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It is the main purpose of this paper to study shortened recurrence relations for generalized Bernoulli numbers and polynomials attached to χ, χ being a primitive Dirichlet character, in which some of the preceding numbers or polynomials are completely excluded. As a result, we are able to establish several kinds of such type recurrences by generalizing some known identities on classical Bernoulli numbers and polynomials such as Saalschutz–Gelfand and von Ettingshausen–Stern's formulas. Furthermore, we discuss shortened recurrence relations for special values of the Riemann zeta and its allied functions.

In this paper, by applying a certain Saalsch ütz–Gelfand type formulafor generalized Bernoulli numbers, we first studyshortenedrecurrence relations forthe values of the DirichletL-function attached to a primitive character at non-positive integers and then derive some detreminantal expressions for these values.Further, we discuss several kinds of infinite sums involving the Riemann zeta valuesat negative odd integers and establish some closed formulas for them by making useof special identities related to the reciprocals of central binomial numbers.

We prove convolution identities of arbitrary orders for Bernoulli and Euler polynomials, i.e., sums of products of a fixed but arbitrary number of these polynomials. They differ from the more usual convolutions found in the literature by not having multinomial coefficients as factors. This generalizes a special type of convolution identity for Bernoulli numbers which was first discovered by Yu. Matiyasevich.

Extending known results on integrals of products of two or three Bernoulli polynomials with limits of integration 0 and 1, we obtain identities for such integrals with limits of integration 0 and x, for a variable x. As applications we obtain certain quadratic and cubic identities for Bernoulli polynomials.

Let p be an odd prime and q p ( a ) = ( a p − 1 − 1 ) / p be the Fermat quotient with base a, p ∤ a . The main purpose of this paper is to investigate the fourth power problem of q p ( 2 ) and deduce an explicit formula represented by a linear combination of Mirimanoff polynomial values.

(2008). Reciprocity Relations for Bernoulli Numbers. The American Mathematical Monthly: Vol. 115, No. 3, pp. 237-244.

We extend Euler's well-known quadratic recurrence relation for Bernoulli numbers, which can be written in symbolic notation as ( B 0 + B 0 ) n = − n B n − 1 − ( n − 1 ) B n , to obtain explicit expressions for ( B k + B m ) n with arbitrary fixed integers k , m ⩾ 0 . The proof uses convolution identities for Stirling numbers of the second kind and for sums of powers of integers, both involving Bernoulli numbers. As consequences we obtain new types of quadratic recurrence relations, one of which gives B 6 k depending only on B 2 k , B 2 k + 2 , … , B 4 k .

In this paper we shall mainly study additive representations of integers prime to the first m primes as a sum of some integers having a peculiar property. The conjectures of Goldbach and twin primes are also observed in connection with these representations of integers.

Letp be an odd prime and\(\mathbb{F}_p \) the finite field withp elements. In the present paper we shall investigate the number of points of certain quadratic hypersurfaces in the vector space\(\mathbb{F}_p^n \) and derive explicit formulas for them. In addition, we shall show that the class number of the real quadratic field\(\mathbb{Q}(\sqrt p )\) (wherep≡1 (mod 4)) over the field ℚ of rational numbers can be expressed by means of these formulas.

Letp⩾5 be a prime,Γa set of certain positive integers, andGa cyclic group of orderp−1. In this paper we mainly investigate the special subideal I Γ(p) (depending onΓ) of the Stickelberger ideal I (p) in the group ringRp= Z p[G] (where Z p= Z /p Z ) and find various kinds of bases of I Γ(p) regarding as a vector space over Z pwhich are intimately connected with the Kummer type system of congruences.

Special congruences of Kummer type are investigated in the connection with the Stickelberger subideals.

In this paper we shall investigate Giuga’s conjecture which asserts an interesting characterization of prime numbers, just as Wilson’s Theorem. Some variations and consequences of the Giuga congruence are discussed by means of Bernoulli numbers. In addition, we shall study various quotients relating to the integers satisfying the Giuga congruence.

We prove a new convolution identity for sums of products of two Bernoulli polynomials. This can be rewritten to obtain a reciprocity relation for a related sum. The proof uses some results on Stirling numbers of both kinds which are of independent interest. In particular, a class of polynomials related to the Stirling numbers of the second kind turns out to be a useful tool.

One can tell that 17 = 23 + 32 19 = 24 + 3 37 = 52 + 22.3 47 = 2 52 3 53 = 32 * 7 _ 2 * 5 97 = 3 5 7 23 are all prime, at a glance, since we have written each n = A + B where each prime < 4 divides exactly one of A and B (and thus n is coprime with every prime < 4). This strange procedure is thoroughly investigated in [1]; in general, it is quite a challenge to so write a given prime n since the product of the primes < 4 is around e{l +° (l )} 5ln . A similar but more complicated method to establish the primality of n goes as follows: let Pl = 2 < P2 = 3 < *** < Pk be the sequence of primes < 4. Write n in the form

Using Kummer's criteria we show that if the first case of Fermat's last theorem fails for the prime p, then there exist irregular pairs satisfying certain relations.

We consider the equation (*) xp+yp+zp=0, where p is an odd prime and x, y, z are non-zero rational integers prime to each other. The study of the equation (*) is usually subdivided into two cases: (I) p ⫮ xyz, (II) p|xyz. In the present paper we shall treat the case (I) of the equation (*) and derive some consequences from the criteria of Kummer and Mirimanoff.

Letp be an odd prime withp≡1 (mod 4) andQ(√p) be the real quadratic field. Also lete andh denote the fundamental unit and the class number of Q(√p), respectively. The main purpose of this paper is to study the “explicit” expressions ofe h ande 2h , and to discuss the problems related to the conjecture of Ankeny-Artin-Chowla.

The main purpose of this paper is to investigate some basic relations (e.g., Voronoi's and Kummer's congruences) of Bernoulli and Euler numbers by manipulating Euler factors in a natural way.

Let p be an odd prime and Q ( ζ p ) be the cyclotomic field defined by a primitive pth root of unity ζ p . It is well-known that the relative class number h p - of Q ( ζ p ) can be expressed by means of the determinant of Inkeri's matrix. The main purpose of this paper is to study Inkeri's class number formula from the viewpoints of bases of Stickelberger ideals in a certain group ring and some special polynomial values.

Analogues of Fermat quotients for a composite modulusm⩾2 are investigated, with special emphasis on various congruences. In particular, the numbersmfor whichaφ(m)≡1 (mod m2), where gcd(a, m)=1, (“Wieferich numbers with basea”) are completely characterized in terms of the Wieferich primes with basea.

Euler's well-known nonlinear relation for Bernoulli numbers, which can be written in symbolic notation as ( B 0 + B 0 ) n = − n B n − 1 − ( n − 1 ) B n , is extended to ( B k 1 + ⋯ + B k m ) n for m ⩾ 2 and arbitrary fixed integers k 1 , … , k m ⩾ 0 . In the general case we prove an existence theorem for Euler-type formulas, and for m = 3 we obtain explicit expressions. This extends the authors' previous work for m = 2 .