# References — Chapter ᚂ(L): Garden variety (part 2) *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.* ### 4.33 {#r-4.33} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Jinkōki changed again in the 1634 edition; the folded-strip pattern was described by Martin Gardner in 1967, and many of its properties were first published by Chandler Davis and Donald Knuth in 1970; Davis and Knuth’s paper on the dragon curve is of 1970; Sylvester called an odd perfect number “little short of a miracle” in 1888; Lambert proved π irrational in 1761; 22/7 is Archimedes’ upper bound for π from about 250 BC, from 96-sided polygons; Gérard Cordonnier found the plastic ratio in 1924; Hans van der Laan, who came to it in 1928, first called it the fundamental ratio and used the name plastic number from the 1950s; Hans van der Laan discovered his system of proportions in 1928; e^(π√163) as an almost-integer first noticed by Charles Hermite in 1859. Van der Laan’s plastic number describes three-dimensional growth, 1 + a = a³, where the golden section is 1 + a = a². *Status: confirmed* - [Japanese numerals, Wikipedia](https://en.wikipedia.org/wiki/Japanese_numerals) — read raw: “In 1634, there was yet another edition which again changed a few values. The above variation is due to inconsistencies in the latter two editions..” - [Dragon curve, Wikipedia](https://en.wikipedia.org/wiki/Dragon_curve) — read raw: “It was described by Martin Gardner in his Scientific American column Mathematical Games in 1967. Many of its properties were first published by Chandler Davis and Donald Knuth.”; “Chandler Davis (1970), Donald J. Knuth (1970).” - [Perfect number, Wikipedia](https://en.wikipedia.org/wiki/Perfect_number) — read raw: “In 1888, Sylvester stated: ... a prolonged meditation on the subject has satisfied me that the existence of any one such [odd perfect number]—its escape, so to say, from the complex web of conditions which hem it in on all sides—would be little short of a miracle.”; “The first four perfect numbers were the only ones known to early Greek mathematics, and the mathematician Nicomachus noted 8128 as early as around AD 100.” - [Irrational number, Wikipedia](https://en.wikipedia.org/wiki/Irrational_number) — read raw: “Johann Heinrich Lambert proved (1761) that π cannot be rational, and that e^n is irrational if n is rational (unless n = 0)..” - [Pi, Wikipedia](https://en.wikipedia.org/wiki/Pi) — read raw: “The first recorded algorithm for rigorously calculating the value of π was a geometrical approach using polygons, devised around 250 BC by the Greek mathematician Archimedes, implementing the method of exhaustion..” - [Plastic ratio, Wikipedia](https://en.wikipedia.org/wiki/Plastic_ratio) — read raw: “French high school student Gérard Cordonnier discovered the ratio for himself in 1924. In his correspondence with Hans van der Laan a few years later, he called it the radiant number. Van der Laan initially referred to it as the fundamental ratio, using the plastic number from the 1950s onward..” - [arXiv:1805.07178, Appendix C](https://arxiv.org/abs/1805.07178) — read in the paper’s PDF, Appendix C (the abstract page does not carry it): “In 1928, shortly after abandoning his architectural studies and becoming a novice monk of the Benedictine Order, Hans van der Laan discovered a new, unique system of architectural proportions..” - [Heegner number, Wikipedia](https://en.wikipedia.org/wiki/Heegner_number) — read raw: “This number was discovered in 1859 by the mathematician Charles Hermite.”; it gives e^(π√163) “≈ 640 320³ + 744” and “In a 1975 April Fools’ Day article in Scientific American magazine, ‘Mathematical Games’ columnist Martin Gardner made the hoax claim that the number was in fact an integer, and that the Indian mathematical genius Srinivasa Ramanujan had predicted it—hence its name.” - [The Plastic Number ratio, Dom Hans van der Laan foundation](https://domhansvanderlaan.nl/theory-practice/theory/the-plastic-number-ratio/) — the three-dimensional contrast with the golden section: “This growth occurs three-dimensionally. Moreover: the Plastic Number represents the smallest proportional growth of three-dimensional objects.”; “This is different than the golden section, which defines the Fibonacci series, and is defined as 1 + a = a2.” 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. ### 4.34 {#r-4.34} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Ramanujan gave almost-integer examples such as e^(π√58) in a paper of 1913–14 (the volume’s dates; 1913 is its first year); Claude Shannon’s 1950 paper Programming a Computer for Playing Chess gives the 10^120 estimate; Skewes proved in 1933, assuming RH, a crossing below e^(e^(e^79)) < 10^(10^(10^34)); In 1955 Skewes gave an unconditional bound of about 10^(10^(10^964)); Bays and Hudson’s 2000 crossing near 1.398×10^316, sharpened by Saouter and Demichel in 2010 to below about 1.397×10^316; No crossing below 10^19 (Büthe 2015); The 1729 taxi story is Hardy’s own, from his 1940 book on Ramanujan, where he went to see Ramanujan ill at Putney; Kurt Mahler proved the Prouhet–Thue–Morse constant transcendental in 1929; Kruskal’s tree theorem of 1960. G. H. Hardy described Skewes’ number as the largest number that has ever served any definite purpose in mathematics. *Status: confirmed* - [Ramanujan Constant, Wolfram MathWorld](https://mathworld.wolfram.com/RamanujanConstant.html) — read raw: “Although Ramanujan (1913-1914) gave few rather spectacular examples of almost integers (such ), he did not actually mention the particular near-identity given above.”; “The name “Ramanujan’s constant” was coined by Simon Plouffe and derives from an April Fool’s joke played by Martin Gardner (Apr. 1975) on the readers of Scientific American.” - [Shannon number, Wikipedia](https://en.wikipedia.org/wiki/Shannon_number) — read raw: “His influential 1950 paper on it, “Programming a Computer for Playing Chess”; the estimate is “based on an average of about 10^3 possibilities for a pair of moves consisting of a move for White followed by a move for Black, and a typical game lasting about 40 such pairs of moves” (exponent as typeset on the page). - [Skewes’s number, Wikipedia](https://en.wikipedia.org/wiki/Skewes%27s_number) — read raw: “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}}}.”; “Without assuming the Riemann hypothesis, Skewes (1955) later proved that there exists a value of x below e^{e^{e^{e^{7.705}}}}‹10^{10^{10^{964}}}.”; “|2010|| 2.2 || Saouter and Demichel || [exp(727.95132478), exp(727.95134682)] (without RH)”; “by Platt & Trudgian (2014) to 1.39\times 10^{17}, and by Büthe (2015) to 10^{19}..” - [1729 (number), Wikipedia](https://en.wikipedia.org/wiki/1729_(number)) — read raw: “{{cite book |title = Ramanujan | url = https://archive.org/details/pli.kerala.rare.37877 | last = Hardy | first = G. H. | author-link = G. H. Hardy |location = New York | publisher = Cambridge University Press | year = 1940”; the page quotes it (p. 12): “I remember once going to see him when he was ill at Putney. I had ridden in taxi cab No. 1729 and remarked that the number seemed to me rather a dull one.” - [Prouhet–Thue–Morse constant, Wikipedia](https://en.wikipedia.org/wiki/Prouhet%E2%80%93Thue%E2%80%93Morse_constant) — read raw: “The Prouhet–Thue–Morse constant was shown to be transcendental by Kurt Mahler in 1929..” - [Kruskal’s tree theorem, Wikipedia](https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem) — read raw: “The theorem was conjectured by Andrew Vázsonyi and proved by Joseph Kruskal (1960); a short proof was given by Crispin Nash-Williams (1963)..” - [Stanley Skewes and the Skewes Number, H. P. Williams (LSE)](https://personal.lse.ac.uk/williahp/pubs/Stanley%20Skewes%20and%20the%20Skewes%20Number.html) — “the famous mathematician G.H.Hardy described the Skewes number as “the largest number that has ever served any definite purpose in mathematics.”” 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. ### 4.35 {#r-4.35} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Harvey Friedman introduced the TREE function in the 2000s; the tribonacci sequence appears in an 1892 puzzle about sheep; The word ‘tribonacci’ coined in 1963 by Mark Feinberg, aged fourteen, in the Fibonacci Quarterly; Knuth introduced up-arrow notation in ‘Mathematics and Computer Science: Coping with Finiteness’, Science, 1976; Euler proved 2^31 - 1 prime in 1772; cos(2π/17): the number Gauss found in 1796; Wantzel proved in 1837 that the cube root of 2 cannot be constructed with compass and straightedge, which settles the doubling of the cube; Liouville’s constant Σ 10^(−n!) was constructed in 1851 so that it could be proved transcendental, one of the first decimal examples of a transcendental number; Hermite showed in 1873 that e is transcendental, and Lindemann did the same for π in 1882. Friedman defined TREE[k] in a Foundations of Mathematics posting of 28 March 2006. *Status: confirmed* - [Kruskal’s tree theorem, Wikipedia](https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem) — read raw: “By incorporating labels, Friedman defined a far faster-growing function.{{Cite web |last=Friedman |first=Harvey |date=28 March 2006 |title=273:Sigma01/optimal/size |url=http://www.cs.nyu.edu/pipermail/fom/2006-March/010279.html.” - [Generalizations of Fibonacci numbers, Wikipedia](https://en.wikipedia.org/wiki/Generalizations_of_Fibonacci_numbers) — read raw: “In 1892, the sequence of numbers appeared in the solution of a recreational problem, concerning a farmer and the raising of sheep, that was posed by the American mathematician Artemas Martin (1835–1918).”; “The moniker tribonacci appeared much later, not until 1963, and is due to Mark Feinberg, at the time a fourteen-year-old high-school student, who introduced the term in an article in the Fibonacci Quarterly.”; “The most notable historical occurrence of the sequence is connected to Charles Darwin (1809–1882) and his seminal book On the Origin of Species, where the procreation and population growth of elephants is considered as an illustrative example.” - [Knuth’s up-arrow notation, Wikipedia](https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation) — read raw: “introduced by Donald Knuth in 1976.{{cite journal | last =Knuth | first = Donald E.| year=1976|title=Mathematics and Computer Science: Coping with Finiteness |journal=Science | volume=194|issue=4271| pages=1235–1242.” - [2,147,483,647, Wikipedia](https://en.wikipedia.org/wiki/2,147,483,647) — read raw: “The primality of this number was proven by Leonhard Euler, who reported the proof in a letter to Daniel Bernoulli written in 1772..” - [Heptadecagon, Wikipedia](https://en.wikipedia.org/wiki/Heptadecagon) — read raw: “the regular heptadecagon is a constructible polygon (that is, one that can be constructed using a compass and unmarked straightedge): this was shown by Carl Friedrich Gauss in 1796..” - [Doubling the cube, Wikipedia](https://en.wikipedia.org/wiki/Doubling_the_cube) — read raw: “But Pierre Wantzel proved in 1837 that the cube root of 2 is not constructible; that is, it cannot be constructed with straightedge and compass..” - [Transcendental number, Wikipedia](https://en.wikipedia.org/wiki/Transcendental_number) — read raw: “Joseph Liouville first proved the existence of transcendental numbers in 1844, and in 1851 gave the first decimal examples such as the Liouville constant”; “The first number to be proven transcendental without having been specifically constructed for the purpose of proving transcendental numbers’ existence was e, by Charles Hermite in 1873..” - [[FOM] 273:Sigma01/optimal/size, Harvey Friedman, 28 March 2006 (archived)](https://web.archive.org/web/20240918084441/https://cs.nyu.edu/pipermail/fom/2006-March/010279.html) — posted “Tue Mar 28 12:57:25 EST 2006”: “Let TREE[k] be the length n of the longest sequence of trees T_1,...,T_n with vertices labeled from {1,...,k}, where each T_i has at most i vertices, and no T_i is inf preserving and label preserving embeddable into a later T_j.” 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. ### 4.36 {#r-4.36} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Γ(1/4) was proved transcendental by Gregory Chudnovsky around 1976; Graham’s number was published by Martin Gardner in 1977. *Status: confirmed* - [Real geometric transcendence for the Gamma function (Arshay Sheth), arXiv](https://arxiv.org/pdf/2605.15117v1) — read raw: “Gregory Volfovich Chudnovsky, Algebraic independence of constants connected with the exponential and elliptic functions, Dokl. Akad. Nauk Ukrain. SSR Ser. A 1976, no. 8, 698.” - [Graham’s number, Wikipedia](https://en.wikipedia.org/wiki/Graham%27s_number) — read raw: “This weaker upper bound for the problem, attributed to an unpublished work of Graham, was eventually published and named by Martin Gardner in Scientific American in November 1977..” 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. ### 4.37 {#r-4.37} Kruskal’s tree theorem of 1960 says that any infinite sequence of finite trees contains one tree that embeds into a later one, and Harvey Friedman’s finite miniaturized version of the theorem defines the function whose value TREE(3) the book uses. *Status: confirmed* - [Kruskal’s tree theorem, Wikipedia](https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem) — states the theorem and its role in defining Friedman’s TREE function. ### 4.38 {#r-4.38} Hilbert posed the transcendence of a^b, for algebraic a not equal to 0 or 1 and algebraic irrational b, as the seventh of his 1900 list of problems, naming 2^√2 and e^π as his own examples, and Gelfond and Schneider answered it affirmatively and independently in 1934. The list had twenty-three problems. *Status: confirmed* - [Hilbert’s seventh problem, Wikipedia](https://en.wikipedia.org/wiki/Hilbert%27s_seventh_problem) — states the problem, Hilbert’s own examples, and the 1934 resolution. - [Hilbert’s problems, Wikipedia](https://en.wikipedia.org/wiki/Hilbert%27s_problems) — carries the count of the list: “Hilbert’s problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900.” - [Gelfond–Schneider theorem, Wikipedia](https://en.wikipedia.org/wiki/Gelfond%E2%80%93Schneider_theorem) — carries e^π as (−1)^(−i) and the cases the theorem covers: “(−1)^(−i) = 23.14069263…,” with 2^√2, √2^√2 and i^i = e^(−π/2) listed as consequences. ### 4.41 {#r-4.41} The Bernoulli numbers are the rational coefficients Bn in the expansion x/(e^x − 1) = sum Bn x^n/n!, published by Jacob Bernoulli in the Ars Conjectandi of 1713, and they give closed forms for sums of integer powers and for the Riemann zeta function at even integers. Seki Takakazu reached the same numbers independently in Japan in 1712. *Status: confirmed* - [Bernoulli number, Wikipedia](https://en.wikipedia.org/wiki/Bernoulli_number) — gives the defining generating function and the applications to power sums and zeta. ### 4.44 {#r-4.44} The harmonic numbers Hn are the partial sums 1 + 1/2 + ... + 1/n of the harmonic series; Leopold Theisinger proved in 1915 that Hn is an integer only for n = 1, and every harmonic number is rational. The sequence begins 1, 3/2, 11/6, 25/12. Hₙ grows roughly like ln n, so the harmonic series diverges, slowly. *Status: confirmed* - [Harmonic number, Wikipedia](https://en.wikipedia.org/wiki/Harmonic_number) — defines the harmonic numbers and states that H_n is an integer only for n = 1, “a result often attributed to Theisinger”; it gives no year (1915 is on the arXiv line). On growth: “The harmonic numbers roughly approximate the natural logarithm function” … “and thus the associated harmonic series grows without limit, albeit slowly.” - [About the Non-Integer Property of Hyperharmonic Numbers, arXiv:0811.0043](https://arxiv.org/abs/0811.0043) — names Theisinger and the year: “It was proven in 1915 by Leopold Theisinger that the H_n harmonic numbers are never integers.” ### 4.45 {#r-4.45} Lehmer’s problem, posed by Derrick Lehmer in 1933, asks whether there is an algebraic integer with Mahler measure strictly between 1 and the value of Lehmer’s number, about 1.1762808, the smallest Mahler measure above 1 ever found, and no smaller example is known nearly a century later. Lehmer’s number is the largest root of his degree-10 polynomial. *Status: confirmed* - [Mahler measure, Wikipedia](https://en.wikipedia.org/wiki/Mahler_measure) — states Lehmer’s conjecture, that there is a constant μ > 1 such that an irreducible integer polynomial p has either M(p) = 1 or M(p) > μ; Lehmer’s number and its smallest-known status are on the Lehmer’s conjecture and arXiv lines. - [Lehmer’s conjecture, Wikipedia](https://en.wikipedia.org/wiki/Lehmer%27s_conjecture) — gives Lehmer’s 1933 paper, his degree-10 polynomial and its measure, and the open status: “P(x) = x^10 + x^9 − x^7 − x^6 − x^5 − x^4 − x^3 + x + 1”; “M(P(x)) = 1.176280818…”; “Lehmer’s conjecture is not yet completely proved and is still a question of much interest.” - [E. Hironaka, Lehmer’s Problem, McKay’s Correspondence, and 2,3,7, arXiv math/0204040](https://arxiv.org/html/math/0204040v1) — carries the search record: “Boyd and Mossinghoff have done searches up to degree 40, but so far no one has found a monic noncyclotomic integer polynomial with smaller Mahler measure.” ### 4.49 {#r-4.49} The look-and-say sequence starts at 1 and each later term describes the digit runs of the one before, giving 1, 11, 21, 1211, 111221, ...; John Conway analyzed it in a 1986 paper and showed the length of the n-th term grows like Conway’s constant to the n. Conway’s constant is about 1.3036. *Status: confirmed* - [Look-and-say sequence, Wikipedia](https://en.wikipedia.org/wiki/Look-and-say_sequence) — defines the sequence and states Conway’s 1986 growth-rate result. ### 4.54 {#r-4.54} States that the transcendence of Γ(1/5) is open, unlike Γ(1/3) and Γ(1/4). Γ(1/2) = √π and Γ(1/6) are also known to be transcendental. *Status: confirmed* - [Open Diophantine Problems, M. Waldschmidt, arXiv math/0312440](https://arxiv.org/pdf/math/0312440) — states Γ(1/5) is open and Γ(1/2) transcendental: “there is no proof so far that Γ(1/5) is transcendental”; “Lindemann’s result on the transcendence of π which implies that Γ(1/2) = √π is also transcendental” - [Gamma function, Wikipedia](https://en.wikipedia.org/wiki/Gamma_function) — gives the known transcendental values: “It has been proved that Γ(n + r) is a transcendental number and algebraically independent of π for any integer n and each of the fractions r = 1/6, 1/4, 1/3, 2/3, 3/4, 5/6.” - [Lemniscate constant, Wikipedia](https://en.wikipedia.org/wiki/Lemniscate_constant) — ties Γ(1/4) to the lemniscate constant: “ϖ = Γ(1/4)² / (2√(2π))”