fsu.edu/~hironaka/papers/Whatis.pdf) — E10 and knot theory: “Thus, Lehmer’s number is the geometric dilatation of φ, and the spectral radius of the Coxeter element of E10.”; “In particular, Lehmer’s number λ is the homological dilatation of the (-2, 3, 7)-pretzel knot (shown in Fig.1, center), and is the Mahler measure of its Alexander polynomial.” - [Algebraic Number, Wolfram MathWorld](https://mathworld.wolfram.com/AlgebraicNumber.html) — degrees of algebraic numbers: “An example of a complex algebraic number is i, and an example of a real algebraic number is sqrt(2), both of which are of degree 2.”; its table of degrees lists “tribonacci constant 3.” - [Carl Friedrich Gauss, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Gauss/) — the birth date: “Born 30 April 1777 Brunswick, Duchy of Brunswick (now Germany).” (Wikipedia’s Gauss article has him “only 19 years old” when he proved the construction; the diary entry of 30 March 1796 is a month before his nineteenth birthday.) ### 4.9 {#r-4.9} The classical transcendence claims for e, π, logarithms, trigonometric values, 2^√2, e^π, log2(3), i^i, and √2^√2 follow from the cited transcendence theorems. The Dottie number 0.7390851…, the unique real fixed point of cosine in radians, is transcendental by Lindemann–Weierstrass; it is named for a professor of French who found it by pressing cos repeatedly. Gelfond proved e^π transcendental in 1929; most combinations of e and π, π^e among them, are not known to be transcendental. Hilbert posed his seventh problem in 1900 and gave 2^√2 as one of its two explicit examples. All values of i^i are e^(−π/2)·e^(−2kπ), all real, the principal one e^(−π/2). Lindemann’s proof that π is transcendental in 1882, extending the work of Hermite, is what showed squaring the circle impossible. ln 2 ≈ 0.6931 is the sum of the alternating harmonic series and is transcendental by Lindemann–Weierstrass. log2(3) ≈ 1.58496 is the information in one trit, in bits, and it is irrational because a positive power of 2 is even and a power of 3 is odd. The universal parabolic constant P = ln(1 + √2) + √2 ≈ 2.2955, the ratio for any parabola of the latus-rectum segment’s arc length to the focal parameter, is transcendental by Lindemann–Weierstrass. A constructible number is algebraic, so π’s transcendence means the circle cannot be squared. e^(π√163) is transcendental by Gelfond–Schneider, like every e^(π√n) with n a positive integer. A non-constant algebraic function of a transcendental number is transcendental, so 2/π is transcendental because π is. *Status: confirmed* - [Wikipedia, Lindemann–Weierstrass theorem](https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem) — states the theorem — if α₁, …, αₙ are algebraic and linearly independent over ℚ then e^α₁, …, e^αₙ are algebraically independent over ℚ — records that Lindemann proved the e^α case in 1882 and Weierstrass the general statement in 1885, and derives the transcendence of e and of π, and adds: “A slight variant on the same proof will show that if α is a non-zero algebraic number then sin(α), cos(α), tan(α) and their hyperbolic counterparts are also transcendental.” (The logarithms are on the Natural logarithm of 2 line.) - [Wikipedia, Gelfond–Schneider theorem](https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem) — “If a and b are algebraic numbers with a ∉ {0,1} and b irrational, then any value of a^b is a transcendental number”; “originally proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider”; and it “answers affirmatively Hilbert’s seventh problem.” - [Dottie number, Wikipedia](https://en.wikipedia.org/wiki/Dottie_number) — value, fixed point, transcendence, and name: “D = 0.739085133215160641655312087673...”; “It is the single real-valued fixed point of the cosine function”; “It is also a transcendental number because of the Lindemann–Weierstrass theorem.”; “It originates from a professor of French named Dottie who observed the number by repeatedly pressing the cosine button on her calculator.” - [Elliptic Functions and Transcendence (survey, PDF), Michel Waldschmidt](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/SurveyTrdceEllipt2005.pdf) — Gelfond’s 1929 proof for e^π: “The transcendence of e^π had already been proved in 1929 by A.O. Gel’fond.”; and on e^(π√163) (Remark 14), with τ = (1 + i√163)/2: “the class number of the imaginary quadratic field Q(τ) is 1, we have j(τ) = −(640 320)³ and |j(τ) − 1/q − 744| < 10^−12”. - [Transcendental number, Wikipedia](https://en.wikipedia.org/wiki/Transcendental_number) — e–π combinations mostly open: “Most nontrivial combinations of two or more transcendental numbers are themselves not known to be transcendental or even irrational,” listing π^e among them. - [Hilbert’s seventh problem, Wikipedia](https://en.wikipedia.org/wiki/Hilbert%27s_seventh_problem) — the year of the problem: “Hilbert’s seventh problem is one of David Hilbert’s list of open mathematical problems posed in 1900.” - [Gelfond–Schneider constant, Wikipedia](https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_constant) — 2^√2 as Hilbert’s test case: “In the address he gave two explicit examples, one of them being the Gelfond–Schneider constant 2^√2.”; also “which was proved to be a transcendental number by Rodion Kuzmin in 1930.” - [Exponentiation, Wikipedia](https://en.wikipedia.org/wiki/Exponentiation) — the branches of i^i, i^i = e^(−π/2) e^(−2kπ): “So, all values of i^i are real, the principal one being e^(−π/2) ≈ 0.2079.” - [Constructible number, Wikipedia](https://en.wikipedia.org/wiki/Constructible_number) — on squaring the circle: “It was not until 1882 that Ferdinand von Lindemann rigorously proved its impossibility, by extending the work of Charles Hermite and proving that π is a transcendental number.” - [Natural logarithm of 2, Wikipedia](https://en.wikipedia.org/wiki/Natural_logarithm_of_2) — value ln 2 ≈ 0.693 147 180 559…, the series 1 − 1/2 + 1/3 − …, and transcendence: “This is the well-known “alternating harmonic series.””; “By the Lindemann–Weierstrass theorem, the natural logarithm of any natural number other than 0 and 1 (more generally, of any positive algebraic number other than 1) is a transcendental number.” - [Ternary numeral system, Wikipedia](https://en.wikipedia.org/wiki/Ternary_numeral_system) — the trit in bits: “One trit is equivalent to log2 3 (about 1.58496) bits of information.” - [Irrational number, Wikipedia](https://en.wikipedia.org/wiki/Irrational_number) — why log2 3 is irrational: “The number 2 raised to any positive integer power must be even (because it is divisible by 2) and the number 3 raised to any positive integer power must be odd (since none of its prime factors will be 2).” - [Universal parabolic constant, Wikipedia](https://en.wikipedia.org/wiki/Universal_parabolic_constant) — value, definition and transcendence: “It is defined as the ratio, for any parabola, of the arc length of the parabolic segment formed by the latus rectum to the focal parameter.”; “P = ln(1+√2)+√2 = 2.29558714939…”; “Hence P is transcendental.” ### 4.10 {#r-4.10} Champernowne’s and the Thue–Morse constants are transcendental, so are the Komornik–Loreti and paperfolding constants, and the digit-constructed constants are described by their stated sequences. The paperfolding constant 0.85073618… has the regular paperfolding sequence 1101100111001001… as its binary digits; the sequence is automatic and not ultimately periodic, and the constant is transcendental. The Thue–Morse sequence starts from 0 and repeatedly appends the complement of what has been written so far. *Status: confirmed* - [OEIS A066716, Champernowne-related constant record](https://oeis.org/A066716) — records Mahler’s transcendence result for the base-b Champernowne constant. - [OEIS A010060, Thue–Morse sequence](https://oeis.org/A010060/internal) — confirms the binary sequence and the transcendence statement for its associated constant. - [Wikipedia, Komornik–Loreti constant](https://en.wikipedia.org/wiki/Komornik%E2%80%93Loreti_constant) — q = 1.787231650…, the smallest base in which 1 has a unique expansion, defined by Vilmos Komornik and Paola Loreti in 1998; Jean-Paul Allouche and Michel Cosnard proved it transcendental in the American Mathematical Monthly 107 (2000), 448–449; its digits are the Thue–Morse sequence: “The Komornik–Loreti constant is the value q such that” 1 = Σ t_{k}/q^{k}, “where t_{k} is the Thue–Morse sequence.” - [Adamczewski and Bugeaud, On the complexity of algebraic numbers I: expansions in integer bases](https://arxiv.org/abs/math/0511674) — the abstract states that irrational automatic and morphic numbers are transcendental, which is the theorem that settles the paperfolding constant; published as Annals of Mathematics 165 (2007), 547–565. - [Regular paperfolding sequence, Wikipedia](https://en.wikipedia.org/wiki/Regular_paperfolding_sequence) — value, digits, not periodic: “This number is known as the paperfolding constant and has the value,” 0.85073618820186...; “The paperfolding word 1101100111001001..., which is created by concatenating the terms of the regular paperfolding sequence”; “A paperfolding sequence is not ultimately periodic.” - [Automatic sequence, Wikipedia](https://en.wikipedia.org/wiki/Automatic_sequence) — “Both the Baum–Sweet sequence (OEIS: A086747) and the regular paperfolding sequence (OEIS: A014577) are automatic.” - [Borwein and Coons, Transcendence of the Gaussian Liouville number and relatives, arXiv (2008)](https://arxiv.org/abs/0806.1694) — the same binary digits, proved transcendental: “define the Gaussian Liouville number by γ:=0.110110011100100111011...”; “we prove that the Gaussian Liouville number and its relatives are transcendental.” - [Thue–Morse sequence, Wikipedia](https://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence) — the construction: “the binary sequence (an infinite sequence of 0s and 1s) that can be obtained by starting with 0 and successively appending the Boolean complement of the sequence obtained thus far.” ### 4.11 {#r-4.11} The lemniscate constant, Gauss’s constant and Cahen’s constant have the stated definitions, values and theorem attributions. Siegel (1932) and Schneider both proved it through periods of elliptic functions; Schneider’s paper is dated 1937 by Wikipedia and 1936 by Waldschmidt, so the card gives no year for it. Corrections welcome. *Status: confirmed* - [Wikipedia, Lemniscate constant](https://en.wikipedia.org/wiki/Lemniscate_constant) — ϖ ≈ 2.622 was proved transcendental by Carl Ludwig Siegel in 1932 and again by Theodor Schneider in 1937; Gauss’s constant is G = ϖ/π ≈ 0.8346268 = 1/M(1,√2), and Chudnovsky proved π and ϖ algebraically independent. - [Wikipedia, Cahen’s constant](https://en.wikipedia.org/wiki/Cahen%27s_constant) — C ≈ 0.643410546288, the alternating sum of 1/(s_i − 1) over Sylvester’s sequence s_i (1/1 − 1/2 + 1/6 − 1/42 + …); Eugène Cahen introduced it and proved it irrational in 1891, and Davison and Shallit proved it transcendental in 1991 from its continued fraction. - [Wolfram MathWorld, Conway’s Constant](https://mathworld.wolfram.com/ConwaysConstant.html) — confirms the look-and-say constant used in the algebraic brace. - [Wikipedia, Particular values of the gamma function](https://en.wikipedia.org/wiki/Particular_values_of_the_gamma_function) — credits G. V. Chudnovsky with proving Γ(1/3) and Γ(1/4) transcendental, and Yuri Nesterenko (1996) with the algebraic independence of Γ(1/4), π and e^π; Chudnovsky’s own independence result (π and ϖ, 1975) is on the Lemniscate constant line. - [Elliptic Functions and Transcendence (survey, PDF), Michel Waldschmidt](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/SurveyTrdceEllipt2005.pdf) — Siegel and Schneider on periods: “The first transcendence result on periods of elliptic functions was proved by C.L. Siegel [199] as early as 1932.”; “Theorem 18 (Schneider, 1936) Assume that the invariants g2 and g3 of ℘ are algebraic. Then for any non-zero period ω of ℘, the numbers ω and η(ω) are transcendental.” ### 4.12 {#r-4.12} Γ(1/4) and Γ(1/3) are transcendental, because Chudnovsky proved that Γ(1/4) and π are algebraically independent; the result was first published in the 1970s and collected in his 1984 monograph. The lemniscate constant ϖ was proved transcendental by Siegel in 1932 and Gauss’s constant by Schneider in 1941, before Chudnovsky; Γ(1/2) = √π is transcendental with π. *Status: judgement* - [Wikipedia, Particular values of the gamma function](https://en.wikipedia.org/wiki/Particular_values_of_the_gamma_function) — confirms Chudnovsky’s authorship of the algebraic-independence result but gives no 1984/1976 date. - [Michel Waldschmidt, Elliptic Functions and Transcendence (survey, PDF)](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/SurveyTrdceEllipt2005.pdf) — “In the 1970’s G.V. Chudnovsky proved strong results of algebraic independence,” including that π and Γ(1/4), and π and Γ(1/3), are algebraically independent; it cites “Chudnovsky’s lecture at the Helsinki ICM in 1978” (G. V. Chudnovsky, Algebraic independence of values of exponential and elliptic functions, Proc. ICM Helsinki 1978, vol. 1, 339–350); on Γ(1/5): “at least two of the three numbers π, Γ(1/5) and Γ(2/5) are algebraically independent,” and among the “Further open problems,” “the algebraic independence of three numbers among π, Γ(1/5), Γ(2/5), e^(π√5).” - [Michel Waldschmidt, Les travaux de G. V. Čudnovskiĭ sur les nombres transcendants, Séminaire Bourbaki 488 (1975/76), Numdam](http://www.numdam.org/item/SB_1975-1976__18__274_0/) — the contemporary account of the result: “Γ(1/4) et π sont algébriquement indépendants,” and likewise Γ(1/3) and π. - [Arshay Sheth, Real geometric transcendence for the Gamma function](https://arxiv.org/html/2605.15117) — “Γ(1/3) and Γ(1/4) are known to be transcendental by the work of Chudnovsky”, citing Dokl. Akad. Nauk Ukrain. SSR Ser. A 1976, no. 8, 698–701, 767. - [Lemniscate constant, Wikipedia](https://en.wikipedia.org/wiki/Lemniscate_constant) — earlier proofs: “The lemniscate constant ϖ and Todd’s first lemniscate constant A were proven transcendental by Carl Ludwig Siegel in 1932 and later by Theodor Schneider in 1937 and Todd’s second lemniscate constant B and Gauss’s constant G were proven transcendental by Theodor Schneider in 1941.” - [Transcendental number, Wikipedia](https://en.wikipedia.org/wiki/Transcendental_number) — under “Numbers proven to be transcendental:,” “Values of the gamma function of rational numbers that are of the form Γ(n/2), Γ(n/3), Γ(n/4) or Γ(n/6).” Editorial note: Three dates are in circulation for one result, so the cards give none of them as a bare year. The first publication is the Doklady paper of 1976, which is what the transcendence literature cites; Wikipedia’s article on the lemniscate constant dates the same theorem to 1975; and the American Mathematical Society monograph that collects it, Contributions to the Theory of Transcendental Numbers, is 1984. “In the 1970s, collected in 1984” is true under all three, and that is what the cards say. Ruled, 10 September 2026: the first publication is the Doklady paper of 1976. Chudnovsky’s own 1980 paper cites it by title, issue and pages, and Waldschmidt, writing on precisely this theorem, dates it to 1976; Wikipedia’s 1975 is an undated secondary assertion carrying no reference. The cards keep the decade wording, which needs no year, and this entry carries 1976 where a date is wanted. Reference 17.24 carries the same ruling. ### 4.13 {#r-4.13} The transcendental numerator is exactly zero because sin π=0, regardless of the status of its other displayed factors; its approximate nonzero parenthetical value is an own computation. π is the smallest positive number at which the sine function equals zero. *Status: confirmed* - [Wikipedia, Lindemann–Weierstrass theorem](https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem) — confirms π is transcendental and the standard exponential consequences. - **Own computation** (the author's own work) — The numerator contains sin(π) as a multiplicative factor, and sin(π)=0; evaluating the remaining displayed parenthetical with standard high-precision arithmetic gives approximately 2.4, but it cannot affect the product. - [Pi, Wikipedia](https://en.wikipedia.org/wiki/Pi) — sin π = 0 with π the first positive zero: “π is also the smallest positive number at which the sine function equals zero, and the difference between consecutive zeroes of the sine function.” Editorial note: The rider deliberately mixes algebraic, proven-transcendental and status-open ingredients; no claim that the rider itself is transcendental is made. ### 4.14 {#r-4.14} The displayed formula has four zero numerators and nonzero denominators, so its chapter factor is zero; the denominator positivity check is an own computation from the displayed terms. *Status: confirmed* - **Own computation** (the author's own work) — 1729−1729=0; 3·(1/3)−1=0; 0.999…−1=0; sin(π)=0. Every denominator is a sum/product of positive displayed values, hence nonzero, so the product is 0·0·0·0=0. ### 4.15 {#r-4.15} The Laplace limit is about 0.6627434 and is the unique real x with x·e^√(1+x²)/(1+√(1+x²)) = 1; it is transcendental, since if it were algebraic then s = √(1+x²) would be a nonzero algebraic number and e^s = (1+s)/x would be algebraic, which Hermite–Lindemann forbids. Francesco Carlini had the limit before Laplace computed 0.66195 in 1827, and Cauchy sharpened it to 0.66274 in 1829. *Status: judgement* - [Wolfram MathWorld, Laplace Limit](https://mathworld.wolfram.com/LaplaceLimit.html) — gives the defining equation and the value 0.6627434…, and says nothing about the number’s arithmetic. - [Wikipedia, Laplace limit](https://en.wikipedia.org/wiki/Laplace_limit) — gives the same equation and 0.66274 34193 49181 58097…, and the history: Carlini had the limit five years before Laplace, who calculated 0.66195 in 1827, and Cauchy gave 0.66274 in 1829. It too says nothing about irrationality or transcendence. - [Michel Waldschmidt, Introduction to Diophantine methods: irrationality and transcendence](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/IntroductionDiophantineMethods.pdf) — Theorem 2.1, Hermite–Lindemann: “let β be a non-zero algebraic number. Then e^β is transcendental.” This is the premise the deduction uses. - **Own computation** (the author's own work) — At x = 0.66274341934918158097474209710925, evaluating x·exp(√(1+x²))/(1+√(1+x²)) to fifty digits gives 1 exactly at that precision; s = √(1+x²) = 1.19967864…, which is not zero, so the theorem applies. Editorial note: No reference work states this number’s arithmetic nature. MathWorld and Wikipedia both give the equation and the value and stop there, and Wikipedia’s list of transcendental numbers does not name it. The status printed here is therefore a deduction from Hermite–Lindemann rather than a result quoted from a source, which is why the card sets the whole argument out in three lines for the reader to check. It is short, it uses nothing but the defining equation and a nineteenth-century theorem, and transcendence carries irrationality with it. ### 4.16 {#r-4.16} Carmichael’s 561 passes the Fermat congruence for all bases coprime to it, while the usual b^(n−1)=1 test is only stated for coprime bases. The priority runs Václav Šimerka, who found the first seven in 1885 in a Czech journal that went unnoticed; A. Korselt, who gave the criterion in 1899 without an example; Robert Carmichael, who published 561 in 1910 and gave the numbers their name; and Alford, Granville and Pomerance, who proved in 1994 that there are infinitely many. *Status: confirmed* - [Wikipedia, Carmichael number](https://en.wikipedia.org/wiki/Carmichael_number) — gives the coprime-base formulation and the equivalent b^n≡b formulation. Re-read raw on 17 September 2026 for the priority: “The first seven Carmichael numbers, from 561 to 8911, were all found by the Czech mathematician Václav Šimerka in 1885” and his work “remained unnoticed”; “Theorem (A. Korselt 1899)” and “he did not give any examples”; “In 1910, Carmichael himself also published the smallest such number, 561, and the numbers were later named after him”; “In 1994 W. R. (Red) Alford, Andrew Granville and Carl Pomerance ... show that there really do exist infinitely many Carmichael numbers”. ### 4.17 {#r-4.17} Cohn proved that the only square Fibonacci numbers are 0, 1 and F12=144, so 144 is the only one after 1. The sequence first appears, as a count of rabbits, in Leonardo of Pisa’s Liber Abaci of 1202. *Status: confirmed* - [Wolfram MathWorld, Fibonacci Number](https://mathworld.wolfram.com/FibonacciNumber.html) — cites Cohn (1964) and states that the only square Fibonacci numbers are 1 and F12=144 under the convention beginning 1,1. - [Fibonacci sequence, Wikipedia](https://en.wikipedia.org/wiki/Fibonacci_sequence) — carries the 1202 date and the rabbits: “The Fibonacci sequence first appears in the book Liber Abaci (The Book of Calculation, 1202) by Fibonacci, where it is used to calculate the growth of rabbit populations.” ### 4.18 {#r-4.18} The largest known prime is M136279841, found by Luke Durant’s GIMPS search on 12 October 2024, with 41,024,320 digits. *Status: confirmed* - [GIMPS, M136279841 press release](https://www.mersenne.org/primes/?press=M136279841) — confirms the discoverer, date, exponent and digit count, and “It is only the 52nd known Mersenne prime ever discovered.” ### 4.19 {#r-4.19} Hilbert’s 1900 seventh-problem examples were 2^√2 and e^π; √2^√2 is a related number covered by Gelfond–Schneider. √2^√2 is also the number in Dov Jarden’s non-constructive proof that an irrational number raised to an irrational power can be rational. *Status: confirmed* - [Wikipedia, Hilbert’s seventh problem](https://en.wikipedia.org/wiki/Hilbert%27s_seventh_problem) — names Hilbert’s examples 2^√2 and e^π. - [Wikipedia, Gelfond–Schneider theorem](https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem) — confirms the theorem used for √2^√2. - [Irrational number, Wikipedia](https://en.wikipedia.org/wiki/Irrational_number) — the classic proof: “Dov Jarden gave a simple non-constructive proof that there exist two irrational numbers a and b, such that a^b is rational:” ### 4.20 {#r-4.20} A real number is constructible with compass and straightedge exactly when it lies at the top of a finite tower of quadratic extensions of ℚ, so its degree over ℚ is a power of 2; Wantzel proved this in 1837 and deduced the impossibility of doubling the cube and of trisecting the general angle. A degree that is a power of 2 is necessary but not sufficient for constructibility. Non-constructible numbers are still computable: every real algebraic number, and π, can be approximated to any precision by an algorithm. *Status: confirmed* - [J. S. Milne, Fields and Galois Theory](https://www.jmilne.org/math/Books/FT0.pdf) — Read 12 September 2026: Theorem 1.37 and Corollaries 1.38–1.39 give the successive-square-root criterion, algebraicity, and the impossibility of constructing ∛2; Remark 3.27 explains why power-of-two degree alone is insufficient. - [Constructible number, Wikipedia](https://en.wikipedia.org/wiki/Constructible_number) — (§ Algebraic properties, § History, § Impossible constructions): a real is constructible exactly when it “lies in a field at the top of a finite tower of real quadratic extensions,” with [ℚ(γ):ℚ] = 2ʳ; “Pierre Wantzel proved algebraically that the problems of doubling the cube and trisecting the angle are impossible to solve using only compass and straightedge”; and the two minimal polynomials, x³ − 2 for ∛2 and 8x³ − 6x − 1 for cos(π/9), both of degree 3; and “However, this condition is not sufficient - there exist field extensions whose degree is a power of two, but which cannot be factored into a sequence of quadratic extensions.” - [MacTutor, Pierre Wantzel](https://mathshistory.st-andrews.ac.uk/Biographies/Wantzel/) — the biography for the 1837 paper in which the criterion and the two impossibility results appear. - [Computable number, Wikipedia](https://en.wikipedia.org/wiki/Computable_number) — “The computable numbers include the specific real numbers which appear in practice, including all real algebraic numbers, as well as e, π, and many other transcendental numbers.” ### 4.21 {#r-4.21} Joseph Liouville proved in 1844 that transcendental numbers exist and gave the first explicit decimal example, Liouville’s constant, in 1851; Georg Cantor proved in 1874 that the real algebraic numbers are countable while the reals are not, so almost all real numbers are transcendental. Cantor’s 1874 proof is constructive: applied to a list of the real algebraic numbers, it produces a transcendental one. *Status: confirmed* - [Transcendental number, Wikipedia (§ History)](https://en.wikipedia.org/wiki/Transcendental_number) — “Joseph Liouville first proved the existence of transcendental numbers in 1844, and in 1851 gave the first decimal examples such as the Liouville constant”; “In 1874 Georg Cantor proved that the algebraic numbers are countable and the real numbers are uncountable”; and “almost all real and complex numbers are transcendental, since the algebraic numbers are countable, while the real numbers and complex numbers are both uncountable.” - [Cantor’s first set theory article, Wikipedia](https://en.wikipedia.org/wiki/Cantor%27s_first_set_theory_article) — Cantor’s paper of 1874, Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen, proves that the set of real algebraic numbers can be put in one-to-one correspondence with the positive integers and that an interval of reals cannot, which together give a transcendental number in every interval. - [Joseph Liouville, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Liouville/) — he proved “the existence of a transcendental number in 1844 when he constructed an infinite class of such numbers using continued fractions. In 1851 he published results on transcendental numbers removing the dependence on continued fractions.” Editorial note: The chapter’s opening history sentence previously carried 4.9, which states only that the classical transcendence claims follow from the Lindemann-Weierstrass and Gelfond-Schneider theorems; neither Liouville’s priority nor Cantor’s counting argument is in it, and Cantor’s 1874 carried no footnote at all. This entry carries both. ### 4.22 {#r-4.22} In the workbench aside: 8 and 9 are the only consecutive perfect powers, conjectured by Catalan in a note published in 1844 and proved by Mihăilescu in 2002; and 4900 is, apart from 1, the only perfect square that is also a square pyramidal number. *Status: confirmed* - [Wolfram MathWorld, Catalan’s Conjecture](https://mathworld.wolfram.com/CatalansConjecture.html) — “The conjecture made by Belgian mathematician Eugene Charles Catalan in 1844” that 8 and 9 are the only consecutive powers, with Mihailescu’s proof sent to several mathematicians on 18 April 2002; the original note is cited as Catalan, J. reine angew. Math. 27, 192, 1844. - [Catalan’s conjecture, Wikipedia](https://en.wikipedia.org/wiki/Catalan%27s_conjecture) — “conjectured by the mathematician Eugène Charles Catalan in 1842 and proven in 2002 by Preda Mihăilescu,” with the same primary note cited as J. Reine Angew. Math. 27 (1844), 192, and the proof published in 2004. - [Square pyramidal number, Wikipedia (§ Cannonball problem)](https://en.wikipedia.org/wiki/Square_pyramidal_number) — “Besides 1, there is only one other number that has this property: 4900, which is both the 70th square number and the 24th square pyramidal number”; the first complete proof that no others exist is G. N. Watson’s, 1918. - **Own computation** (the author's own work) — Enumerating the square pyramidal numbers n(n+1)(2n+1)/6 for n up to 200 and testing each for squareness gives exactly n = 0, 1 and 24, that is 0, 1 and 4900, so 4900 is the only one after 1. Editorial note: Two comparable sources date the conjecture differently — MathWorld says 1844, Wikipedia’s lead says 1842 — while both cite the same primary note, Crelle 27 (1844), 192. The page keeps 1844, the year of the publication both sources point at; no tilde is needed because the disagreement is over an undated secondary assertion, not over two datable events. The aside’s superlative for 4900 is corrected to “the only square after 1,” the same form this chapter’s fib144 card already uses for 144. ### 4.23 {#r-4.23} The numbers left on the workbench have the stated definitions and values: 153 is the sum of the cubes of its digits; 2520 is the least common multiple of 1 through 10; the chessboard holds 2⁶⁴ − 1 grains; the centillion is 10³⁰³ on the short scale and 10⁶⁰⁰ on the long, and is the highest -illion name the major English dictionaries include; Theodorus proved √3 irrational; the supergolden ratio is the real root of x³ = x² + 1; 2^(1/12) is the equal-tempered semitone; τ = 2π; the golden angle is 2π/φ² and separates the florets of a sunflower; the magic angle is arccos(1/√3), the tilt of magic-angle spinning NMR; nonzero values of arccos at algebraic arguments are transcendental, and so is any nonzero algebraic multiple of π. Turning by the golden angle keeps successive elements off any common radial line; Vogel’s sunflower model of 1979 places floret n at n × 137.508°. √3, a square root of a positive integer, is constructible. *Status: confirmed* - [Narcissistic number, Wikipedia](https://en.wikipedia.org/wiki/Narcissistic_number) — gives the base-10 example 153 = 1³ + 5³ + 3³. - [2520 (number), Wikipedia](https://en.wikipedia.org/wiki/2520_(number)) — the smallest number divisible by all integers from one to ten, their least common multiple. - [Wheat and chessboard problem, Wikipedia](https://en.wikipedia.org/wiki/Wheat_and_chessboard_problem) — the total number of grains is 2^64 − 1, or 18,446,744,073,709,551,615. - [Names of large numbers, Wikipedia](https://en.wikipedia.org/wiki/Names_of_large_numbers) — lists centillion as 10^303 (short scale) and 10^600 (long scale), and says centillion appears to be the highest name ending in -illion included in the dictionaries it surveys. - [Theodorus of Cyrene, Wikipedia](https://en.wikipedia.org/wiki/Theodorus_of_Cyrene) — quotes Plato’s Theaetetus: Theodorus showed the square roots of the non-square numbers from 3 up to 17 irrational; his method is unknown. - [Square root of 3, Wikipedia](https://en.wikipedia.org/wiki/Square_root_of_3) — value 1.7320508075688772…, also called Theodorus’s constant. - [Supergolden ratio, Wikipedia](https://en.wikipedia.org/wiki/Supergolden_ratio) — the unique real solution of x³ = x² + 1, 1.465571231876768…. - [Twelfth root of two, Wikipedia](https://en.wikipedia.org/wiki/Twelfth_root_of_two) — 1.0594630943592952646; in the equal-tempered chromatic scale each note is 2^(1/12) times the frequency of the one below. - [Turn (angle), Wikipedia](https://en.wikipedia.org/wiki/Turn_(angle)) — one turn is 2π radians, proposed to be written τ, citing Michael Hartl’s Tau Manifesto (2010); 1 turn = τ rad ≈ 6.283185307 rad. - [Golden angle, Wikipedia](https://en.wikipedia.org/wiki/Golden_angle) — ≈ 137.50776°, 360(2 − φ) degrees or 2π/φ² radians; the angle separating the florets on a sunflower; Vogel (1979) is in its reading list, and his model is on the Fermat’s spiral line. - [Magic Angle, Wolfram MathWorld](https://mathworld.wolfram.com/MagicAngle.html) — θ_m = cos^(-1)(1/√3) = 54.7356103… degrees. - [Magic angle spinning, Wikipedia](https://en.wikipedia.org/wiki/Magic_angle_spinning) — the sample is spun at the magic angle, about 54.74°, to the magnetic field in solid-state NMR. - [Transcendental number, Wikipedia](https://en.wikipedia.org/wiki/Transcendental_number) — lists nonzero values of arcsin and arccos at algebraic arguments as transcendental by Lindemann–Weierstrass, and says a non-constant algebraic function of π, such as 5π, is transcendental. - [Phyllotaxis, Wikipedia](https://en.wikipedia.org/wiki/Phyllotaxis) — why successive elements never line up: “Since three golden arcs add up to slightly more than enough to wrap a circle, this guarantees that no two leaves ever follow the same radial line from center to edge.” - [Fermat’s spiral, Wikipedia](https://en.wikipedia.org/wiki/Fermat%27s_spiral) — Vogel’s model: “The full model proposed by H. Vogel in 1979 is r = c√n, θ = n × 137.508°, where θ is the angle, r is the radius or distance from the center, and n is the index number of the floret and c is a constant scaling factor.” and “The angle 137.508° is the golden angle which is approximated by ratios of Fibonacci numbers.” - [Constructible number, Wikipedia](https://en.wikipedia.org/wiki/Constructible_number) — carries √3’s constructibility as a square root of a positive number: “The algebraically constructible real numbers are the subset of the real numbers that can be described by formulas that combine integers using the operations of addition, subtraction, multiplication, multiplicative inverse, and square roots of positive numbers.” ### 4.24 {#r-4.24} The chapter’s small-integer history is as stated: Christian Kramp introduced the notation n! in 1808; Plato in the Laws proposes 5040 as the number of citizens of a polis because it divides so many ways, being divisible by every number from 1 to 12 except 11, and it has 60 divisors in all; the name primorial was coined by Harvey Dubner, in Factorial and primorial primes, Journal of Recreational Mathematics 19 (1987), 197-203; and Euclid gives the first known definition of the golden ratio, as a line cut in extreme and mean ratio. 30 is the largest number all of whose totatives other than 1 are prime (Schatunowsky’s theorem of 1893). *Status: confirmed* - [Factorial, Wikipedia](https://en.wikipedia.org/wiki/Factorial) — read as raw HTML: “The notation n! for factorials was introduced by the French mathematician Christian Kramp in 1808.” - [5040 (number), Wikipedia](https://en.wikipedia.org/wiki/5040_(number)) — read as raw HTML: “Plato mentions in his dialogue Laws that 5040 is a convenient number to use for dividing many things (including both the citizens and the land of a city-state or polis) into lesser parts, making it an ideal number for the number of citizens (heads of families) making up a polis,” and “this number can be divided by all the (natural) numbers from 1 to 12 with the single exception of 11.” The page gives the reason Plato gives, which is divisibility rather than the divisor count; 5040 = 2⁴·3²·5·7 has 5·3·2·2 = 60 divisors, which is the book’s own arithmetic. - [Primorial, Wikipedia](https://en.wikipedia.org/wiki/Primorial) — read as raw HTML: “The name ‘primorial’, coined by Harvey Dubner, draws an analogy to primes similar to the way the name ‘factorial’ relates to factors,” and its reference list gives Dubner, Harvey (1987), “Factorial and primorial primes,” J. Recr. Math. 19, 197-203. - [Golden ratio, Wikipedia](https://en.wikipedia.org/wiki/Golden_ratio) — read as raw HTML: “The golden ratio was called the extreme and mean ratio by Euclid,” and Euclid’s Elements (c. 300 BC) “contains its first known definition,” “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.” - [Hala King and Victor Pambuccian, Generalized Schatunowsky theorem in a weak arithmetic, arXiv:2506.08256](https://arxiv.org/abs/2506.08256) — abstract: “Schatunowsky’s 1893 theorem, that 30 is the largest number all of whose totatives are primes, has been recently generalized by Kaneko and Nakai.” Editorial note: Added 17 September 2026. Six cards in this chapter hung datable history on r-4.1, whose claim line covers the chapter’s elementary identities and type classifications and whose three sources are the DLMF Bernoulli tables, MathWorld on algebraic numbers and Brocard’s problem. None of that history was in it. Two of the six, #sixty and #twenty, were re-pointed at this chapter’s own r-2.3 and r-2.4; #sqrt2’s Pythagorean tradition already stood on r-8.9; the remaining three, #fact7, #primorial5 and #phi, are what this entry carries. Note what the Plato source does and does not say: the reason he gives is that 5040 divides by every number from 1 to 12 but 11, not that it has 60 divisors, so the card now says both and attributes only the first to him. ### 4.25 {#r-4.25} The chapter epigraph is Kronecker’s, said at the 1886 Berlin meeting of natural scientists and reported by Heinrich Weber in the Jahresbericht der Deutschen Mathematiker-Vereinigung of 1893. *Status: confirmed* - [Leopold Kronecker, Wikiquote](https://en.wikiquote.org/wiki/Leopold_Kronecker) — read as raw HTML: “Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk. God made the integers, all the rest is the work of man. In a lecture to the Berliner Naturforschung-Versammlung, 1886; quoted in the obituary ‘Leopold Kronecker’, by H[einrich] Weber, in Jahresbericht der Deutsche Mathematiker Vereinigung, 1893, page 19,” which is the page’s footer line word for word. Editorial note: Added 17 September 2026 to source the chapter epigraph, which carried no marker. This is one of the four secondhand epigraphs where the book asserts provenance rather than quoting a printed original, so the reporter is named in the claim line as well as in the footer. ### 4.26 {#r-4.26} A simple continued fraction writes a real number as a₀ + 1/(a₁ + 1/(a₂ + ⋯)) with integer partial quotients; the finite truncations are its convergents. The convergents give a systematic sequence of rational approximations. The universal equal-denominator superlative is not asserted: for example, 8/5 = [1;1, 1, 2] has initial convergent 1/1, but 2/1 is closer with the same denominator. The expansion terminates exactly for rationals, so an infinite expansion is a proof of irrationality. *Status: confirmed* - [Simple continued fraction, Wikipedia](https://en.wikipedia.org/wiki/Simple_continued_fraction) — The earlier annotation used this page for the procedure and described a best-approximation superlative too broadly. That universal equal-denominator assertion is no longer used in this entry: the direct rational counterexample in its claim and the NIST definitions below control the book’s wording. - [Continued Fraction, MathWorld](https://mathworld.wolfram.com/ContinuedFraction.html) — the same simple-continued-fraction notation (convergents are only linked from it, under See also). - [NIST DLMF, §1.12 Continued Fractions](https://dlmf.nist.gov/1.12) — Section 1.12(ii), equations 1.12.3–4, defines the finite truncations as convergents. Its indexing begins at zero; the book counts the initial integer truncation as first. The equal-denominator counterexample in the claim is direct rational arithmetic. ### 4.27 {#r-4.27} A Salem number is a real algebraic integer greater than 1 whose other conjugates all lie in the closed unit disc with at least one on the unit circle; the minimal polynomial is therefore reciprocal, of even degree at least 4, and every Salem number is a unit. The smallest known Salem number is the largest real root of Lehmer’s degree-10 polynomial x¹⁰ + x⁹ − x⁷ − x⁶ − x⁵ − x⁴ − x³ + x + 1, about 1.1762808, which is also the smallest known Mahler measure above 1; whether anything smaller exists is open. A Pisot number is a real algebraic integer greater than 1 all of whose other conjugates lie strictly inside the unit circle. *Status: confirmed* - [Salem number, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Salem_number) — read as raw HTML: “An algebraic integer θ > 1 such that all other Galois conjugates of θ lie inside the closed unit disc |z| ≤ 1, with at least one lying on the boundary”; reciprocity, even degree ≥ 4; “The smallest known Salem number is the number σ₁ = 1.1762808… of degree 10 known as Lehmer’s number… This is also the smallest known value > 1 of the Mahler measure.” - [Salem number, Wikipedia](https://en.wikipedia.org/wiki/Salem_number) — read as raw HTML: the same definition, names them after Raphaël Salem, gives Lehmer’s polynomial explicitly with root about 1.17628, states that every Salem number is a unit and a Perron number, and that it is conjectured to be the smallest. - [Pisot–Vijayaraghavan number, Wikipedia](https://en.wikipedia.org/wiki/Pisot%E2%80%93Vijayaraghavan_number) — the definition: “In mathematics, a Pisot–Vijayaraghavan number (or Pisot number or PV number) is a real algebraic integer greater than 1, all of whose Galois conjugates are less than 1 in absolute value.” ### 4.28 {#r-4.28} The class number of an imaginary quadratic field ℚ(√−d) is 1 exactly when its ring of integers has unique factorization; by the Stark–Heegner theorem there are exactly nine such d — 1, 2, 3, 7, 11, 19, 43, 67 and 163. Gauss conjectured the list; Heegner proved it in 1952 up to a minor gap, and Baker and Stark proved it independently in 1966. When the class number is 1 the j-invariant at the corresponding point is an integer, which is why e^(π√163) misses an integer so narrowly. *Status: confirmed* - [Heegner number, Wikipedia](https://en.wikipedia.org/wiki/Heegner_number) — read as raw HTML: “Heegner numbers are square-free positive integers d such that the imaginary quadratic field ℚ(√−d) has class number 1. Equivalently, the ring of algebraic integers… has unique factorization”; “there are precisely nine Heegner numbers: 1, 2, 3, 7, 11, 19, 43, 67, and 163… conjectured by Gauss and proved up to minor flaws by Kurt Heegner in 1952. Alan Baker and Harold Stark independently proved the result in 1966”; and “if the imaginary quadratic extension ℚ(τ) has class number 1… the j-invariant is an integer.” - [Class Number, MathWorld](https://mathworld.wolfram.com/ClassNumber.html) — the class number as the order of the ideal class group, and its being 1 exactly for unique factorization. ### 4.29 {#r-4.29} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: the sixteen-year-old Nicolò Paganini found the amicable pair 1184, 1210, published in the Atti of the Turin Academy for 1866–67 (MathWorld dates the find 1866, Wikipedia 1867); Jacob Bernoulli published the Bernoulli numbers in the Ars Conjectandi of 1713 (Seki a year earlier); Pierre Wantzel proved in 1837 that doubling the cube with compass and straightedge is impossible; Conway chained-arrow notation comes from The Book of Numbers by Conway and Guy, 1996; David Champernowne, an undergraduate, showed in 1933 that the constant is normal in base 10; Kurt Mahler proved Champernowne’s constant transcendental in 1937; Conway’s paper on the look-and-say sequence, The Weird and Wonderful Chemistry of Audioactive Decay, appeared in Eureka 46 in January 1986 and was reprinted by Springer in 1987; Wantzel proved in 1837 that the polygons outside Gauss’s criterion (the heptagon first) cannot be constructed; e first appears in (Jacob) Bernoulli’s 1683 study of compound interest. By Kummer’s theorem the Bernoulli numerators decide which primes are regular, and for a regular prime p Fermat’s equation of degree p has no solution in nonzero integers. *Status: confirmed* - [Amicable Pair, Wolfram MathWorld](https://mathworld.wolfram.com/AmicablePair.html) — read raw: “In 1866, 16-year old B. Nicolò I. Paganini found the small amicable pair (1184, 1210) which had eluded his more illustrious predecessors (Paganini 1866-1867; Dickson 2005, p. 47).”; “In 1636, Fermat found the pair (17296, 18416) and in 1638, Descartes found (9363584, 9437056), although these results were actually rediscoveries of numbers known to Arab mathematicians. By 1747, Euler had found 30 pairs, a number which he later extended to 60.” - [Bernoulli number, Wikipedia](https://en.wikipedia.org/wiki/Bernoulli_number) — read raw: “Bernoulli’s result was published posthumously in Ars Conjectandi in 1713. Seki Takakazu independently discovered the Bernoulli numbers and his result was published a year earlier, also posthumously, in 1712.”; “The Bernoulli numbers are related to Fermat’s Last Theorem (FLT) by Kummer’s theorem, which says: If the odd prime p does not divide any of the numerators of the Bernoulli numbers B₂, B₄, ..., Bₚ₋₃ then xᵖ + yᵖ + zᵖ = 0 has no solutions in nonzero integers. Prime numbers with this property are called regular primes.” (sub- and superscripts as typeset on the page) - [Doubling the cube, Wikipedia](https://en.wikipedia.org/wiki/Doubling_the_cube) — read raw: “The nonexistence of a compass-and-straightedge solution was finally proven by Pierre Wantzel in 1837..” - [Conway chained arrow notation, Wikipedia](https://en.wikipedia.org/wiki/Conway_chained_arrow_notation) — read raw: “John H. Conway & Richard K. Guy, The Book of Numbers, 1996, p.59-62.” - [Champernowne constant, Wikipedia](https://en.wikipedia.org/wiki/Champernowne_constant) — read raw: “It is named after economist and mathematician D. G. Champernowne, who published it as an undergraduate in 1933.”; “K. Mahler, Arithmetische Eigenschaften einer Klasse von Dezimalbrüchen, Proc. Konin. Neder. Akad. Wet. Ser. A. 40 (1937), p. 421–428..” - [Look-and-say sequence, Wikipedia](https://en.wikipedia.org/wiki/Look-and-say_sequence) — read raw: “title=The Weird and Wonderful Chemistry of Audioactive Decay |journal=Eureka |date=January 1986 |volume=46 |pages=5–16.” - [Constructible polygon, Wikipedia](https://en.wikipedia.org/wiki/Constructible_polygon) — read raw: “A full proof of necessity was given by Pierre Wantzel in 1837..” - [e (mathematical constant), Wikipedia](https://en.wikipedia.org/wiki/E_(mathematical_constant)) — read raw: “Jacob Bernoulli discovered this constant in 1683, while studying a question about compound interest.” - [Amicable numbers, Wikipedia](https://en.wikipedia.org/wiki/Amicable_numbers) — dates the find a year later than MathWorld: “The second smallest pair, (1184, 1210), was discovered in 1867 by 16-year-old B. Nicolò I. Paganini (not to be confused with the composer and violinist), having been overlooked by earlier mathematicians.” Editorial note: Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim. ### 4.30 {#r-4.30} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: the letter e appears in a letter Euler wrote to Goldbach in 1731 (MacTutor calls this its first appearance; Wikipedia has Euler using the letter from 1727 or 1728, in an unpublished paper on explosive forces in cannons); Charles Hermite proved e transcendental in 1873, the first number proved transcendental without having been specifically constructed for the purpose; Eddington’s claim was made in his Tarner lectures of 1938, published in 1939; Fermat conjectured in 1640 that all numbers 2^(2^n)+1 are prime; Euler factored F5 = 641 x 6700417 in 1732; Johann Gustav Hermes spent ten years on the 65537-gon construction, finished in 1894; the fast-growing hierarchy of Martin Löb and Stanley Wainer, 1970; Gauss found in 1799 that the reciprocal of the arithmetic-geometric mean of 1 and sqrt 2 matches 2/π times the lemniscate integral; Theodor Schneider proved Gauss’s constant transcendental in 1941. In the same passage Eddington claims the same number of electrons as protons. *Status: confirmed* - [The number e, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/HistTopics/e/) — read raw: “Whatever the reason, the notation e made its first appearance in a letter Euler wrote to Goldbach in 1731..” - [e (mathematical constant), Wikipedia](https://en.wikipedia.org/wiki/E_(mathematical_constant)) — read raw: “It was the first number to be proved transcendental without having been specifically constructed for this purpose (compare with Liouville number); the proof was given by Charles Hermite in 1873..” - [Eddington number, Wikipedia](https://en.wikipedia.org/wiki/Eddington_number) — read raw: “During a course of lectures that he delivered in 1938 as Tarner Lecturer at Trinity College, Cambridge, Eddington averred that:.” - [Some historical notes on proving Fermat numbers and cofactors to be composite, 64ordle.au](https://64ordle.au/fermat/historical/) — read raw: “Thus in the second half of 1640 we find Fermat writing at least three letters to Frénicle directly, and at least once more to Mersenne..” - [Fermat number, Wikipedia](https://en.wikipedia.org/wiki/Fermat_number) — read raw: “Fermat’s conjecture was refuted by Leonhard Euler in 1732.” - [65537-gon, Wikipedia](https://en.wikipedia.org/wiki/65537-gon) — read raw: “the first explicit construction of a regular 65537-gon was given by Johann Gustav Hermes (1894). The construction is very complex; Hermes spent 10 years completing the 200-page manuscript.”; “Although it was known to Carl Friedrich Gauss by 1801 that the regular 65537-gon was constructible.” - [Hierarchies of number-theoretic functions. I (Löb and Wainer), Archiv für mathematische Logik und Grundlagenforschung, Springer](https://link.springer.com/article/10.1007/BF01967649) — read raw: “Arch math Logik 13, 39–51 (1970). https://doi.org/10.1007/BF01967649.” - [Elliptic functions: introduction course, lecture 2 (the AGM), Vladimir G. Tkachev, Linköping University](https://users.mai.liu.se/vlatk48/teaching/lect2-agm.pdf) — read raw: “By May 30th, 1799, Gauss had observed, purely computationally, that” 1/M(1, √2) and (2/π)∫₀¹ dt/√(1 − t⁴) “agreed to at least eleven (!) decimal places.” - [Lemniscate constant, Wikipedia](https://en.wikipedia.org/wiki/Lemniscate_constant) — read raw: “Todd’s second lemniscate constant B and Gauss’s constant G were proven transcendental by Theodor Schneider in 1941..” Editorial note: Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim. ### 4.31 {#r-4.31} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Hilbert set his problems (the seventh among them) in 1900; Kasner’s nine-year-old nephew Milton Sirotta named the googol in 1920; Kasner published the name in Mathematics and the Imagination in 1940; googolplex was named at the same time in 1920 as the googol; Leopold Theisinger proved in 1915 that H_n is an integer only for n = 1; Wilhelm Ackermann’s three-argument function of 1928 is an early version of the hyperoperations; Reuben Goodstein defined the sequence and coined the Greek names (tetration, pentation...) in 1947; D. R. Kaprekar found the 6174 routine in 1949; Knuth introduced up-arrow notation in 1976, in Science. Kasner’s nephew was asked to name the number, and named the googolplex at the same time. The hyperoperation sequence as a whole is a version of Ackermann’s original three-argument function. A googol exceeds the roughly 10^80 elementary particles in the observable universe. *Status: confirmed* - [Hilbert’s seventh problem, Wikipedia](https://en.wikipedia.org/wiki/Hilbert%27s_seventh_problem) — read raw: “Hilbert’s seventh problem is one of David Hilbert’s list of open mathematical problems posed in 1900..” - [Googol, Wikipedia](https://en.wikipedia.org/wiki/Googol) — read raw: “The term was coined in 1920 by nine-year-old Milton Sirotta (1911–1981), nephew of American mathematician Edward Kasner.”; “Kasner popularized the concept in his 1940 book Mathematics and the Imagination.”; “Carl Sagan pointed out that the total number of elementary particles in the observable universe is around 10^80 (the Eddington number)” (exponent as typeset on the page). - [Googolplex, Wikipedia](https://en.wikipedia.org/wiki/Googolplex) — read raw: “In 1920, Edward Kasner’s nine-year-old nephew, Milton Sirotta, coined the term googol.” - [Bemerkung über die harmonische Reihe (L. Theisinger), Monatshefte für Mathematik und Physik, Springer](https://link.springer.com/article/10.1007/BF01999444) — read raw: “Monatsh. f. Mathematik und Physik 26, 132–134 (1915). https://doi.org/10.1007/BF01999444.” - [Hyperoperation, Wikipedia](https://en.wikipedia.org/wiki/Hyperoperation) — read raw: “In 1928, Wilhelm Ackermann defined a 3-argument function”; “In his 1947 paper, Reuben Goodstein introduced the specific sequence of operations that are now called hyperoperations, and also suggested the Greek names tetration, pentation, etc., for the extended operations beyond exponentiation.” - [Kaprekar Routine, Wolfram MathWorld](https://mathworld.wolfram.com/KaprekarRoutine.html) — read raw: “The Kaprekar routine is an algorithm discovered in 1949 by D. R. Kaprekar for 4-digit numbers.” - [Knuth’s up-arrow notation, Wikipedia](https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation) — read raw: “Knuth’s up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976..” - [Edward Kasner, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Kasner/) — quoting Mathematics and the Imagination: “The name “googol” was invented by a child (Dr Kasner’s nine-year-old nephew) who was asked to think up a name for a very big number, namely, 1 with a hundred zeros after it.”; “At the same time that he suggested “googol” he gave a name for a still larger number: “Googolplex.”” - [Yang, Li, Feng and Jiang, On the integrality of the first and second elementary symmetric functions, AIMS Mathematics 2(4) (2017)](https://www.aimspress.com/fileOther/PDF/Math/Math-02-00682.pdf) — the first name and year: “The first published proof went back to 1915 by Leopold Theisinger.” Editorial note: Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim. ### 4.32 {#r-4.32} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Ackermann’s function of 1928 (preceding Goodstein’s naming of tetration, pentation, hexation in 1947); Lehmer’s degree-10 polynomial and its root 1.17628 found by Derrick Lehmer in 1933; Gauss’s diary records his results on the lemniscate in 1798, and the relation between the arithmetic-geometric mean of 1 and √2 and π/ϖ in May 1799; Gauss proved in 1799 that M(1, √2) = π/ϖ; Liouville announced in 1844 that numbers approximated too well by rationals must be transcendental; Liouville gave the constant Σ10^(−n!) in his full paper of 1851, the first decimal constant proven transcendental; the Bignum Bakeoff, which Ralph Loader’s program won, was a contest of December 2001; Jinkōki first printed in 1627 with many errors; Jinkōki corrected in the 1631 edition. Ralph Loader’s program won the Bignum Bakeoff, whose entries were C programs of at most 512 characters; it diagonalizes over the calculus of constructions and returns D^5(99). *Status: confirmed* - [Hyperoperation, Wikipedia](https://en.wikipedia.org/wiki/Hyperoperation) — read raw: “In 1928, Wilhelm Ackermann defined a 3-argument function \phi(a, b, n) which gradually evolved into a 2-argument function known as the Ackermann function..” - [Lehmer’s Mahler Measure Problem, Wolfram MathWorld](https://mathworld.wolfram.com/LehmersMahlerMeasureProblem.html) — read raw: “Lehmer’s Mahler measure problem is an unsolved problem in mathematics attributed to Lehmer (1933) that concerns the minimum Mahler measure for a univariate polynomial that is not a product of cyclotomic polynomials..” - [B. Sury, The Arithmetico-geometric Mean of Gauss, Resonance, August 2000](https://www.ias.ac.in/article/fulltext/reso/005/08/0072-0083) — read raw: “Gauss’s 92nd entry made in 1798 in his famous mathematical diary reads “I have obtained most elegant results concerning the lemniscate, which surpasses all expectation.” - [Arithmetic–geometric mean, Wikipedia](https://en.wikipedia.org/wiki/Arithmetic%E2%80%93geometric_mean) — read raw: “In 1799, Gauss proved that M(1,√2) = π/ϖ where ϖ is the lemniscate constant..” - [Joseph Liouville, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Liouville/) — read raw: “led him to prove the existence of a transcendental number in 1844 when he constructed an infinite class of such numbers using continued fractions”; “In 1851 he published results on transcendental numbers removing the dependence on continued fractions. In particular he gave an example of a transcendental number, the number now named the Liouvillian number 0 . 1100010000000000000000010000 ....” - [David Moews, BIGNUM BAKEOFF contest rules (the organiser’s own file, dated 10 Dec 2001)](http://djm.cc/bignum-rules-posted.txt) — read raw: “Entries must be e-mailed to dmoews@xraysgi.ims.uconn.edu and received before 23:59:59 PST, December 31, 2001..” - [Japanese numerals, Wikipedia](https://en.wikipedia.org/wiki/Japanese_numerals) — read raw: “The initial edition was published in 1627 and had many errors, most of which were fixed in the 1631 edition. In 1634, there was yet another edition which again changed a few values.”; “Variation is due to the Jinkōki (塵劫記), Japan’s oldest mathematics text.” Its table of large numbers runs in steps of 10⁴ up to 極 (10⁴⁸) and then gives each higher name two values, from “10^52 (or 10^56)” for 恒河沙 to “10^68 (or 10^88)” for 無量大数. - [Transcendental number, Wikipedia](https://en.wikipedia.org/wiki/Transcendental_number) — the first decimal examples: “Joseph Liouville first proved the existence of transcendental numbers in 1844, and in 1851 gave the first decimal examples such as the Liouville constant” - [Liouville number, Wikipedia](https://en.wikipedia.org/wiki/Liouville_number) — the 1844 approximation bound: “In 1844, Joseph Liouville proved a bound showing that there is a limit to how well algebraic numbers can be approximated by rational numbers, and he defined Liouville numbers specifically so that they would have rational approximations better than the ones allowed by this bound.” - [BIGNUM BAKEOFF contest recap, David Moews](http://djm.cc/bignum-results.txt) — “The aim of this contest was to write a C program of 512 characters or less (excluding whitespace) that returned as large a number as possible from main()”; “The winner is Ralph Loader”; “The final and winning entry, {loader.c}, diagonalizes over the Huet-Coquand ‘calculus of constructions’.”; “The program’s return value is D@@5(99).” Editorial note: Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.