The Bernoulli numbers B_n play an important role in several topics of mathematics. These numbers can be defined by the power series

where all numbers B_n are zero with odd index n > 1. The even-indexed rational numbers B_n alternate in sign. First values are

The values can be computed iteratively by the recurrence formula

which can be written symbolically as

The sequences of the numerators and denominators of B_n are A027641 and A027642, respectively.

Jacob Bernoulli Bernoulli1713 (1655-1705) introduced a sequence of rational numbers in his Ars Conjectandi, which was published posthumously in 1713. He used these numbers, later called Bernoulli numbers, to compute the sum of consecutive integer powers.

This formula is given by

where S_n(x) is a polynomial of degree n + 1.

An explicit formula for B_n was derived by Worpitzky Worpitzky1883 in 1883:

using the symbol

with S₂(n, k) being the Stirling numbers of the second kind.

He also gave another formula for S_n(x):

Furthermore, one has by means of iterated forward differences the relation

yielding the double sum

The Bernoulli numbers are connected with the Riemann zeta function

on the positive real axis by Euler's celebrated formula for positive even n, also valid for n = 0:

The functional equation of ζ(s) leads to the following formula for negative integer arguments:

In 1850 Kummer Kummer1850 introduced two classes of odd primes, later called regular and irregular (see, e.g., Hilbert Hilbert1897;Chap.~31).

An odd prime p is called regular if p does not divide the class number of the cyclotomic field where is the set of p-th roots of unity; otherwise irregular. Kummer then proved that Fermat's Last Theorem is true, that is

has no solution in positive integers x, y, and z, for the case when the exponent n is a regular prime.

He also provided an equivalent definition concerning Bernoulli numbers:

If p does not divide any of the numerators of the Bernoulli numbers B₂, B₄, …, B_p−3, then p is regular.

The irregular primes below 100 are 37, 59, and 67; see A000928.

In 1915 Jensen Jensen1915 proved that infinitely many irregular primes p exist with the restriction p ≡ 3 (mod 4). Carlitz Carlitz1954 later gave a short (and weaker) proof without any restriction on p.

Unfortunately, it is still an open question whether infinitely many regular primes exist. 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.

The pair (p, ℓ) is called an irregular pair, if p divides the numerator of B_ℓ where ℓ is even and 2 ≤ ℓ ≤ p − 3.

The index of irregularity i(p) is defined to be the number of such pairs belonging to p. If i(p) = 0, then p is regular, otherwise irregular.

The first irregular pairs are (37, 32), (59, 44), and (67, 58). The irregular prime p = 157 is the least prime with i(p) = 2: (157, 62), (157, 110).

The denominator of B_n for positive even n is given by the famous von Staudt-Clausen theorem, independently found by von Staudt Staudt1840 and Clausen Clausen1840 in 1840:

As a consequence, the denominator is squarefree and divisible by 6.

Given a Bernoulli number B_n with n even, Rado Rado1934 showed that there exist infinitely many even m such that

implying that the numbers B_m have the same denominator as B_n.

A special case is given for n = 2p, where p is an odd prime p ≡ 1 (mod 3):

See A112772, which is a subsequence of A051222; the sequence of the increasing denominators is A090801.

The unsigned numerator of the divided Bernoulli number B_n/n for positive even n equals 1 only for n = 2, 4, 6, 8, 10, 14; otherwise the numerator consists of a product of powers of irregular primes:

Since B_n/n is a p-integer for all primes p with p − 1 not dividing n, the structure of the numerator of B_n is given by

The additional left product is a trivial factor of B_n that divides n, see A300711.

For the signed numerators of B_n and B_n/n for even n see A000367 and A001067, respectively.

The Kummer congruences describe the most important arithmetical properties of the Bernoulli numbers, which give a modular relation between these numbers.

Let φ denote Euler's totient function. Let n and m be positive even integers and p be a prime with p − 1 ∤ n.

If n ≡ m (mod φ(p^r)) where r ≥ 1, then

Furthermore,

In 1851 Kummer Kummer1851 originally introduced these congruences without the Euler factors (1 − pⁿ⁻¹) and hence with restrictions on r and n. He showed that the second congruence holds for n > r, whereas the first congruence was derived from the latter only for r = 1 (in these cases the Euler factors vanish). Subsequently, these congruences were widely generalized by several authors (see, e.g., Fresnel Fresnel1967).

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. One kind of their constructions deals with p-adic zeta functions defined in certain residue classes; for a detailed theory see Koblitz Koblitz1996;Chap. II.

For a prime p and even n ≥ 2 define the zeta function

Let p ≥ 5 and ℓ ∈ {2, 4, …, p − 3} be fixed. Define the p-adic zeta function on by

for p-adic integers s by taking any sequence (t_ν)_{ν ≥ 1} of nonnegative integers that p-adically converges to s. Indeed, this function is well-defined and has the following properties.

At nonnegative integer arguments the function ζ_(p,ℓ)(s) interpolates values of the function ζ_p(1 − n). The Kummer congruences then state for r ≥ 1 that

when s ≡ s' (mod p^r−1) for nonnegative integers s and s'.

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.

The p-adic zeta function ζ_(p,ℓ)(s) can be written as a special Mahler expansion (Kellner Kellner2007):

with integral coefficients

One has the relation

Condition for the existence of a unique simple zero (Kellner Kellner2007):

If (p, ℓ) is an irregular pair and a₁ ∈ , that is

then the p-adic zeta function ζ_(p,ℓ)(s) has a unique simple zero ξ_(p,ℓ) ∈ .

So far, no irregular pair (p, ℓ) has been found that the non congruence relation above holds as a congruence.

Example: In the case (p, ℓ) = (37, 32) one computes that

For more p-adic digits see Kellner Kellner2007 and A299468.

The irregular pairs of higher order describe the first appearance of higher powers of irregular prime factors of B_n/n.

An irregular pair (p, n) of order r has the property that p^r divides B_n/n with n < φ(p^r) = (p − 1)p^r−1. For r = 1 this gives the usual definition of irregular pairs, since the condition p divides B_n/n is then equal to p divides B_n.

A zero of the p-adic zeta function ζ_(p,ℓ)(s) describes the irregular pairs (p, n) of higher order with n ≡ ℓ (mod p − 1), and vice versa (Kellner Kellner2007).

For example, one obtains for the irregular pair (37, 32) that

and

Irregular pairs of higher order can be effectively and easily computed using Bernoulli numbers with small indices. By this means one can even predict the extremely huge index of the first occurrence of the power 37³⁷ as listed above; see A251782.

Under the assumption that every p-adic zeta function ζ_(p,ℓ)(s) has a unique simple zero ξ_(p,ℓ) in case (p, ℓ) is an irregular pair, one has for even n ≥ 2 (Kellner Kellner2007):

where

and |·|_p is the ultrametric p-adic absolute value.

The denominator can be described by poles (always lying at ξ_(p,0) = 0) and the numerator by zeros of p-adic zeta functions. Equivalently, the formula reads for the Bernoulli numbers:

The first product gives the trivial factor, the second product describes the product over irregular prime powers, and the third product yields the denominator of B_n.

Moreover, the formulas are valid for all irregular pairs (p, ℓ) with

This follows by computations of irregular pairs and cyclotomic invariants in that range by Hart, Harvey, and Ong HartHarveyOng2017. So far, no counterexample is known.

Let h(d) denote the class number of the imaginary quadratic field of discriminant d < −4. There is the following connection with the Bernoulli numbers due to Carlitz Carlitz1953.

If p > 3 is a prime with p ≡ 3 (mod 4), then

using the well-known relation that h(−p) < p. This implies that p cannot divide the above Bernoulli number. Therefore, an irregular pair (p, (p + 1)/2) cannot exist when p ≡ 3 (mod 4).

The Minkowski-Siegel mass formula states for positive integers n = 2k with 8 ∣ n that

where the sum runs over all even unimodular lattices Λ in dimension n and Aut(Λ) is the automorphism group of Λ.

The products of (divided) Bernoulli numbers with explicit asymptotic constants (Kellner Kellner2009) are given by

with

where is the Glaisher-Kinkelin constant A074962 and is the product over all Riemann zeta values at even positive integer arguments A080729.

The Bernoulli polynomials B_n(x) can be defined by the generating function

and are given by the formula

which can be written symbolically as

The constant term of these polynomials is the Bernoulli number

The polynomial S_n(x) of degree n + 1, giving the sum of the nth powers of consecutive integers from 1 up to x − 1 in case x is a positive integer, satisfies the relation

The denominator of B_n(x) − B_n, the nth Bernoulli polynomial without constant term, is given by the remarkable formula (Kellner and Sondow KellnerSondow2017, Kellner Kellner2017, and A195441)

where s_p(n) denotes the sum of the base-p digits of n. The finite product can be written with explicit sharp bounds as

where

If {·} denotes the fractional part and n ≥ 1 is a fixed integer, then there is the surprising relation (Kellner Kellner2017b):

The denominator of the nth Bernoulli polynomial B_n(x) can be described by a similar formula (Kellner and Sondow KellnerSondow2018, A144845):

Further properties are given by Kellner and Sondow KellnerSondow2018. Using the notation

we have the relations

where

This implies a sequence of divisibilities

Furthermore, the denominators obey the rules

As a consequence, the quotients

are integral for odd and even n (A286516 and A286517), respectively.

The Euler numbers E_n may be defined by the power series of the hyperbolic secant function

which is an even function implying that all E_n = 0 with odd index n. The even-indexed numbers E_n are integers and alternate in sign. The first values (A028296) are

The values can be computed iteratively for even n ≥ 2 by the recurrence formula

which can be written symbolically as

A prime p is called E-irregular, if p divides at least one of the Euler numbers E₂,  E₄, …, E_p−3; otherwise p is E-regular.

The pair (p, ℓ) is called an E-irregular pair, if p divides E_ℓ where ℓ is even and 2 ≤ ℓ ≤ p − 3. The index of E-irregularity i_E(p) is defined to be the number of such pairs belonging to p.

The first E-irregular pairs are (19, 10), (31, 22), and (43, 12); see A120337. The E-irregular prime p = 241 is the least prime with i_E(p) = 2: (241, 210), (241, 238).

In 1954 Carlitz Carlitz1954 proved that infinitely many E-irregular primes exist. Later Ernvall Ernvall1975 showed the more specialized result that infinitely many E-irregular primes p ≢ ±1 (mod 8) exist.

As in the case of the Bernoulli numbers, it is still an open question whether infinitely many E-regular primes exist.

For the Euler numbers one can state a similar conjectural formula as in the case of the Bernoulli numbers, though it is a bit more complicated.

One may conjecturally state for even n ≥ 2 that

where ξ_(p,ℓ) is the unique simple zero of a certain p-adic L-function associated with an E-irregular pair (p, ℓ) ∈ when ℓ ≠ 0, respectively, with a rare exceptional prime p with (p, 0) ∈ in case ℓ = 0.

Let h(d) denote the class number of the imaginary quadratic field of discriminant d < −4. Due to Carlitz Carlitz1953 one has the following connection with the Euler numbers.

If p is a prime with p ≡ 1 (mod 4), then

using the well-known relation h(−4p) < p. Therefore p cannot divide the above Euler number. Consequently, an E-irregular pair (p, (p − 1)/2) cannot exist when p ≡ 1 (mod 4).

CalcBn V3.0 is a multi-threaded program for computing the Bernoulli numbers via the Riemann zeta function. It uses special optimizations such that the main part of calculation can be performed by integer arithmetic. CalcBn depends on the GMP library, so it is recommended to use the latest version of GMP with possible optimizations for the current hardware. The source code of CalcBn for 32/64-bit Linux and Windows complies with C++11. It is released under the terms of the GNU Public License.

Usage: CalcBn [option] index
  -v, --version   version and copyright
  -t, --thread n  use n (1..32) threads
  -s, --suppress  no output of result
  -d, --digit     print number of digits
  -T, --time      print timing
  -c, --check     check result
  index           even index (2..10^7)
                    

Factorizations of the numerators of the Bernoulli numbers, respectively, the Euler numbers with even index 2 to 10 000. Computed prime factors are less than one million, except for known greater prime factors.

Computed irregular pairs (p, ℓ) for primes below 20 000.

Number of digits of the numerator / denominator: 4767554 / 24.

More than 4.7 million digits are omitted in the middle of the numerator! (Computed by CalcBn V1.0 on Dec. 16, 2002.)

Number of digits of the numerator / denominator: 7415484 / 55.

More than 7.4 million digits are omitted in the middle of the numerator! (Computed by CalcBn V1.2 on Feb. 8, 2003.)

Number of digits of the numerator / denominator: 10137147 / 31.

More than 10 million digits are omitted in the middle of the numerator! (Computed by CalcBn V1.2 on Feb. 10, 2003.)