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# Glossary cards — Chapter ᚃ(F): The local park

*Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#group-T5, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.*

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

*Unresolved constants, and uncomputable and infeasible numbers: Legendre’s constant, the not-even-known-irrational, TREE(3) against Rayo’s number, Chaitin’s Ω, and BB(6). Each card: its symbol, type, one-line headline, the longer note, the reference entries it cites and the cards it links to.*

### the letter Fearn — `ᚃ` {#g-og-fern}

*Ogham letter*

Ogham letter name meaning ‘alder’; the Auraicept makes it the shield-wood of the warrior bands.

Fern means ‘alder-tree’, from Primitive Irish *wernā, which is why the letter’s original sound value was [w] rather than [f]. The three Old Irish word-oghams, the Bríatharogaim ascribed to Morann mac Moín, to Mac ind Óc, and to Cú Chulainn, give airenach fían, “vanguard of warriors”; comét lachta, “milk container”; and dín cridi, “protection of the heart.” The Auraicept explains the first of the three: “the van of the warrior-bands, that is, alder, for thereof are the shields.” It is one of the five names the medieval tradition did not need to translate. Modern tree lore and ogham divination descend from the medieval plant glosses by way of Robert Graves’s The White Goddess of 1948, which read the letter order as an ancient seasonal calendar of tree magic. Celtic scholarship rejects that reconstruction; the associations in circulation today are modern, not ancient.

Cites: [2.15](ch02-refs.md#r-2.15), [2.16](ch02-refs.md#r-2.16), [2.17](ch02-refs.md#r-2.17), [2.19](ch02-refs.md#r-2.19)

### Copeland–Erdős constant — `Copeland–Erdős` {#g-cope}

*Status open*

0.235711131719…, the primes concatenated; normal in base 10, hence irrational, transcendence unknown.

The Copeland–Erdős constant 0.235711131719… is the primes written one after another as decimals. Arthur Copeland and Paul Erdős proved in 1946 that it is normal in base 10, and a normal number is automatically irrational, since a rational’s digits repeat. Whether it is transcendental is open, which is why it is tinted as status open and why it stands among the unresolved constants in Chapter ᚃ(F); it also appears in Chapter ᚂ(L) as the hand-built cousin under Champernowne’s constant.

Cites: [5.4](ch05-refs.md#r-5.4)

See also: [Champernowne’s constant](ch04-cards.md#g-champ), [a normal number](ch05-cards.md#g-normalnum)

### Legendre’s constant — `Legendre’s` {#g-legendre}

*Integer or rational*

Conjectured to be about 1.08366 in 1808 and proven to be exactly 1 in 1899.

In 1808 Legendre proposed that the number of primes up to x is about x/(ln x − B) with B ≈ 1.08366, and B was called Legendre’s constant. In 1899 de la Vallée Poussin proved that the best value of B is exactly 1. That is why Legendre’s constant minus 1 is the zero of the unresolved-constants column.

Cites: [5.1](ch05-refs.md#r-5.1)

See also: [the prime-counting function](ch18-cards.md#g-primecount)

### Euler–Mascheroni constant — `γ` {#g-gamma}

*Status open*

About 0.5772, and not even known to be irrational, one of the embarrassments of number theory.

The Euler–Mascheroni constant γ ≈ 0.5772 is the limit of 1 + ½ + … + 1/n − ln n, introduced by Euler in 1734. Lorenzo Mascheroni computed it to 32 decimal places in 1790, of which only the first 19 were right; the error was caught by von Soldner in 1809. It appears throughout analysis and number theory, and it is not known whether it is irrational, let alone transcendental; Aptekarev observed in 2009 that at least one of γ and the Gompertz constant is irrational, and Rivoal in 2012 that at least one is transcendental. The flagship of the not-even-known-irrational brace.

Cites: [5.30](ch05-refs.md#r-5.30), [5.6](ch05-refs.md#r-5.6)

See also: [Gompertz constant](ch05-cards.md#g-gompertz)

### Feigenbaum’s δ — `δ` {#g-feig}

*Status open*

About 4.6692, the universal period-doubling ratio of chaos; irrationality unproven.

Feigenbaum’s constant δ ≈ 4.6692 is the universal ratio at which period-doubling bifurcations approach chaos, found numerically by Mitchell Feigenbaum in 1975 and proven universal by Lanford in 1982. It is the same for every one-dimensional map with a single quadratic maximum, which is why it counts as a constant of nature within mathematics. Whether it is irrational is unknown.

Cites: [5.30](ch05-refs.md#r-5.30), [5.7](ch05-refs.md#r-5.7)

### Feigenbaum’s α — `α_{F}` {#g-feigA}

*Status open*

Its magnitude is about 2.5029; the signed convention is approximately −2.5029. Its irrationality remains open.

Feigenbaum’s second constant describes the spatial scaling of successive period-doubling forks. A common signed convention gives α_{F} ≈ −2.502907875; many expositions quote its magnitude. This book displays |α_{F}| when a positive denominator value is intended. The universality statement concerns one-dimensional maps with a single quadratic maximum, the same class named on the δ card; Lanford’s 1982 computer-assisted proof established it for that class. Its irrationality is unknown.

Cites: [5.30](ch05-refs.md#r-5.30), [5.7](ch05-refs.md#r-5.7)

See also: [Feigenbaum’s δ](ch05-cards.md#g-feig)

### Catalan’s constant — `G` {#g-catalan}

*Status open*

About 0.9160; whether it is even irrational is unknown.

Catalan’s constant G = 1 − 1/9 + 1/25 − 1/49 + … ≈ 0.9160, named after Eugène Catalan, who studied it in 1865. It appears in combinatorics and in the volumes of hyperbolic solids, and it is not known whether it is irrational. This is the G of Chapter ᚃ(F); Gauss’s constant, also written G and proven transcendental, is in Chapter ᚂ(L).

Cites: [5.30](ch05-refs.md#r-5.30), [5.8](ch05-refs.md#r-5.8)

See also: [Gauss’s constant](ch04-cards.md#g-gaussc)

### Khinchin’s constant — `K` {#g-khin}

*Status open*

About 2.6854, the geometric mean of almost every continued fraction’s terms; status open.

Khinchin’s constant K ≈ 2.6854 is the geometric mean of the partial quotients of the continued fraction of almost every real number, proven by Aleksandr Khinchin in ~1934 to exist and to be the same for almost all numbers. It is not known to be irrational; it is not even known whether K’s own continued fraction has this property.

Cites: [5.26](ch05-refs.md#r-5.26), [5.9](ch05-refs.md#r-5.9)

See also: [continued fractions and convergents](ch04-cards.md#g-contfrac)

### Glaisher–Kinkelin constant — `A` {#g-glaisher}

*Status open*

About 1.2824; status open.

The Glaisher–Kinkelin constant A ≈ 1.2824 appears in the asymptotics of the hyperfactorial 1¹ · 2² · 3³ · … · nⁿ, studied by Hermann Kinkelin in 1860 and James Glaisher in 1878, and it is connected to the derivative of the zeta function at −1. Its irrationality is open.

Cites: [5.26](ch05-refs.md#r-5.26), [5.9](ch05-refs.md#r-5.9)

See also: [Riemann zeta function](ch18-cards.md#g-zeta)

### Brun’s constant — `B_{2}` {#g-brun}

*Status open*

The sum of reciprocals of twin primes, estimated at about 1.9022; rational if there are finitely many twin primes.

Brun’s constant B₂ ≈ 1.9022 is the sum of the reciprocals of the twin primes, (1/3 + 1/5) + (1/5 + 1/7) + (1/11 + 1/13) + …. Viggo Brun proved in 1919 that the sum converges, so twin primes are sparse even if infinite. If there are finitely many twin primes it is a rational number, so its status is tied to the twin prime conjecture; its computation by Nicely in 1994 famously exposed the Pentium division bug. Its familiar digits are an extrapolation from the twin primes below 10^{16}; what is proved is only that it lies between 1.84 and 2.29. This B₂ is Brun’s constant, not the Bernoulli number B₂ = ⅙ of Chapter ᚂ(L).

Cites: [5.10](ch05-refs.md#r-5.10), [5.30](ch05-refs.md#r-5.30)

See also: [the Bernoulli number B₂](ch04-cards.md#g-bern2)

### Erdős–Borwein constant — `E_{EB}` {#g-eb}

*Status open*

About 1.6067, proven irrational in 1948; transcendence open.

The Erdős–Borwein constant E_{EB} = Σ 1/(2ⁿ − 1) ≈ 1.6067, the sum of the reciprocals of the Mersenne numbers. Paul Erdős proved it irrational in 1948, and Peter Borwein gave a second proof in 1992; whether it is transcendental is open, so it stands in the irrational-but-unclassified brace.

Cites: [5.2](ch05-refs.md#r-5.2)

See also: [Mersenne numbers](ch05-cards.md#g-mersnum)

### Gompertz constant — `Gompertz` {#g-gompertz}

*Status open*

About 0.5963; at least one of it and γ is transcendental (Rivoal, 2012), and nobody knows which.

The Gompertz constant ≈ 0.5963 is the integral of e^(−x)/(1 + x) from 0 to ∞, named by Le Lionnais for its role in survival analysis, after Benjamin Gompertz, whose 1825 law of human mortality gives the Gompertz curve, and connected to Euler’s divergent series 1 − 1 + 2 − 6 + 24 − …. Aptekarev (2009) observed that at least one of γ and this constant is irrational, and Rivoal (2012) that at least one is transcendental; nobody knows which.

Cites: [5.31](ch05-refs.md#r-5.31), [5.6](ch05-refs.md#r-5.6)

See also: [Euler–Mascheroni constant](ch05-cards.md#g-gamma)

### Mills’ constant — `Mills’` {#g-mills}

*Status open*

About 1.3064, the smallest θ with ⌊θ^(3ⁿ)⌋ prime for every n; proved irrational by Kota Saito in 2024, transcendence open; its digits are known only under the Riemann Hypothesis.

Mills’ constant θ ≈ 1.3064 is the smallest real number such that the integer part of θ^(3ⁿ) is prime for every n, from William Mills’ 1947 theorem that such a number exists. Its digits are known only under the Riemann Hypothesis, which is needed to guarantee a prime between consecutive cubes; the number itself is defined without it, and Kota Saito proved it irrational in 2024 with no such assumption (Mathematika 71, 2025). Whether it is transcendental is open: Saito showed that either θ is transcendental or some power θ^(3^m) is a cubic Pisot number, and that transcendence would follow from Cramér’s conjecture or from the Density Hypothesis. Status open.

Cites: [5.11](ch05-refs.md#r-5.11), [5.31](ch05-refs.md#r-5.31)

See also: [the Riemann Hypothesis](ch18-cards.md#g-rh)

### the unclassified combinations of π and e — `π + e, π·e, π^{e}` {#g-pie}

*Status open*

π + e ≈ 5.86, π·e ≈ 8.54, and π^e ≈ 22.46; the irrationality of each is open, though at least one of π + e and π·e must be transcendental.

π + e ≈ 5.8599, π·e ≈ 8.5397, and π^e ≈ 22.459 are the simplest combinations of the two most famous transcendentals whose status is unknown: none is known to be irrational. Because π and e are both roots of x² − (π + e)x + πe, at least one of π + e and πe must be transcendental, but which one is open. Contrast e^π, which Gelfond–Schneider settles and which lives in Chapter ᚂ(L).

Cites: [5.12](ch05-refs.md#r-5.12)

See also: [Gelfond’s constant](ch04-cards.md#g-epi), [the Gelfond–Schneider theorem](ch04-cards.md#g-gelfschn)

### twin prime constant — `C_{2}` {#g-twinprime}

*Status open*

About 0.6602, the constant in the Hardy–Littlewood twin prime conjecture; status open.

The twin prime constant C₂ = Π (1 − 1/(p − 1)²) over odd primes ≈ 0.6602, the constant in Hardy and Littlewood’s 1923 conjecture for how many twin primes lie below x. Its irrationality is open.

Cites: [5.26](ch05-refs.md#r-5.26), [5.9](ch05-refs.md#r-5.9)

### Meissel–Mertens constant — `Meissel–Mertens` {#g-mm}

*Status open*

About 0.2615, the γ of the primes; status open.

The Meissel–Mertens constant M ≈ 0.2615 is the limit of the sum of 1/p over primes up to n minus ln ln n, the primes’ own version of γ, established by Meissel in 1866 and Mertens in ~1874. Not known to be irrational.

Cites: [5.31](ch05-refs.md#r-5.31), [5.6](ch05-refs.md#r-5.6)

See also: [Euler–Mascheroni constant](ch05-cards.md#g-gamma)

### reciprocal Fibonacci constant — `ψ` {#g-recipfib}

*Status open*

About 3.3599, proven irrational in 1989; transcendence open.

The reciprocal Fibonacci constant ψ = 1/1 + 1/1 + 1/2 + 1/3 + 1/5 + 1/8 + … ≈ 3.3599. Richard André-Jeannin proved it irrational in 1989; whether it is transcendental is open, so it stands beside the Erdős–Borwein constant in the irrational-but-unclassified brace.

Cites: [5.3](ch05-refs.md#r-5.3)

See also: [Erdős–Borwein constant](ch05-cards.md#g-eb)

### Golomb–Dickman constant — `Golomb–Dickman` {#g-gd}

*Status open*

About 0.6243, the expected share of a random permutation taken up by its longest cycle, as the permutation grows; status open.

The Golomb–Dickman constant λ ≈ 0.6243 is the expected fraction of n taken up by the longest cycle of a random permutation of n objects, as n grows, and equally the expected size of the largest prime factor of a random integer on a logarithmic scale; found by Golomb in 1964 and, in the number-theoretic form, by Dickman in 1930. Status open.

Cites: [5.26](ch05-refs.md#r-5.26), [5.9](ch05-refs.md#r-5.9)

### Landau–Ramanujan constant — `Landau–Ramanujan` {#g-lr}

*Status open*

About 0.7642, the constant in the count of numbers that are sums of two squares; status open.

The Landau–Ramanujan constant K ≈ 0.7642 is the constant in the count of integers up to x that are sums of two squares, about K·x/√(ln x), proven by Landau in 1908 and independently found by Ramanujan. Its irrationality is open.

Cites: [5.26](ch05-refs.md#r-5.26), [5.9](ch05-refs.md#r-5.9)

### Apéry’s constant — `ζ(3)` {#g-apery}

*Status open*

About 1.2021, the sum of the reciprocal cubes; Apéry proved it irrational in 1978, and whether it is transcendental is open.

Apéry’s constant ζ(3) = 1 + 1/8 + 1/27 + … ≈ 1.2020569, the zeta function of Chapter ᚑ(O) at 3. Euler found ζ(2) = π²/6 in 1734 by the usual account and read it to the St Petersburg Academy in December 1735, the year other accounts give for the result; no closed form for ζ(3) has ever been found; Roger Apéry stunned the field in 1978 by proving it irrational with methods that had been available for 200 years. Its transcendence is open, which puts it in the irrational-but-unclassified brace here as well as in Chapter ᚑ(O)’s denominator. Status open.

Cites: [5.30](ch05-refs.md#r-5.30), [5.5](ch05-refs.md#r-5.5)

See also: [Riemann zeta function](ch18-cards.md#g-zeta), [the Basel value](ch18-cards.md#g-zeta2)

### zeta of five — `ζ(5)` {#g-zeta5}

*Status open*

About 1.0369, the sum of the reciprocal fifth powers; not known to be irrational, though at least one of ζ(5), ζ(7), ζ(9), ζ(11) is (Zudilin, 2001).

ζ(5) = 1 + 1/32 + 1/243 + … ≈ 1.0369278. Apéry’s method does not reach it: Rivoal proved in 2000 that infinitely many odd zeta values are irrational, and Wadim Zudilin in 2001 that at least one of ζ(5), ζ(7), ζ(9), and ζ(11) is, but no single one of them has been settled. The even values are all rational multiples of powers of π, and so transcendental; the odd ones are the open frontier. Status open.

Cites: [5.13](ch05-refs.md#r-5.13), [5.31](ch05-refs.md#r-5.31)

### pi to the pi and e to the e — `π^{π} + e^{e}` {#g-pipi}

*Status open*

About 36.46 and 15.15; neither is known to be irrational, though Schanuel’s conjecture would make both transcendental.

π^π ≈ 36.4622 and e^e ≈ 15.1543 are the two simplest self-powers of the famous transcendentals, and neither is known to be irrational; Gelfond–Schneider needs an algebraic base, and π and e are not. Schanuel’s conjecture of the 1960s would settle these, π + e, π·e, and π^e in one stroke, and remains open. Status open.

Cites: [5.12](ch05-refs.md#r-5.12), [5.24](ch05-refs.md#r-5.24), [5.31](ch05-refs.md#r-5.31)

See also: [the Gelfond–Schneider theorem](ch04-cards.md#g-gelfschn), [the unclassified combinations of π and e](ch05-cards.md#g-pie), [Schanuel’s conjecture](ch05-cards.md#g-schanuel)

### Lévy’s constant — `γ_{L}` {#g-levy}

*Status open*

About 3.2758, e^(π²/(12 ln 2)), the growth rate of the denominators of almost every continued fraction; irrationality open.

Lévy’s constant e^(π²/(12 ln 2)) ≈ 3.275823 is the limit of the nth root of the nth convergent denominator of the continued fraction of almost every real number. Khinchin proved in ~1935 that the limit exists and is the same for almost all numbers, and Paul Lévy found the closed form in 1936; Khinchin’s own constant beside it is the geometric mean of the partial quotients. Nothing is known about whether it is irrational. Status open.

Cites: [5.26](ch05-refs.md#r-5.26), [5.9](ch05-refs.md#r-5.9)

See also: [continued fractions and convergents](ch04-cards.md#g-contfrac), [Khinchin’s constant](ch05-cards.md#g-khin)

### the Fransén–Robinson constant — `F` {#g-fransen}

*Status open*

About 2.8078, the area under 1/Γ(x) from 0 to ∞; irrationality open.

The Fransén–Robinson constant F = ∫₀^∞ dx/Γ(x) ≈ 2.8077702, the area under the reciprocal gamma function of Chapter ᚐ(A), computed to high precision by Herman P. Robinson, who reached 36 places by Newton–Cotes quadrature, and by Arne Fransén, who reached 65 by Euler–Maclaurin summation. It is the continuous analogue of e = Σ 1/n!, and it has no known closed form and no known irrationality proof. Status open.

Cites: [5.26](ch05-refs.md#r-5.26), [5.9](ch05-refs.md#r-5.9)

See also: [Euler’s number](ch04-cards.md#g-e), [the gamma function](ch17-cards.md#g-gammafn)

### Bernstein’s constant — `β` {#g-bernstein}

*Status open*

About 0.2802, the limit of how badly polynomials of degree n must miss |x| on [−1, 1], times n; irrationality open.

Bernstein’s constant β ≈ 0.2801694 governs how well |x| can be approximated by polynomials on [−1, 1]: Sergei Bernstein showed in 1913 or 1914 that the best degree-n error times n tends to a limit, and computed it to a few digits; Varga and Carpenter found in 1985 that it is not 1/(2√π), which Bernstein had suspected. Its irrationality is open. Status open.

Cites: [5.26](ch05-refs.md#r-5.26), [5.9](ch05-refs.md#r-5.9)

### Viswanath’s constant — `1.1319…` {#g-viswanath}

*Status open*

The growth rate of a Fibonacci sequence in which each step adds or subtracts at random; found by Divakar Viswanath in 1999, status open.

Viswanath’s constant ≈ 1.1319882 is the rate at which the absolute values of a random Fibonacci sequence, xₙ = ±xₙ₋₁ ± xₙ₋₂ with the signs chosen by coin flips, grow almost surely; Divakar Viswanath computed it in 1999, using the Stern–Brocot tree and a fractal measure. Nothing is known about its irrationality. Status open.

Cites: [5.26](ch05-refs.md#r-5.26), [5.9](ch05-refs.md#r-5.9)

### Niven’s constant — `c_{N}` {#g-niven}

*Status open*

About 1.7052, the average over all integers of the largest exponent in their factorization; irrationality open.

Niven’s constant ≈ 1.7052111 is the average, over the integers up to x as x grows, of the largest exponent appearing in the prime factorization; Ivan Niven found it in 1969 as 1 + Σ (1 − 1/ζ(k)) over k ≥ 2, so it is built from the zeta function of Chapter ᚑ(O). Its irrationality is open. Status open.

Cites: [5.26](ch05-refs.md#r-5.26), [5.9](ch05-refs.md#r-5.9)

See also: [Riemann zeta function](ch18-cards.md#g-zeta)

### Artin’s constant — `C_{Artin}` {#g-artin}

*Status open*

About 0.3740, the conjectured density of primes for which 2 is a primitive root; irrationality open.

Artin’s constant Π (1 − 1/(p(p − 1))) over primes ≈ 0.3739558 is the density, conjectured by Emil Artin in 1927, of primes p for which 2 generates every nonzero residue mod p, as in the cyclic number 142857 of Chapter ᚂ(L), where 10 is such a root of 7. Hooley proved the conjecture in 1967 assuming the generalized Riemann Hypothesis; unconditionally it is open, and so is the constant’s irrationality. Status open.

Cites: [5.26](ch05-refs.md#r-5.26), [5.9](ch05-refs.md#r-5.9)

See also: [the cyclic number](ch04-cards.md#g-cyclic), [the Riemann Hypothesis](ch18-cards.md#g-rh)

### Sierpiński’s constant — `K_{S}` {#g-sierpinski}

*Status open*

About 2.5850, the limit of a sum counting the ways integers up to n are sums of two squares, once π ln n is taken away; irrationality open.

Sierpiński’s constant K_{S} ≈ 2.5849817 is the limit of the sum Σ r₂(k)/k over k ≤ n minus π ln n, where r₂ counts representations as a sum of two squares; Wacław Sierpiński found it in 1907, and it is built from π, γ, and the Gamma function at ¼. It is not known to be irrational. Status open.

Cites: [5.26](ch05-refs.md#r-5.26), [5.9](ch05-refs.md#r-5.9)

See also: [Euler–Mascheroni constant](ch05-cards.md#g-gamma), [the gamma function](ch17-cards.md#g-gammafn), [pi](ch04-cards.md#g-pi), [the Brahmagupta–Fibonacci two-square identity](ch21-cards.md#g-twosq)

### the MRB constant — `MRB` {#g-mrb}

*Status open*

About 0.1879, the limit of the alternating sums of nth roots of n; found by Marvin Ray Burns in 1999, status open.

The MRB constant ≈ 0.1878596 is the limit of the even partial sums of Σ (−1)ᵏ k^(1/k), the sums of the first 2n terms; the full series diverges, since its terms tend to ±1, but the even partial sums settle down, and their limit is the constant. Marvin Ray Burns found it in 1999 and it has since been computed to millions of digits without any closed form or irrationality proof. Status open.

Cites: [5.26](ch05-refs.md#r-5.26), [5.9](ch05-refs.md#r-5.9)

### Somos’ quadratic recurrence constant — `σ` {#g-somos}

*Status open*

About 1.6617, the limit of √(1·√(2·√(3·…))), found by Michael Somos in 1999; irrationality open.

Somos’ quadratic recurrence constant σ = √(1·√(2·√(3·√(4·…)))) ≈ 1.6616879, the infinite nested radical Michael Somos studied in 1999; it equals Π n^(1/2ⁿ) and has expressions through the polylogarithm. It is not known to be irrational. Status open.

Cites: [5.26](ch05-refs.md#r-5.26), [5.9](ch05-refs.md#r-5.9)

See also: [the polylogarithm](ch17-cards.md#g-polylog)

### the Kepler–Bouwkamp constant — `ρ_{KB}` {#g-keplerb}

*Status open*

About 0.1149, the radius of the circle left after inscribing a triangle, a square, a pentagon, and every polygon after them inside a unit circle, each inside the last; irrationality open.

The Kepler–Bouwkamp constant ≈ 0.1149420 is the radius of the circle that remains when a unit circle has a triangle inscribed in it, a circle inscribed in the triangle, a square in that circle, a circle in the square, a pentagon, and so on forever: the product of cos(π/n) over n ≥ 3. Kepler drew the first steps in 1596 when modeling the planets, in the Mysterium Cosmographicum, and Christoffel Jacob Bouwkamp computed the limit in 1965. Not known to be irrational. Status open.

Cites: [5.26](ch05-refs.md#r-5.26), [5.9](ch05-refs.md#r-5.9)

### Backhouse’s constant — `B_{N}` {#g-backhouse}

*Status open*

About 1.4561, the growth rate hidden in the power series whose coefficients are the primes; found by Nigel Backhouse in 1995, status open.

Backhouse’s constant ≈ 1.4560749 is the limit of the ratios of successive coefficients of 1/P(x), where P(x) = 1 + 2x + 3x² + 5x³ + 7x⁴ + … has the primes as coefficients; Nigel Backhouse noticed the limit in 1995 and Philippe Flajolet proved it exists. It has no irrationality proof. Status open.

Cites: [5.26](ch05-refs.md#r-5.26), [5.9](ch05-refs.md#r-5.9)

### Schanuel’s conjecture — `Schanuel` {#g-schanuel}

*Status open*

Stephen Schanuel’s unproved conjecture on exponentials, which would settle π + e, π·e, π to the e, π to the π, and e to the e in one stroke.

Schanuel’s conjecture says that if z₁, …, zₙ are complex numbers linearly independent over the rationals, then the field ℚ(z₁, …, zₙ, e^z₁, …, e^zₙ) has transcendence degree at least n over ℚ. Stephen Schanuel formulated it and Serge Lang published it in 1966. It would make e and π algebraically independent, and so settle π + e, π·e, π^e, π^π, and e^e—every unclassified combination in this chapter’s second brace—in one stroke. It remains one of the major unsolved problems of transcendental number theory, and nothing in this chapter assumes it. Status open.

Cites: [5.24](ch05-refs.md#r-5.24)

See also: [the unclassified combinations of π and e](ch05-cards.md#g-pie), [pi to the pi and e to the e](ch05-cards.md#g-pipi)

### a normal number — `freq(b) = 1/10^|b|` {#g-normalnum}

*Status open*

Every block of digits turns up as often as chance says it should. Almost every real is normal, and almost no specific one is known to be.

A number is normal in a base when every finite block of digits appears in its expansion with exactly the frequency chance would assign it, and absolutely normal when that is true in every base at once. Émile Borel introduced the idea in 1909 and proved that almost every real number is normal—the exceptions form a set of measure zero—which is the classic case of knowing a property holds almost everywhere while being unable to certify it anywhere in particular. Sierpiński managed to specify an absolutely normal number in 1917, and Becher and Figueira produced a computable one in 2002, but whether π, e, or √2 is normal is open. The one thing normality does settle at once is irrationality: a rational expansion is eventually periodic, and a periodic expansion cannot give every block its share. Status open.

Cites: [5.28](ch05-refs.md#r-5.28)

See also: [almost everywhere](ch20-cards.md#g-aeverywhere), [Euler’s number](ch04-cards.md#g-e), [pi](ch04-cards.md#g-pi), [square root of two](ch04-cards.md#g-sqrt2)

### Mersenne numbers — `2ⁿ − 1` {#g-mersnum}

*Function or operator*

A Mersenne number is any number of the form 2ⁿ − 1, named for Marin Mersenne, and it can only be prime when n itself is prime, which makes hunting for Mersenne primes one of the oldest games in number theory.

Mersenne numbers are 2ⁿ − 1 for n = 1, 2, 3, …; primality of n is necessary but not sufficient for 2ⁿ − 1 to be prime. The Erdős–Borwein constant is the sum of the reciprocals of all Mersenne numbers, Σ 1/(2ⁿ − 1), which Paul Erdős proved irrational in 1948. The largest known prime of any kind is almost always a Mersenne prime, such as the record-holder 2¹³⁶²⁷⁹⁸⁴¹ − 1, because a fast primality test, the Lucas–Lehmer test, exists just for this family.

Cites: [5.40](ch05-refs.md#r-5.40)

See also: [Erdős–Borwein constant](ch05-cards.md#g-eb)

### uncomputable — `uncomputable` {#g-uncomputable}

*Function or property*

A precisely defined quantity or function for which no single algorithm can produce answers to every requested accuracy or input.

In this book, uncomputability means that no general method exists, as for the Busy Beaver function or the digits of a fixed Chaitin constant. It does not mean that no individual finite value can ever be written down. The x in Rose’s number is a stronger case: it is not even definable, so no formula names it and no program prints it.

Cites: [1.1](ch01-refs.md#r-1.1), [5.25](ch05-refs.md#r-5.25)

See also: [the Busy Beaver function](ch09-cards.md#g-busybeaver), [Chaitin’s constant](ch05-cards.md#g-omega)

### infeasibility — `infeasibility` {#g-infeasibility}

*Author clarification*

The condition of being impracticable: possible in principle, but beyond a stated practical limit such as available time, space, or displayable size.

Author’s clarification: this is a glossary term used to name the book’s distinction between an exact finite quantity and a full calculation or expansion that cannot feasibly be carried out. It is not a new kind of number and does not mean mathematical impossibility.

Cites: [5.25](ch05-refs.md#r-5.25)

### sixth Busy Beaver number — `BB(6)` {#g-bb6}

*Integer or rational*

Radó’s 1962 function at 6: a fixed finite integer whose exact value remains unknown; the function n ↦ BB(n), not this one value, is uncomputable.

The Busy Beaver function BB(n), defined by Tibor Radó in 1962, is the largest number of steps a halting Turing machine with n states and two symbols can make when started on a blank tape. No algorithm computes BB(n) for every n. BB(5) = 47,176,870 was formally proved in 2024; BB(6) remains unresolved, and a machine found in June 2025 shows it exceeds 2↑↑↑5 — a tower of twos whose height is itself a tower of twos. The distinction matters: BB as a function is uncomputable, but BB(6) is one fixed natural number and therefore computable in the standard existential sense, even though nobody presently knows its digits. Type: integer, status unresolved.

Cites: [5.19](ch05-refs.md#r-5.19), [5.21](ch05-refs.md#r-5.21), [9.5](ch09-refs.md#r-9.5)

See also: [the fifth Busy Beaver number](ch09-cards.md#g-bb5), [the Busy Beaver function](ch09-cards.md#g-busybeaver), [Turing machine](ch09-cards.md#g-turingm), [uncomputable](ch05-cards.md#g-uncomputable), [Knuth’s up-arrow notation](ch04-cards.md#g-uparrow)

### Rayo’s number — `Rayo’s` {#g-rayo}

*Integer or rational*

The smallest integer larger than every integer nameable in at most a googol symbols of set theory; the winner of the 2007 big-number duel at MIT, claimed at the time as the largest named number.

Rayo’s number was defined by Agustín Rayo on 26 January 2007, in a big-number duel against Adam Elga at MIT, as the smallest positive integer larger than every finite integer denoted by a qualifying expression of first-order set theory using at most a googol symbols. The exact formal-language conditions are essential. It exceeds TREE(3), which has a much shorter description. Like every fixed finite integer it is computable in the technical sense, although no feasible computation could write it out. Type: integer beyond feasible calculation.

Cites: [5.15](ch05-refs.md#r-5.15)

See also: [googol](ch04-cards.md#g-googol), [TREE(3)](ch04-cards.md#g-tree3)

### positive infinity of the extended reals — `∞` {#g-inf}

*Infinity or limit device*

The limit target in n → ∞; infinity labels the process and is not used as an ordinary exponent.

Positive infinity on the extended real line is a limit device, not an ordinary real number. In Chapter ᚃ(F) the formula uses it only in n → ∞ for the limit of x^{n} when 0 < x < 1; that limit is 0.

Cites: [5.17](ch05-refs.md#r-5.17)

### Chaitin’s constant — `Ω` {#g-omega}

*Definable but uncomputable*

The halting probability: definable, uncomputable, and transcendental.

Fix an optimal universal prefix-free computer. Prefix-free means that no valid program is the beginning of another valid program; these programs delimit themselves. Its halting probability is Ω = Σ 2^{−|p|}, summed over its halting programs p. The value depends on the chosen computer. Gregory Chaitin defined the halting probability in 1975. It is a definite real number between 0 and 1 whose first n bits would decide the halting problem for all programs up to about n bits long, so no algorithm can compute it; it is also known to be transcendental, and algorithmically random, which is why it is normal. Its type is uncomputable, which outranks transcendental in this book’s color scheme. This Ω is Chaitin’s, not the Lambert omega constant W(1) of Chapter ᚐ(A).

Cites: [5.18](ch05-refs.md#r-5.18)

See also: [the omega constant](ch17-cards.md#g-W1), [the halting problem, as a bit](ch09-cards.md#g-halting), [Kolmogorov complexity](ch09-cards.md#g-kolmogorov), [uncomputable](ch05-cards.md#g-uncomputable)

### the limit operator — `lim` {#g-limop}

*Function or operator*

The value approached by a sequence or function; here n tends to infinity while an ordinary power tends to zero.

The limit notation used in Chapter ᚃ(F). For every real x with 0 < x < 1, lim_{n→∞}x^{n} = 0. Newton used limiting arguments in the seventeenth century; Cauchy gave the subject systematic definitions in 1821 and Weierstrass supplied the modern epsilon language. In this formula infinity labels a process and is never treated as an ordinary exponent. A function of the framework.

Cites: [5.17](ch05-refs.md#r-5.17), [5.31](ch05-refs.md#r-5.31)

See also: [positive infinity of the extended reals](ch05-cards.md#g-inf)

### the Collatz problem — `n/2 or 3n + 1` {#g-collatz}

*Status open*

Halve it if it is even, triple and add one if it is odd. Does every start reach 1? Verified for every start up to 2.36 × 10²¹, but never proved in general.

Lothar Collatz introduced the rule in the 1930s: halve an even number, replace an odd n by 3n + 1, repeat. The conjecture is that every positive integer eventually reaches 1. The orbits wander up and down so wildly that they are called hailstone sequences, and the conjecture has been checked for every start up to 2.36 × 10²¹ without a general proof appearing. It matters in this chapter because it is not only unsolved but obstructive: the machines that block the sixth Busy Beaver value are Collatz-like, so a problem nobody can settle about arithmetic becomes a problem nobody can settle about a particular six-state machine. Status open.

Cites: [5.29](ch05-refs.md#r-5.29)

See also: [sixth Busy Beaver number](ch05-cards.md#g-bb6)