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# References — Chapter ᚃ(F): The local park

*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ref-ch-5, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*

[Guide for AI readers](index.md) · [The chapter](ch05.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.*

### 5.1 {#r-5.1}

Legendre’s correction was about 1.08366 in 1808 and de la Vallée Poussin proved the best constant was exactly 1 in 1899.

*Status: confirmed*

- [Legendre’s constant, Wikipedia](https://en.wikipedia.org/wiki/Legendre%27s_constant) — “Legendre proposed in 1808 the formula y = x/log(x) − 1.08366”; Chebyshev showed in 1849 that the limit, if it exists, must be 1, and de la Vallée Poussin proved in 1899 that it is.

### 5.2 {#r-5.2}

The Erdős–Borwein constant is irrational and its transcendence is open.

*Status: confirmed*

- [Erdős–Borwein constant, Wikipedia](https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Borwein_constant) — the value 1.606695152415291763…, Paul Erdős’s irrationality proof of 1948 and Peter Borwein’s second proof of 1992. Nothing is asserted about transcendence either way, which is the sense in which the card calls it open.
- [List of mathematical constants, Wikipedia](https://en.wikipedia.org/wiki/List_of_mathematical_constants) — the Erdős–Borwein constant row reads “✗” under ℚ (rational) and “?” under 𝔸 (algebraic), 𝒫 and 𝓔𝓟: not rational, algebraic status unknown.

### 5.3 {#r-5.3}

The reciprocal Fibonacci constant is about 3.3599 and irrational, with transcendence open.

*Status: confirmed*

- [Reciprocal Fibonacci constant, Wikipedia](https://en.wikipedia.org/wiki/Reciprocal_Fibonacci_constant) — the value 3.359885666243177553…, and Richard André-Jeannin’s 1989 proof of irrationality, settling a conjecture of Erdős, Graham and Carlitz. The transcendence results there are for the Fibonacci zeta function at even arguments, not for ψ itself.
- [List of mathematical constants, Wikipedia](https://en.wikipedia.org/wiki/List_of_mathematical_constants) — the Reciprocal Fibonacci constant row reads “✗” under ℚ (rational) and “?” under 𝔸 (algebraic) and 𝒫: irrational, algebraic status unknown.

### 5.4 {#r-5.4}

Copeland–Erdős is the concatenation of the primes, normal in base 10 and therefore irrational. Its transcendence has not been proved.

*Status: confirmed*

- [Copeland–Erdős constant, Wikipedia](https://en.wikipedia.org/wiki/Copeland%E2%80%93Erd%C5%91s_constant) — “In base 10, the constant is a normal number, a fact proven by Arthur Herbert Copeland and Paul Erdős” (1946, per the page’s reference), and “It follows that the constant is irrational; this can also be proven with Dirichlet’s theorem on arithmetic progressions or Bertrand’s postulate.”
- [Copeland–Erdős Constant, Wolfram MathWorld](https://mathworld.wolfram.com/Copeland-ErdosConstant.html) — the decimal 0.23571113171923… and the 1946 normality result; no transcendence claim is made.
- [Marek Wolf, Continued fractions constructed from prime numbers, arXiv:1003.4015](https://arxiv.org/abs/1003.4015) — “Arthur Herbert Copeland and Paul Erdős have shown that CCE is normal, but apparently it is not proved that CCE is transcendental.”

### 5.5 {#r-5.5}

Apéry proved ζ(3) irrational in 1978 and its transcendence is open. No closed form for ζ(3) is known, and Apéry’s proof, greeted with disbelief, used only methods available for 200 years.

*Status: confirmed*

- [Apéry’s constant, Wikipedia](https://en.wikipedia.org/wiki/Ap%C3%A9ry%27s_constant) — the value 1.202056903159594285…, “Roger Apéry, who proved in 1978 that it is an irrational number,” and “Apéry’s constant has not yet been proved transcendental.”
- [Roger Apéry, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Apery/) — “In 1978 he presented a lecture on his proof at the Journées Arithmétiques de Marseille which was greeted with doubt and disbelief.” and “The problem had long been an open question, yet Apéry’s proof only used methods which had been available for 200 years.”
- [Computing ζ(3), John D. Cook](https://www.johndcook.com/blog/2021/08/28/computing-zeta-3/) — “There are closed-form expressions for ζ(s) when s is an even integer but so far not nobody has found one for s = 3.”

### 5.6 {#r-5.6}

Euler’s constant is about 0.5772 and its irrationality is unknown; at least one of it and the Gompertz constant is irrational, which Lagarias calls an observation of A. I. Aptekarev in 2009 and Wikipedia credits to Shidlovski (1962 on its Gompertz constant page, 1959 as Shidlovsky on its Euler–Mascheroni page), and at least one of them is transcendental, which Tanguy Rivoal proved in 2012; and the Meissel–Mertens constant is its analogue over the primes. The Meissel–Mertens constant is not known to be irrational. Le Lionnais named the Gompertz constant for its role in survival analysis; Euler met it in studying divergent series, and it is the Borel-summed value of 1 − 1 + 2 − 6 + 24 − 120 + ⋯.

*Status: confirmed*

- [Euler–Mascheroni constant, Wikipedia](https://en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant) — limit definition, value, and open irrationality; “In 1959 Andrei Shidlovsky proved that at least one of Euler’s constant γ and the Gompertz constant δ is irrational.”
- [Gompertz constant, Wikipedia](https://en.wikipedia.org/wiki/Gompertz_constant) — the value 0.596347362323194074…; “In 1962, Shidlovski proved that at least one of the Euler–Mascheroni constant and the Euler–Gompertz constant is irrational,” improved “in 2012 by Tanguy Rivoal, who proved that at least one of them is transcendental”; “When Euler studied divergent infinite series, he encountered δ via, for example, the above integral representation. Le Lionnais called δ the Gompertz constant because of its role in survival analysis”; and δ is “the regularized value of the summation of alternating factorials of all natural numbers (1 − 1 + 2 − 6 + 24 − 120 + ⋯), which is defined by Borel summation.”
- [Meissel–Mertens constant, Wikipedia](https://en.wikipedia.org/wiki/Meissel%E2%80%93Mertens_constant) — M = 0.261497212847642783…, the limiting difference between the harmonic series over the primes and log log; γ “has an analogous definition involving a sum over all integers (not just the primes).” No arithmetic status is stated.
- [Jeffrey C. Lagarias, Euler’s constant: Euler’s work and modern developments, arXiv:1303.1856 (Bull. AMS 50 (2013) 527–628)](https://arxiv.org/abs/1303.1856) — read raw via the ar5iv rendering of arXiv:1303.1856. Section 3.16: “In 2012 Rivoal proved a result which, as a special case, implies the transcendence of at least one of the Euler constant γ and the Euler-Gompertz constant δ. It uses the Shidlovskii method and improves on an observation of Aptekarev that at least one of γ and δ must be irrational.” Aptekarev’s paper is On linear forms containing the Euler constant, arXiv:0902.1768, 28 February 2009.
- [List of mathematical constants, Wikipedia](https://en.wikipedia.org/wiki/List_of_mathematical_constants) — the Meissel–Mertens constant row reads “?” under ℚ (rational), 𝔸 (algebraic) and 𝒫: status unknown; the row’s year column reads “1866 & 1873.”

Editorial note: Corrected 17 September 2026. The entry credited the irrationality result to Shidlovski in 1962, following Wikipedia’s Gompertz constant article, while the #gamma and #gompertz cards credited Aptekarev in 2009. The cards were right. Lagarias’s survey, which is the source that Wikipedia article itself footnotes for the sentence, says Rivoal’s 2012 theorem “improves on an observation of Aptekarev that at least one of γ and δ must be irrational”. Wikipedia’s Euler’s constant article makes the same attribution to Shidlovski and dates it 1959 rather than 1962, so the two tertiary pages disagree with each other as well as with the survey they cite. This is not a tie to record: a refereed survey in the Bulletin of the AMS is the better source and is the one the others are quoting. Shidlovskii is not absent from the story — Rivoal’s proof uses his method, and Lagarias dates his breakthrough on E-functions to 1959 — but the at-least-one-is-irrational observation is Aptekarev’s.

### 5.7 {#r-5.7}

Feigenbaum’s constants describe period-doubling scaling; signed α is negative while the displayed magnitude is positive. The signed value is α ≈ −2.502907875.

*Status: confirmed*

- [Feigenbaum constants, Wikipedia](https://en.wikipedia.org/wiki/Feigenbaum_constants) — δ = 4.669201609102990671…, α = 2.502907875095892822…, and “Both numbers are believed to be transcendental, although they have not been proven to be so. In fact, there is no known proof that either constant is even irrational.”
- [Feigenbaum Constant, Wolfram MathWorld](https://mathworld.wolfram.com/FeigenbaumConstant.html) — “For the logistic map,” followed by displayed equation (8): “alpha = -2.502907875....”

### 5.8 {#r-5.8}

Catalan’s constant has the stated series and value and unknown irrationality. Gauss’s constant, also written G, was proven transcendental by Theodor Schneider in 1941.

*Status: judgement*

- [Catalan’s constant, Wikipedia](https://en.wikipedia.org/wiki/Catalan%27s_constant) — (article last edited 7 September 2026): “It is not known whether G is irrational, let alone transcendental,” and G described as “arguably the most basic constant whose irrationality and transcendence (though strongly suspected) remain unproven.”
- [Zhi-Wei Sun, Catalan’s constant is irrational (preprint)](https://arxiv.org/abs/2609.04176) — posted 3 September 2026 to the arXiv’s unvetted math.GM section, “In this paper we prove the irrationality of G via using suitable weights”; not refereed. Re-read raw on 17 September 2026: the record still shows “[Submitted on 3 Sep 2026]” and a single version, v1, in math.GM. There is no v2, no withdrawal and no journal reference, so nothing supersedes the reference literature and the book’s “irrationality unknown” stands.
- [Lemniscate constant, Wikipedia](https://en.wikipedia.org/wiki/Lemniscate_constant) — “Todd’s second lemniscate constant B and Gauss’s constant G were proven transcendental by Theodor Schneider in 1941.”

Editorial note: A claimed proof that Catalan’s constant is irrational was posted as a preprint on 3 September 2026, in the arXiv section reserved for unvetted work. It has not been refereed, objections were raised within days, and the reference literature is unchanged, so Catalan’s constant stays among the numbers not known to be irrational. If the proof stands, G rises one brace.

### 5.9 {#r-5.9}

None of the constants in the second brace — Khinchin’s constant, the Glaisher–Kinkelin, Golomb–Dickman, Landau–Ramanujan, Lévy, Fransén–Robinson, Bernstein, Viswanath, Niven, Artin and Sierpiński constants, the twin prime constant, the MRB constant, Somos’ nested root, the Kepler–Bouwkamp radius and Backhouse’s constant — has been proved irrational. Nor have Catalan’s, Brun’s, the Gompertz or the Meissel–Mertens constant, which the same field note names with them. Whether Khinchin’s constant itself has Khinchin’s geometric mean in its own continued fraction has not been established either.

*Status: judgement*

- [Khinchin’s constant, Wikipedia](https://en.wikipedia.org/wiki/Khinchin%27s_constant) — K₀ = 2.685452001065306445…, the geometric mean of the partial quotients of almost every real number, and “The algebraic properties of Khinchin’s constant itself, e. g. whether it is a rational, algebraic irrational, or transcendental number, are also not known.”; “Many well-known numbers, such as π, the Euler–Mascheroni constant γ, and Khinchin’s constant itself, based on numerical evidence, are thought to be among the numbers for which the limit … converges to Khinchin’s constant. However, none of these limits has been rigorously established.”
- [Lévy’s constant, Wikipedia](https://en.wikipedia.org/wiki/L%C3%A9vy%27s_constant) — e^(π²/(12 ln 2)) = 3.275822918721811159…, the growth rate of the denominators of almost every continued fraction; no arithmetic status is stated.
- [Somos’ quadratic recurrence constant, Wikipedia](https://en.wikipedia.org/wiki/Somos%27_quadratic_recurrence_constant) — σ = 1.661687949633594121… as the nested root √(1√(2√(3…))); no arithmetic status is stated.
- [Twin prime, Wikipedia](https://en.wikipedia.org/wiki/Twin_prime) — “Define the twin prime constant C₂ as C₂ = ∏(1−1/(p−1)²) ≈ 0.660161815846869573…,” the factor in the first Hardy–Littlewood conjecture.
- [List of mathematical constants, Wikipedia](https://en.wikipedia.org/wiki/List_of_mathematical_constants) — the table marks the arithmetic status of each constant in this group with a question mark, and its rows for Catalan’s constant, Brun’s constant, the Gompertz constant and the Meissel–Mertens constant also read “?” in the rational column. Its year column gives 1922 for the twin prime constant, where MathWorld cites the 1923 paper.

Editorial note: Khinchin’s constant is the one of this group whose arithmetic status a reference work states in as many words. For the others no source asserts a status either way: “status open” on their cards records the absence of a proof, not a theorem, which is the honest reading of a chapter about numbers nobody has classified.

### 5.10 {#r-5.10}

Brun’s constant is the convergent sum over twin primes. Its decimal value is an extrapolated estimate; the proved bounds are 1.840503 < B₂ < 2.288513.

*Status: confirmed*

- [Brun’s constant, Wikipedia](https://en.wikipedia.org/wiki/Brun%27s_constant) — twin-prime reciprocal sum, convergence, and rationality conditional on finiteness; “While the exact value of B 2 is unknown, it is thought to be around 1.9,” and of the 2002 value 1.902160583104: “The last is based on extrapolation from the sum 1.830484424658... for the twin primes below 10 16.”
- [D. J. Platt and T. S. Trudgian, Improved bounds on Brun’s constant, arXiv:1803.01925](https://arxiv.org/abs/1803.01925) — “We improve the unconditional bounds on Brun’s constant to 1.840503 < B < 2.288513.”

### 5.11 {#r-5.11}

Mills’ constant is irrational, proved unconditionally by Kota Saito in 2024; its transcendence is open, and only its digits depend on the Riemann Hypothesis.

*Status: confirmed*

- [Saito, Mills’ constant is irrational](https://arxiv.org/abs/2404.19461) — “We define Mills’ constant as the smallest real number ξ satisfying this property… we show that Mills’ constant is irrational. Furthermore, we obtain partial results on the transcendency of this number.” Submitted 30 April 2024, revised 8 May 2025; no hypothesis is assumed. Journal reference on the page: “Mathematika 71 (2025), no. 3, Paper No. e70027.” Read in the paper’s PDF: “Theorem 1.2. Either ξ3 is transcendental, or there exists a positive integer m such that ξ3^(3^m) is a Pisot number of degree 3”; “assuming Cramér’s conjecture, ξ3 is transcendental.”
- [Mills’ constant, Wikipedia](https://en.wikipedia.org/wiki/Mills%27_constant) — “it is known to be irrational,” citing Saito, and “Its value is unproven, but if the Riemann hypothesis is true, it is approximately 1.3063778838630806904686144926…”; “Caldwell and Cheng used this method to compute 6850 base 10 digits of Mills’ constant under the assumption that the Riemann hypothesis is true.”
- [Saito, Transcendency of variants of Mills’ constant](https://arxiv.org/abs/2508.16068) — a 2025 sequel proving Mills’ constant transcendental under the Density Hypothesis for the Riemann zeta function, and transcendental outright for several variants; the unconditional question is left open.

Editorial note: The irrationality proof assumes nothing; the Riemann Hypothesis enters only in computing the constant’s digits, where it supplies a prime between consecutive cubes. Transcendence is genuinely open: Saito proves that either the constant is transcendental or some power of it is a cubic Pisot number, and that transcendence would follow from Cramér’s conjecture or from the Density Hypothesis.

### 5.12 {#r-5.12}

At least one of π+e and πe is transcendental, but which one is unknown. π + e ≈ 5.8599, π·e ≈ 8.5397 and π^e ≈ 22.459. π^π ≈ 36.4622 and e^e ≈ 15.1543.

*Status: confirmed*

- [Transcendental number, Wikipedia](https://en.wikipedia.org/wiki/Transcendental_number) — “Since the field of algebraic numbers is algebraically closed and e and π are roots of the polynomial x²−(e+π)x+eπ, at least one of the numbers eπ and e+π is transcendental”; π+e, π·e, π^π, e^e and π^e are listed as conjectured transcendental, none of them proved even irrational.
- [A059742, OEIS](https://oeis.org/A059742) — “Decimal expansion of e + Pi.”: 5.8598744820488384738… (value checked by computation; OEIS blocks automated fetches)
- [A019609, OEIS](https://oeis.org/A019609) — “Decimal expansion of Pi*e.”: 8.5397342226735670654… (value checked by computation; OEIS blocks automated fetches)
- [Gelfond’s constant, Wikipedia](https://en.wikipedia.org/wiki/Gelfond%27s_constant) — “The decimal expansion of π^e is given by π^e = 22.45915771836104547342...”
- [A073233, OEIS](https://oeis.org/A073233) — “Decimal expansion of Pi^Pi.”: 36.462159607207911770… (value checked by computation; OEIS blocks automated fetches)
- [A073226, OEIS](https://oeis.org/A073226) — “Decimal expansion of e^e.”: 15.154262241479264189… (value checked by computation; OEIS blocks automated fetches)

### 5.13 {#r-5.13}

At least one of ζ(5), ζ(7), ζ(9), and ζ(11) is irrational without identifying which. ζ(5) ≈ 1.0369278 and is not itself known to be irrational; extending Apéry’s proof has produced no result on any specific odd value; each even value ζ(2n) is a rational multiple of π^(2n).

*Status: confirmed*

- [Zudilin odd zeta result](https://doi.org/10.1070/RM2001v056n04ABEH000427) — theorem statement says at least one of ζ(5), ζ(7), ζ(9), ζ(11) is irrational.
- [Particular values of the Riemann zeta function, Wikipedia](https://en.wikipedia.org/wiki/Particular_values_of_the_Riemann_zeta_function) — table row ζ(5): “1.0369277551433699263...”; “The relationship between zeta at the positive even integers and powers of pi may be written as a_n ζ(2n) = π^(2n) b_n where a_n and b_n are coprime positive integers for all n.”
- [Transcendental number, Wikipedia](https://en.wikipedia.org/wiki/Transcendental_number) — “in particular Apéry’s constant ζ(3), which is known to be irrational. For the other numbers ζ(5), ζ(7), ζ(9), ... even this is not known.”
- [Apéry’s constant, Wikipedia](https://en.wikipedia.org/wiki/Ap%C3%A9ry%27s_constant) — “Many people have tried to extend Apéry’s proof that ζ(3) is irrational to other values of the Riemann zeta function with odd arguments. Although this has so far not produced any results on specific numbers, it is known that infinitely many of the odd zeta constants ζ(2n + 1) are irrational”

### 5.15 {#r-5.15}

Rayo’s number is a fixed finite integer defined by a googol-symbol set-theory description, beyond feasible calculation. Rayo’s opponent in the duel was Adam Elga, and Rayo’s entry won; like every fixed finite integer the number is computable in the technical sense. Its size vastly exceeds TREE(3).

*Status: confirmed*

- [Rayo’s number, Wikipedia](https://en.wikipedia.org/wiki/Rayo%27s_number) — records the 2007 MIT duel, googol-symbol definition, and fixed-finite-integer distinction; “At the time, it was claimed to be the largest named number, as its size vastly exceeds even other notable large numbers such as Graham’s number and TREE(3).”
- [Big Number Duel, Agustín Rayo (MIT)](https://web.mit.edu/arayo/www/bignums.html) — read raw: “On January 26th 2007, Adam Elga and I got together to see who could come up with a bigger finite number.” Under the heading “The winning entry” it sets out the definition by at most a googol symbols of first-order set theory that is now called Rayo’s number.
- [Computable number, Wikipedia](https://en.wikipedia.org/wiki/Computable_number) — read as raw wikitext: “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.” Every integer is algebraic, so every fixed integer is computable.

### 5.16 {#r-5.16}

TREE(3) is the TREE-function value associated with Kruskal’s tree theorem.

*Status: confirmed*

- [Kruskal’s tree theorem, Wikipedia](https://en.wikipedia.org/wiki/Kruskal%27s_tree_theorem) — confirms the theorem context behind TREE(3).

### 5.17 {#r-5.17}

For 0<x<1, x^n tends to zero as n tends to infinity. Positive infinity is one of two elements adjoined to the reals, greater than every real number, so that limits at infinity can be defined; it is not itself a real number. On the extended real line 0 × (±∞) is usually left undefined, though probability and measure theory often define it as 0.

*Status: confirmed*

- [Limit laws, OpenStax Calculus](https://openstax.org/books/calculus-volume-1/pages/2-3-the-limit-laws) — states the limit laws for limits as x approaches a, among them the “Power law for limits”; the fact that x^n tends to zero for 0<x<1 is on the Sequences line below.
- [5.1 Sequences, Calculus Volume 2, OpenStax](https://openstax.org/books/calculus-volume-2/pages/5-1-sequences) — read raw: “We summarize these results regarding the geometric sequence {rⁿ}: rⁿ → 0 if 0 < r < 1, rⁿ → 1 if r = 1, rⁿ → ∞ if r > 1.”
- [Extended real number line, Wikipedia](https://en.wikipedia.org/wiki/Extended_real_number_line) — read as raw wikitext: “the extended real number system is obtained from the real number system ℝ by adding two elements denoted +∞ and −∞ that are respectively greater and lower than every real number.” and “Adjoining the elements +∞ and −∞ to ℝ enables a definition of “limits at infinity””; and, with its math as rendered: “The expressions ∞ − ∞, 0 × (±∞), and ±∞/±∞ (called indeterminate forms) are usually left undefined.” and “However, in the context of probability or measure theory, 0 × ±∞ is often defined as 0.”

### 5.18 {#r-5.18}

Chaitin’s Ω, introduced by Gregory Chaitin in 1975, is the halting probability of a fixed optimal universal prefix-free computer and is uncomputable, algorithmically random and transcendental; normality follows from the randomness rather than being separately stated. Ω is a real number between 0 and 1, and its first N bits would settle the halting problem for all programs up to N bits long. Even so, Calude and colleagues computed the first 64 bits of one such Ω, for a particular universal machine, in 2002.

*Status: confirmed*

- [Chaitin’s Constant, Wolfram MathWorld](https://mathworld.wolfram.com/ChaitinsConstant.html) — defines the halting probability and states Chaitin constants are uncomputable and transcendental; and “Calude et al. (2002) computed the first 64 bits of Chaitin’s constant for a certain universal Turing machine.”
- [Gregory Chaitin, The Limits of Reason, free preprint](https://arxiv.org/pdf/math/0611740) — The technical discussion on PDF page 11 defines the fixed universal self-delimiting Turing machine: no extension of a valid program is valid. Prefix-free programs make the probability sum converge; the precise value depends on the chosen machine.
- [Chaitin’s constant, Wikipedia](https://en.wikipedia.org/wiki/Chaitin%27s_constant) — read as raw wikitext: “this sum converges to a real number between 0 and 1”; “Knowing the first N bits of Ω, one could calculate the halting problem for all programs of a size up to N.”; and, after “It is algorithmically random,” “As a consequence, it is a normal number, which means that its digits are equidistributed as if they were generated by tossing a fair coin.”

### 5.19 {#r-5.19}

BB(n) is the maximum halting time over n-state machines; BB(6) is fixed but currently unknown. As one fixed natural number, BB(6) is computable in the standard sense, though its value is unknown; only the function n ↦ BB(n) is uncomputable.

*Status: confirmed*

- [Antihydra, bbchallenge](https://bbchallenge.org/antihydra) — Antihydra is an unresolved six-state halting problem that might determine BB(6); the page does not claim permanent unknowability.
- [BB(6), BusyBeaverWiki](https://wiki.bbchallenge.org/wiki/BB(6)) — BB(6) is the unsolved sixth value of the Busy Beaver function; the best lower bound, as of 24 September 2026, is: the champion found by mxdys in June 2025 proves S(6) > Σ(6) > 10↑↑10↑↑10↑↑8 (the bound as restated in base 10 by 23 September 2026), which exceeds 2↑↑↑5 = 2↑↑2↑↑2↑↑4.
- [Computable number, Wikipedia](https://en.wikipedia.org/wiki/Computable_number) — read as raw wikitext: “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.” Every integer is algebraic, so every fixed integer is computable.

### 5.20 {#r-5.20}

Antihydra is unresolved rather than proved unknowable.

*Status: judgement*

- [Antihydra, bbchallenge](https://bbchallenge.org/antihydra) — Antihydra is an unresolved six-state halting problem that might determine BB(6); the page does not claim permanent unknowability.
- [Antihydra, BusyBeaverWiki](https://wiki.bbchallenge.org/wiki/Antihydra) — Antihydra is classified as a BB(6) Cryptid and carries the unsolved problem “Does Antihydra run forever?”; the page was last edited 9 May 2026 and records no resolution.

Editorial note: Antihydra halts or does not; nobody has proved which. The machine runs a Collatz-like process, every attempt at a proof either way has failed so far, and if it halts it may be the machine that fixes BB(6). That is an open problem, not a proved impossibility, and the chapter says nobody yet knows rather than nobody ever will.

### 5.21 {#r-5.21}

BB(5)=47,176,870 has been proved while BB(6) remains unresolved.

*Status: confirmed*

- [Antihydra, bbchallenge](https://bbchallenge.org/antihydra) — Antihydra is an unresolved six-state halting problem that might determine BB(6); the page does not claim permanent unknowability.
- [BB(6), BusyBeaverWiki](https://wiki.bbchallenge.org/wiki/BB(6)) — BB(6) is “the unsolved 6th value of the Busy Beaver function” (BB(5) = 47,176,870 is carried by the Antihydra page of this entry). Re-read on 24 September 2026 (these counts change often): mxdys’s holdout list “has 855 machines up to equivalence and 1802 machines not considering equivalence as of mid-September 2026,” with Antihydra one of the cryptids among them; the champion found by mxdys in June 2025 proves S(6) > Σ(6) > 10↑↑10↑↑10↑↑8 (the bound as restated in base 10 by 23 September 2026), which exceeds 2↑↑↑5 = 2↑↑2↑↑2↑↑4. BB(6) is not waiting on Antihydra alone.

### 5.22 {#r-5.22}

Ω lies between 0 and 1 and BB(6) is positive, so the denominator is positive and finite.

*Status: confirmed*

- [Chaitin’s Constant, Wolfram MathWorld](https://mathworld.wolfram.com/ChaitinsConstant.html) — defines the halting probability and states Chaitin constants are uncomputable and transcendental.
- [Antihydra, bbchallenge](https://bbchallenge.org/antihydra) — Antihydra is an unresolved six-state halting problem that might determine BB(6); the page does not claim permanent unknowability.
- **Own computation** (the author's own work) — The displayed factor is 0 divided by positive sums; its value is 0. The limit uses 0<x<1, so its denominator is positive and finite.

### 5.23 {#r-5.23}

The displayed chapter factor is zero because both numerator columns are zero over positive denominators.

*Status: confirmed*

- **Own computation** (the author's own work) — The displayed factor is 0 divided by positive sums; its value is 0. The limit uses 0<x<1, so its denominator is positive and finite.

### 5.25 {#r-5.25}

“Uncomputable” names the absence of a general algorithm, while “infeasibility” names impracticability rather than impossibility. A real number is computable when some algorithm approximates it to any requested accuracy; Radó’s Busy Beaver functions and the digits of a Chaitin constant are standard uncomputable examples.

*Status: confirmed*

- [Wiktionary, infeasibility](https://en.wiktionary.org/wiki/infeasibility) — defines the noun as “the state of being infeasible or impractical.”
- [Cambridge English Dictionary, “infeasible”](https://dictionary.cambridge.org/us/dictionary/english/infeasible) — “ fetched: defines it as not able to be achieved and lists “impracticable” and “unworkable” among its synonyms.
- **Author’s terminology choice** (the author's own work) — “ “Infeasibility” is retained because the chapter needs the established word for something possible in principle but not practically performable; “impossibility” would erase that distinction.
- [Computable number, Wikipedia](https://en.wikipedia.org/wiki/Computable_number) — read as raw wikitext: “A real number a is computable if it can be approximated by some computable function f: ℕ → ℤ in the following manner: given any positive integer n, the function produces an integer f(n) such that:,” and “There are many arithmetically definable, non-computable real numbers.”
- [Busy beaver, Wikipedia](https://en.wikipedia.org/wiki/Busy_beaver) — read as raw wikitext: “In his original 1962 paper, Radó defined two functions related to the busy beaver game: the score function Σ(n) and the shifts function S(n).” and “He proved that both of these functions were noncomputable, because they each grew faster than any computable function.”
- [Chaitin’s constant, Wikipedia](https://en.wikipedia.org/wiki/Chaitin%27s_constant) — read as raw wikitext: “Each halting probability is a normal and transcendental real number that is not computable, which means that there is no algorithm to compute its digits.”

### 5.24 {#r-5.24}

Schanuel’s conjecture is unproved, was published by Serge Lang in 1966, and would settle the transcendence of e + π, eπ, π to the e, π to the π and e to the e.

*Status: confirmed*

- [Schanuel’s conjecture, Wikipedia](https://en.wikipedia.org/wiki/Schanuel%27s_conjecture) — “Given any set of n complex numbers {z₁,...,zₙ} that are linearly independent over ℚ, the field extension ℚ(z₁,...,zₙ, e^z₁,...,e^zₙ) has transcendence degree at least n over ℚ”; the conjecture is Stephen Schanuel’s and “was published by Serge Lang in 1966”; it remains a major unsolved problem of transcendental number theory, and it would give the transcendence of e + π, eπ, e^e, π^e and π^π.

Editorial note: The conjecture is the reason the second brace has a ceiling rather than a floor: one proof would move every combination of π and e in this chapter at once. Nothing in this chapter assumes it.

### 5.26 {#r-5.26}

The origins of the second brace’s constants are as the cards state. Khinchin considered the geometric-mean limit in ~1934 (sources differ: MathWorld gives 1934, NUMDAM’s record of Compositio Mathematica 1 gives 1935; corrections welcome) and it holds for almost all reals; Kinkelin (1860) and Glaisher (1878) studied the hyperfactorial constant, which is the derivative of the Riemann zeta function at -1; Dickman (1930) studied the largest prime factor and Golomb (1964) the longest cycle of a random permutation; Landau proved the sum-of-two-squares count in 1908 and Ramanujan stated it independently; Khinchin showed the convergent-denominator limit exists in ~1935 (sources differ: Wikipedia’s Lévy’s constant says 1935 in its text but cites a paper of 1936; corrections welcome) and Paul Levy found the closed form in 1936; Herman P. Robinson computed the reciprocal-gamma integral to 36 places and A. Fransen to 65; Sergei Bernstein gave his limit in ~1913 and conjectured it was 1/(2 root pi), which Varga and Carpenter disproved in 1985; Divakar Viswanath computed the random-Fibonacci growth rate in 1999 using the Stern-Brocot tree; Ivan Niven found his constant in 1969; Emil Artin stated his conjecture to Helmut Hasse on 27 September 1927 and Christopher Hooley proved it in 1967 under the generalized Riemann hypothesis; Waclaw Sierpinski published his constant in 1907; the twin prime constant is the factor in Hardy and Littlewood’s conjecture of 1923; Marvin Ray Burns published the MRB constant in 1999; Michael Somos studied his nested radical in 1999; Kepler drew the inscribed polygons in the Mysterium Cosmographicum of 1596 and C. J. Bouwkamp computed the limit in 1965; and Nigel Backhouse conjectured his limit in 1995, with Philippe Flajolet proving it exists. In base 10 the prime 7 is such a case: 10 is a primitive root modulo 7, which gives the cyclic number 142857. Sources differ on the year of Bernstein’s paper: MathWorld dates Acta Mathematica 37 to 1913, while the publisher’s record and Varga and Carpenter’s abstract give 1914. Corrections welcome. Fransén’s first name, Arne, is given on the publisher’s page for his 1979 BIT paper. Kepler drew the inscribed polygons while seeking the geometry of the planetary orbits. The MRB constant has been computed to millions of digits. Sierpiński’s constant is the limit of the sum of r₂(k)/k for k up to n, minus π ln n.

*Status: confirmed*

- [Wolfram MathWorld, Khinchin’s Constant](https://mathworld.wolfram.com/KhinchinsConstant.html) — read as raw HTML: “Khinchin (1934) considered the limit of the geometric mean … Amazingly, except for a set of measure 0, this limit is a constant independent of x,” and its reference list gives Khintchine, A., “Metrische Kettenbruchprobleme,” Compositio Math. 1, 361-382, 1934. Note the page adds “as proved in Kac (1959),” so the 1934 paper is where the limit was considered and a later proof is credited too; the card’s “proven by Khinchin in 1934” follows the usual attribution rather than MathWorld’s finer one.
- [A. Khintchine, Metrische Kettenbruchprobleme, Compositio Mathematica 1, NUMDAM](https://www.numdam.org/item/CM_1935__1__361_0/) — the journal’s digitised record gives 1935, where MathWorld above gives 1934: “Compositio Mathematica, Tome 1 (1935), pp. 361-382.”
- [Wolfram MathWorld, Glaisher-Kinkelin Constant](https://mathworld.wolfram.com/Glaisher-KinkelinConstant.html) — read as raw HTML: the constant “is the derivative of the Riemann zeta function (Kinkelin 1860; Jeffrey 1862; Glaisher 1877, 1878, 1893, 1894; Voros 1987),” with the two papers given as Kinkelin, “Über eine mit der Gammafunktion verwandte Transcendente …,” J. reine angew. Math. 57, 122-158, 1860, and Glaisher, J. W. L., “On the Product 1¹·2²·3³…,” Messenger Math. 7, 43-47, 1878.
- [Wolfram MathWorld, Golomb-Dickman Constant](https://mathworld.wolfram.com/Golomb-DickmanConstant.html) — read as raw HTML: “Golomb (1964) computed an approximation … to the constant … known as the Golomb constant or Golomb-Dickman constant” and “Dickman (1930) investigated the probability that the greatest prime factor of a random integer between 1 and n satisfies …,” with the papers given as Golomb, S. W., “Random Permutations,” Bull. Amer. Math. Soc. 70, 747, 1964, and Dickman, K., “On the Frequency of Numbers Containing Prime Factors of a Certain Relative Magnitude,” Arkiv för Mat., Astron. och Fys. 22A, 1-14, 1930.
- [Wolfram MathWorld, Landau-Ramanujan Constant](https://mathworld.wolfram.com/Landau-RamanujanConstant.html) — read as raw HTML: the count of integers below x that are sums of two squares behaves as stated “as proved by Landau (1908),” and “Ramanujan independently stated the theorem in the slightly different form.”
- [Lévy’s constant, Wikipedia](https://en.wikipedia.org/wiki/L%C3%A9vy%27s_constant) — read as raw HTML: “In 1935, the Soviet mathematician Aleksandr Khinchin showed that the denominators qₙ of the convergents of the continued fraction expansions of almost all real numbers satisfy lim qₙ^(1/n) = e^β” and “Soon afterward, in 1936, the French mathematician Paul Lévy found the explicit expression for the constant, namely e^β = e^(π²/(12 ln 2)) = 3.275822918721811159787681882….” Its footnote for Khinchin’s result, read as raw wikitext, gives 1936: “Zur metrischen Kettenbruchtheorie,” Compositio Mathematica 3, No. 2, 275–285 (1936).
- [Fransén-Robinson constant, Wikipedia](https://en.wikipedia.org/wiki/Frans%C3%A9n%E2%80%93Robinson_constant) — read as raw HTML: “The value was computed to 36 decimal places by Herman P. Robinson using 11 point Newton-Cotes quadrature, to 65 digits by A. Fransén using Euler-Maclaurin summation, and to 80 digits by Fransén and S. Wrigge using Taylor series and other methods.” Neither this page nor MathWorld’s gives a year for Robinson’s computation; MathWorld’s earliest Fransén citation is “Accurate Determination of the Inverse Gamma Integral,” BIT 19, 137-138, 1979. The page also reads “The constant is somewhat close to Euler’s number e = 2.71828...,” approximating the integral by the sum of 1/Γ(n), “and this sum is the standard series for e”; and “It is however unknown whether F can be expressed in closed form in terms of other known constants.”
- [Wolfram MathWorld, Bernstein’s Constant](https://mathworld.wolfram.com/BernsteinsConstant.html) — read as raw HTML: “Bernstein showed that … He conjectured that the lower limit was 1/(2√π). However, this was disproven by Varga and Carpenter (1987) and Varga (1990),” with Bernstein’s own paper given as “Sur la meilleure approximation de |x| par les polynomes de degrés donnés,” Acta Math. 37, 1-57, 1913.
- [R. S. Varga and A. J. Carpenter, On the Bernstein conjecture in approximation theory, Constructive Approximation 1 (1985), 333-348](https://doi.org/10.1007/BF01890040) — the publisher’s page gives “Published: December 1985” and “Volume 1, pages 333–348 (1985),” by Richard S. Varga and Amos J. Carpenter. Its abstract also reads: “by means of numerical calculations, Bernstein determined, in the same paper, the following upper and lower bounds for β: 0.278<β<0.286.”
- [R. S. Varga and A. J. Carpenter, On a conjecture of S. Bernstein in approximation theory, Mathematics of the USSR-Sbornik 57 (1987), 547–560, Math-Net.Ru](https://doi.org/10.1070/sm1987v057n02abeh003086) — the same paper and abstract in the translation journal: “Mathematics of the USSR-Sbornik, 1987, Volume 57, Issue 2, Pages 547–560,” “Received: 27.03.1985” and “Original paper language: Russian.” This is the 1987 MathWorld cites, so the book’s 1985 is the first publication.
- [Random Fibonacci sequence, Wikipedia](https://en.wikipedia.org/wiki/Random_Fibonacci_sequence) — read as raw HTML: “In 1999, Divakar Viswanath showed that the growth rate of the random Fibonacci sequence is equal to 1.1319882487943…, a mathematical constant that was later named Viswanath’s constant,” and the method “uses Furstenberg’s formula for the Lyapunov exponent of a random matrix product and integration over a certain fractal measure on the Stern-Brocot tree.”
- [Wolfram MathWorld, Niven’s Constant](https://mathworld.wolfram.com/NivensConstant.html) — read as raw HTML: “Niven (1969) also proved that … where Niven’s constant is given by …,” the paper being Niven, Ivan, “Averages of exponents in factoring integers,” Proc. Amer. Math. Soc., 1969.
- [Artin’s conjecture on primitive roots, Wikipedia](https://en.wikipedia.org/wiki/Artin%27s_conjecture_on_primitive_roots) — read as raw HTML: “The conjecture was made by Emil Artin to Helmut Hasse on September 27, 1927, according to the latter’s diary”; “In 1967, Christopher Hooley published a conditional proof for the conjecture, assuming certain cases of the generalized Riemann hypothesis”; and “Without the generalized Riemann hypothesis, there is no single value of a for which Artin’s conjecture is proved.”
- [OEIS A062089, decimal expansion of Sierpiński’s constant](https://oeis.org/A062089) — read as raw HTML: its link list gives the original paper as Wacław Sierpiński, “O sumowaniu szeregu Σ τ(n) f(n), gdzie τ(n) oznacza liczbę rozkładów liczby n na sumę kwadratów dwóch liczb całkowitych,” Prace Matematyczno-Fizyczne, Vol. 18, No. 1 (1907), pp. 1-59.
- [Wolfram MathWorld, Twin Primes Constant](https://mathworld.wolfram.com/TwinPrimesConstant.html) — read as raw HTML: its reference list gives Hardy, G. H. and Littlewood, J. E., “Some Problems of ‘Partitio Numerorum.’ III. On the Expression of a Number as a Sum of Primes,” Acta Math. 44, 1-70, 1923, which is the conjecture the constant belongs to. Wikipedia’s List of mathematical constants gives the constant the year 1922 instead; sources differ, corrections welcome.
- [MRB constant, Wikipedia](https://en.wikipedia.org/wiki/MRB_constant) — read as raw HTML: “The constant is named after its discoverer, Marvin Ray Burns, who published his discovery of the constant in 1999,” and “Burns had initially called the constant ‘rc’ for root constant but, at Simon Plouffe’s suggestion, the constant was renamed.”
- [Wolfram MathWorld, Somos’s Quadratic Recurrence Constant](https://mathworld.wolfram.com/SomossQuadraticRecurrenceConstant.html) — read as raw HTML: the constant “has closed-form solution … where Li is a polylogarithm, Φ is a Lerch transcendent,” and its reference list gives Somos, M., “Several Constants Related to Quadratic Recurrences,” unpublished note, 1999.
- [OEIS A085365, decimal expansion of the Kepler-Bouwkamp constant](https://oeis.org/A085365) — read as raw HTML: “Named after the German astronomer and mathematician Johannes Kepler (1571-1630) and the Dutch mathematician Christoffel Jacob Bouwkamp (1915-2003),” with the two primary items in its link list given as Johannes Kepler, Mysterium Cosmographicum, Tübingen, 1596, see p. 39, and C. J. Bouwkamp, “An infinite product,” Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Series A, Vol. 68 (1965), pp. 40-46.
- [Backhouse’s constant, Wikipedia](https://en.wikipedia.org/wiki/Backhouse%27s_constant) — read as raw HTML: the limit of the ratios of successive coefficients of 1/P(x) “was conjectured to exist by Backhouse, and later proven by Philippe Flajolet,” the conjecture being cited to Backhouse 1995.
- [Cyclic number, Wikipedia](https://en.wikipedia.org/wiki/Cyclic_number) — read as raw wikitext: “For example, the case b = 10, p = 7 gives the cyclic number 142857” and “this sequence is the set of primes p such that b is a primitive root modulo p. A conjecture of Emil Artin is that this sequence contains 37.395..% of the primes.”
- [S. Bernstein, Sur la meilleure approximation de |x| par des polynomes de degrés donnés, Acta Mathematica 37, SpringerLink](https://link.springer.com/article/10.1007/BF02401828) — read raw: “Published: December 1914” and “Volume 37, pages 1–57 (1914).” The Varga–Carpenter abstract (doi 10.1007/BF01890040, already cited) reads “the famous Russian mathematician S. Bernstein in 1914 established the existence of a positive constant β.”
- [Accurate determination of the inverse gamma integral, BIT 19 (1979), 137-138, SpringerLink](https://link.springer.com/article/10.1007/BF01931232) — read raw: the author line gives “Arne Fransén,” of the “National Defence Research Institute, Division 1, Section 144, Stockholm 80, Sweden.”
- [Mysterium Cosmographicum, Wikipedia](https://en.wikipedia.org/wiki/Mysterium_Cosmographicum) — read as raw wikitext: “he realized that regular polygons bound one inscribed and one circumscribed circle at definite ratios, which, he reasoned, might be the geometrical basis of the universe. After failing to find a unique arrangement of polygons that fit known astronomical observations (even with extra planets added to the system), Kepler began experimenting with 3-dimensional polyhedra.”
- [Try to beat these MRB constant records!, Wolfram Community](https://community.wolfram.com/groups/-/m/t/366628) — read raw, Marvin Ray Burns’s record thread: “Over 4 million digits of CMRB were finished on Wed 16 Jan 2019, 19:55:20,” and a later run ends “Enter MRB1 to print 6532491 digits.”
- [Sierpiński’s constant, Wikipedia](https://en.wikipedia.org/wiki/Sierpi%C5%84ski%27s_constant) — read as raw wikitext: “One way of defining it is as the following limit: K = lim n→∞ [Σ k=1..n r₂(k)/k − π ln n] where r₂(k) is a number of representations of k as a sum of the form a² + b² for integer a and b.” Its closed form uses Γ(1/4), π and γ.

Editorial note: Added 17 September 2026 for the B6 pass. Sixteen cards in the second brace cited r-5.9 alone, and r-5.9 states exactly one proposition, that none of these constants has been proved irrational. Everything else those cards say - roughly twenty-five dates and discoverers - was carried by nothing. This entry carries it; r-5.9 keeps its one clean proposition. Three things the sourcing settled. (1) Bernstein: the book says Varga and Carpenter disproved the 1/(2 root pi) conjecture in 1985 and MathWorld says 1987. Crossref shows both, Constructive Approximation 1 (1985) and Mathematics of the USSR-Sbornik 57 (1987), which is the Russian version of the same result, so 1985 is the first publication and the book is right; do not change it to 1987. (2) Fransén-Robinson: no source found gives a year for the computation at all, and the earliest Fransén paper any of them cites is 1979, so the card’s “in 1978” was removed rather than sourced. (3) Khinchin: MathWorld says Khinchin considered the limit in 1934 and credits a proof to Kac (1959); the card keeps the usual attribution, and the difference is recorded in the source note above rather than hidden.

### 5.27 {#r-5.27}

The chapter epigraph is Hilbert’s, the closing words of his retirement address to the Society of German Scientists and Physicians at Koenigsberg on 8 September 1930; shortly afterward he read a four-minute version of its finale on German radio, the recording known as his radio address.

*Status: confirmed*

- [David Hilbert, Wikipedia](https://en.wikipedia.org/wiki/David_Hilbert) — read as raw HTML: “The epitaph on his tombstone in Göttingen consists of the famous lines he spoke at the conclusion of his retirement address to the Society of German Scientists and Physicians on 8 September 1930 … Wir müssen wissen. Wir werden wissen. Translation: We must know. We shall know.” On Gödel: “The day before Hilbert pronounced these phrases at the 1930 annual meeting of the Society of German Scientists and Physicians, Kurt Gödel — in a round table discussion during the Conference on Epistemology held jointly with the Society meeting — [announced the incompleteness theorem].”
- [James T. Smith, David Hilbert’s Radio Address (San Francisco State University; also in MAA Convergence)](https://www.smith-at-sfsu.net/Documents/HilbertRadio/HilbertRadio.pdf) — read as extracted PDF text: on 8 September 1930 Hilbert addressed the Society of German Natural Scientists and Physicians in Königsberg; “Shortly afterward, he read on German radio a four-minute version of the finale of his speech”; the German transcription ends “Wir müssen wissen, Wir werden wissen” and the English translation “We must know, We will know.”

Editorial note: Added 17 September 2026 to source the chapter epigraph, which carried no marker. Two small notes. The source translates the second sentence “We shall know” where the page has “We will know”; both are current and the page’s is kept. And the footer’s “announced at the same meeting” is right in substance — the Conference on Epistemology was held jointly with the Society meeting — but Gödel spoke the day before Hilbert, which the footer does not say. On 29 September 2026 Ian kept “radio” in the footer (IAN-LIST 6), and Smith’s edition of the recording was added as the source that carries it.

### 5.28 {#r-5.28}

A number is normal in a base when every finite block of digits occurs in its expansion with the frequency chance would give it, and absolutely normal when that holds in every base. Émile Borel introduced the notion in 1909 and proved with the Borel–Cantelli lemma that almost all reals are normal; Wacław Sierpiński showed in 1917 that a particular absolutely normal number can be specified, and Becher and Figueira proved in 2002 that a computable one exists. Normality in base 10 implies irrationality, and the Copeland–Erdős constant is normal in base 10.

*Status: confirmed*

- [Normal number, Wikipedia](https://en.wikipedia.org/wiki/Normal_number) — read as raw HTML: “The concept of a normal number was introduced by Émile Borel (1909). Using the Borel–Cantelli lemma, he proved that almost all real numbers are normal… Wacław Sierpiński (1917) showed that it is possible to specify a particular such number. Becher and Figueira (2002) proved that there is a computable absolutely normal number.”
- [Normal Number, MathWorld](https://mathworld.wolfram.com/NormalNumber.html) — the digit-frequency definition, and the standing state of ignorance about π, e and √2.

Editorial note: The chapter needs only the one-way implication: a number normal in base 10 cannot be rational, because a rational expansion is eventually periodic and a periodic expansion cannot give every block its fair share. The converse fails, and nothing here claims it.

### 5.29 {#r-5.29}

The Collatz conjecture takes a positive integer, halves it when even and replaces it by 3n + 1 when odd, and asserts that the sequence always reaches 1. It is named after Lothar Collatz, who introduced the idea in 1937, two years after his doctorate; the sequences are also called hailstone sequences. It has been verified for all positive integers up to 2.36 × 10²¹ and no general proof is known. Sources differ on the date: Wikipedia gives 1937, while MathWorld calls the exact date of origin unclear and says Collatz traced it to investigations begun in his student years, 1928 to 1933. Corrections welcome. Collatz-like Cryptid machines such as Antihydra block the value of BB(6).

*Status: confirmed*

- [Collatz conjecture, Wikipedia](https://en.wikipedia.org/wiki/Collatz_conjecture) — read as raw HTML: “The conjecture has been shown to hold for all positive integers up to 2.36 × 10²¹, but no general proof has been found. It is named after the mathematician Lothar Collatz, who introduced the idea in 1937, two years after receiving his doctorate,” and gives “hailstone sequence” as the other name.
- [Collatz Conjecture, MathWorld](https://mathworld.wolfram.com/CollatzConjecture.html) — the same rule and the conjecture’s open status, and on the date: “Its exact date of origin is unclear, although Collatz traced its development to investigations of iterations of arithmetic functions begun during his student years from 1928 to 1933.”
- [OEIS A006577, number of halving and tripling steps to reach 1 in the 3x + 1 problem](https://oeis.org/A006577) — the step counts themselves, for the record that the orbit lengths are ordinary integers even though the conjecture is open.
- [BB(6), BusyBeaverWiki](https://wiki.bbchallenge.org/wiki/BB(6)) — read raw: “With the discovery of the Cryptid machine Antihydra in June 2024, we now know that we must solve a Collatz-like problem in order to solve BB(6) and thus BB(6) is Hard.”

### 5.30 {#r-5.30}

Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Euler found zeta(2) = pi^2/6 in 1734; Euler’s solution of the Basel problem is dated 1734 by the usual account and was read to the St Petersburg Academy on 5 December 1735; MacTutor and the Euler Archive date the result itself to 1735; Viggo Brun proved in 1919 that the sum of reciprocals of twin primes converges; Nicely’s computation of Brun’s constant in 1994 exposed the Pentium division bug; Eugene Catalan studied the constant in 1865; Feigenbaum found delta numerically in 1975; Lanford proved universality in 1982; The constant gamma was introduced by Euler in 1734; Mascheroni’s error was caught by von Soldner in 1809.

*Status: confirmed*

- [Basel problem, Wikipedia](https://en.wikipedia.org/wiki/Basel_problem) — read raw: “It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 in The Saint Petersburg Academy of Sciences..”
- [Brun’s theorem, Wikipedia](https://en.wikipedia.org/wiki/Brun%27s_theorem) — read raw: “Brun’s theorem was proved by Viggo Brun in 1919.”
- [Pentium FDIV bug, Wikipedia](https://en.wikipedia.org/wiki/Pentium_FDIV_bug) — read raw: “The bug was discovered in 1994 by Thomas R. Nicely, a professor of mathematics at Lynchburg College..”
- [Catalan’s constant, Wikipedia](https://en.wikipedia.org/wiki/Catalan%27s_constant) — read raw: “Catalan’s constant was named after Eugène Charles Catalan, who found quickly-converging series for its calculation and published a memoir on it in 1865.”; “In low-dimensional topology, Catalan’s constant is 1/4 of the volume of an ideal hyperbolic octahedron”; “In combinatorics and statistical mechanics, it arises in connection with counting domino tilings, spanning trees, and Hamiltonian cycles of grid graphs.”
- [Feigenbaum constants, Wikipedia](https://en.wikipedia.org/wiki/Feigenbaum_constants) — read raw: “Feigenbaum made this discovery in 1975, and he officially published it in 1978.”; “The first proof of the universality of the Feigenbaum constants was carried out by Oscar Lanford—with computer-assistance—in 1982.”
- [Euler’s constant, Wikipedia](https://en.wikipedia.org/wiki/Euler%27s_constant) — read raw: “The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled De Progressionibus harmonicis observationes (Observations on harmonic progressions; Eneström Index 43).”
- [J. C. Lagarias, Euler’s constant: Euler’s work and modern developments (arXiv:1303.1856), ar5iv](https://ar5iv.labs.arxiv.org/html/1303.1856) — read raw: “However in 1809 Johann von Soldner [275], while preparing the first book on the logarithmic integral function”; on Mascheroni: “In 1790, in a book proposing to answer some problems raised by Euler, Mascheroni [213] introduced the notation A for this constant”; “he gave a numerical expression for it to 32 decimal places”; “Von Soldner’s answer disagreed with Mascheroni’s in the 20-th decimal place.”

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.

### 5.31 {#r-5.31}

Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Benjamin Gompertz proposed his law of human mortality in 1825; Cauchy gave limits systematic definitions in 1821; William Mills’ 1947 theorem that such a number exists; The limit was established by Meissel in 1866; The limit was established by Mertens in 1874 (MathWorld, after Lindqvist and Peetre; Wikipedia’s List of mathematical constants gives 1866 & 1873, so sources differ; corrections welcome); Schanuel’s conjecture dates from the 1960s; Rivoal proved in 2000 that infinitely many odd zeta values are irrational; Zudilin proved in 2001 that at least one of zeta(5), zeta(7), zeta(9), zeta(11) is irrational. Newton used limiting arguments in the seventeenth century: in the Scholium to the Principia in 1687 he had a clear definition of a limit.

*Status: confirmed*

- [J. C. Lagarias, Euler’s constant: Euler’s work and modern developments (arXiv:1303.1856), ar5iv](https://ar5iv.labs.arxiv.org/html/1303.1856) — read raw: “Benjamin Gompertz (1779–1865) was a mathematician and actuary who in 1825 proposed a functional model for approximating mortality tables.”
- [Limit (mathematics), Wikipedia](https://en.wikipedia.org/wiki/Limit_(mathematics)) — read raw: “Augustin-Louis Cauchy in 1821, followed by Karl Weierstrass, formalized the definition of the limit of a function which became known as the (ε, δ)-definition of limit..”
- [Mills’ constant, Wikipedia](https://en.wikipedia.org/wiki/Mills%27_constant) — read raw: “This constant is named after William Harold Mills who in 1947 proved the existence of A based on results of Guido Hoheisel and Albert Ingham on prime gaps..”
- [Mertens Constant, Wolfram MathWorld](https://mathworld.wolfram.com/MertensConstant.html) — read raw: “According to Lindqvist and Peetre (1997), this was shown independently by Meissel in 1866 and Mertens (1874)..”
- [Schanuel’s conjecture, Wikipedia](https://en.wikipedia.org/wiki/Schanuel%27s_conjecture) — read raw: “It is due to Stephen Schanuel and was published by Serge Lang in 1966..”
- [T. Rivoal, La fonction Zeta de Riemann prend une infinite de valeurs irrationnelles aux entiers impairs, arXiv](https://arxiv.org/abs/math/0008051) — read raw: “[Submitted on 7 Aug 2000] Title: La fonction Zeta de Riemann prend une infinite de valeurs irrationnelles aux entiers impairs Authors: Tanguy Rivoal.”
- [W. V. Zudilin, One of the numbers zeta(5), zeta(7), zeta(9), zeta(11) is irrational, Math-Net.Ru](https://www.mathnet.ru/eng/rm427) — read raw: “Russian Mathematical Surveys, 2001, Volume 56, Issue 4, Pages 774–776.”

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.

### 5.40 {#r-5.40}

Defines Mersenne numbers 2^n - 1, states the primality precondition on n, and notes the Erdős–Borwein constant as the sum of their reciprocals. They are named for Marin Mersenne; Paul Erdős proved the Erdős–Borwein constant irrational in 1948; the record prime 2^136279841 − 1 is a Mersenne prime, and the Lucas–Lehmer test makes this family easy to test.

*Status: confirmed*

- [Mersenne prime, Wikipedia](https://en.wikipedia.org/wiki/Mersenne_prime) — defines Mersenne numbers 2^n-1, states that n must be prime for 2^n-1 to be prime, and gives the Lucas–Lehmer primality test used to search them.
- [Erdős–Borwein constant, Wikipedia](https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Borwein_constant) — carries the definition and Erdős’s 1948 irrationality proof: “the sum of the reciprocals of the Mersenne numbers”; “In 1948, Erdős showed that the constant E is an irrational number.”
- [Largest known prime number, Wikipedia](https://en.wikipedia.org/wiki/Largest_known_prime_number) — carries the Mersenne dominance of record primes: “As of October 2024, the seven largest known primes are Mersenne primes. The last 18 record primes were Mersenne primes.”