Bernoulli subshift mod#

Kellner, On irregular prime power divisors of the Bernoulli numbers, Math. Jensen, Om talteoretiske Egenskaber ved de Bernoulliske Tal, Nyt Tidss. Hilbert, Die Theorie der algebraischen Zahlkörper, Jahresber. Harvey, A multimodular algorithm for computing Bernoulli numbers, Math. Ong, Irregular primes to two billion, Math. Fresnel, Nombres de Bernoulli et fonctions L p-adiques, Ann.

Bernoulli subshift mod#

Ernvall, On the distribution mod 8 of the E-irregular primes, Ann. Clausen, Lehrsatz aus einer Abhandlung über die Bernoullischen Zahlen, Astr. Carlitz, The class number of an imaginary quadratic field, Comment. More than 10 million digits are omitted in the middle of the numerator! (Computed by CalcBn V1.2 on Feb. Number of digits of the numerator / denominator: 10137147 / 31. Since ℤ is dense in ℤ p, the function ζ ( p, ℓ)( s), restricted on nonnegative integer arguments, uniquely extends, by means of the Kummer congruences and preserving the interpolation property, to a continuous function on ℤ p. When s ≡ s' ( mod p r− 1) for nonnegative integers s and s'. The Kummer congruences then state for r ≥ 1 that Indeed, this function is well-defined and has the following properties.Īt nonnegative integer arguments the function ζ ( p, ℓ)( s) interpolates values of the function ζ p( 1 − n). Let p ≥ 5 and ℓ ∈ be fixed.įor p-adic integers s by taking any sequence ( t ν) ν ≥ 1 of nonnegative integers that p-adically converges to s. II.įor a prime p and even n ≥ 2 define the zeta function One kind of their constructions deals with p-adic zeta functions defined in certain residue classes for a detailed theory see Koblitz Koblitz1996 Chap. The values ζ( 1 − n) = − B n/ n and the Kummer congruences lead to the construction of p-adic zeta and L-functions, as introduced by Kubota and Leopoldt KubotaLeopoldt1964 in 1964. However, several computations (see, e.g., Hart, Harvey, and Ong HartHarveyOng2017) suggest that about 60% of all primes are regular, which agree with an expected distribution proposed by Siegel Siegel1964. Unfortunately, it is still an open question whether infinitely many regular primes exist. In 1915 Jensen Jensen1915 proved that infinitely many irregular primes p exist with the restriction p ≡ 3 ( mod 4).Ĭarlitz Carlitz1954 later gave a short (and weaker) proof without any restriction on p. The irregular primes below 100 are 37, 59, and 67 see A000928. If p does not divide any of the numerators of the Bernoulli numbers B 2, B 4 , …, B p− 3, then p is regular. He also provided an equivalent definition concerning Bernoulli numbers: Has no solution in positive integers x, y, and z, for the case when the exponent n is a regular prime.

Kummer then proved that Fermat's Last Theorem is true, that is Where is the set of p-th roots of unity otherwise irregular. arXiv: 1850 Kummer Kummer1850 introduced two classes of odd primes, later called regular and irregular (see, e.g., Hilbert Hilbert1897 Chap.~31).Īn odd prime p is called regular if p does not divide the class number of the cyclotomic field 'Every countably infinite group is almost Ornstein'. 'Entropy and isomorphism theorems for actions of amenable groups'.

^ Ornstein, Daniel Weiss, Benjamin (1987).

Transactions of the American Mathematical Society. A Bernoulli scheme is a discrete-time stochastic process where each independent random variable may take on one of N distinct possible values, with the outcome i occurring with probability p i Counterexample Machine'.