Some congruences for Apéry numbers

Congruences for the Apéry numbers are proved which generalize the results and conjectures of Chowla, J. Cowles, and M. Cowles.     

Arithmetic properties of Apery numbers

Let (A_n)_n1 be the sequence of Apéry numbers with a generalterm given by . In thispaper, we prove that both the inequalities (A_n) > c₀ loglog log n and P(A_n) > c₀ (log n log log n)^1/2 hold fora set of positive integers n of asymptotic density 1. Here,(m) is the number of distinct prime factors of m, P(m) is thelargest prime factor of m and c₀ > 0 is an absolute constant.The method applies to more general sequences satisfying botha linear recurrence of order 2 with polynomial coefficientsand certain Lucas-type congruences.     

Some congruences on harmonic numbers and binomial sums

Some new congruences on harmonic numbers are established. In addition, we obtain a congruence of binomial sums, which is a generalization of that of van Hamme and confirms a conjecture of Swisher.     

Some identities and congruences concerning Euler numbers and polynomials

In this paper, using the properties of the moments of p-adic measures, we establish some identities and Kummer likewise congruences concerning Euler numbers and polynomials. In the preliminaries, we introduce the Laplace transform which is an important tool for the determination of the moments of p-adic measures. We also give a sequence n(dn) linked to Euler numbers and which satisfies the same type of congruences and identities as the Euler numbers. At the end, for p=2, we give congruences on Euler numbers involving the sequence n(dn).     

Further congruences involving Bernoulli numbers

Congruences of Voronoi's type for Bernoulli numbers are proved via Bernoulli distributions and connections to some already known congruences of a similar type are briefly discussed.     

Some congruences for Saito-Kurokawa lifts

We give two examples of congruences between Saito-Kurokawa lifts. Moreover, we prove that a certain p-adic limit of Siegel-Eisenstein series of level one becomes a Siegel modular form of level p and trivial character.     

一些包含Fibonacci-Lucas数的恒等式和同余式

给出了一些包含F ibonacci-Lucas数的恒等式和同余式.     

p-adic Proofs of congruences for the Bernoulli numbers

Many of the classical theorems for the Bernoulli numbers, particularly those congruences needed in the study of irregular primes, follow easily from the existence of the (p ? 1)st roots of unity in the ring of p-adic integers. Proofs are given for the von Staudt-Clausen theorem, the theorem of J. C. Adams, the Friedmann-Tamarkine congruence, a theorem of Vandiver, special cases of the congruences of Voronoi, Kummer, and Carlitz, and the congruences of E. Lehmer.     

Some congruences on trioids

We present the least idempotent congruence on the trioid with a commutative operation, the least semilattice congruence on the trioid with an idempotent operation, and the least separative congruence on the trioid with a commutative operation. Also we construct different examples of trioids.     

Some congruences for traces of singular moduli

We address a question posed by Ono [Ken Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series, CBMS Reg. Conf. Ser. Math., vol. 102, Amer. Math. Soc., Providence, RI, 2004, Problem 7.30], prove a general result for powers of an arbitrary prime, and provide an explanation for the appearance of higher congruence moduli for certain small primes. One of our results overlaps but does not coincide with a recent result of Jenkins [Paul Jenkins, p-Adic properties for traces of singular moduli, Int. J. Number Theory 1 (1) (2005) 103-107]. This result essentially coincides with a recent result of Edixhoven [Bas Edixhoven, On the p-adic geometry of traces of singular moduli, preprint, 2005, math.NT/0502213 v1], and we hope that the comparison of the methods, which are entirely different, may reveal a connection between the p-adic geometry and the arithmetic of half-integral weight Hecke operators.     

Some 3-adic congruences for binomial sums

We prove some 3-adic congruences for binomial sums,which were conjectured by Zhi-Wei Sun.For example,for any integer m≡1(mod 3)and any positive integer n,we have31n n.1Xk=01mk 2k kmin{3(n),3(m.1).1},where 3(n)denotes the 3-adic order of n.In our proofs,we use several auxiliary combinatorial identities and a series converging to 0 over the 3-adic field.     

Some congruences of subnormal subgroups

Si discutono congruenze e semicongruenze e del posetR(G), costituito dai sottogruppi normali di un gruppoG. Più dettagliata è l'analisi quandoR(G)/ρ è una catena a quandoR(G) è un reticolo.

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Conferenza tenuta il 16 maggio 1995     

Some curious congruences modulo primes

Let n be a positive odd integer and let p>n+1 be a prime. We mainly derive the following congruence:

     

Euler numbers congruences and Dirichlet L-functions

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In this paper we apply Yamamoto's Theorem [Y. Yamamoto, Dirichlet series with periodic coefficients, in: Proc. Intern. Sympos. “Algebraic Number Theory”, Kyoto, 1976, JSPS, Tokyo, 1977, pp. 275-289] to find the residue modulo a prime power of the linear combination of Dirichlet L-function values L(s,χ) at positive integral arguments s such that s and χ are of the same parity, in terms of Euler numbers, whereby we obtain the finite expressions for short interval character sums. The results obtained generalize the previous results pertaining to the congruences modulo a prime power of the class numbers as the special case of s=1.

Video

For a video summary of this paper, please visit http://www.youtube.com/watch?v=_KAv4FCdVUs.     

Some congruences for 6-colored generalized Frobenius partitions

We prove some congruences discovered by Baruah and Sarmah and by Xia for \(c\phi _6(n)\), the number of 6-colored generalized Frobenius partitions of n.     

Some congruences modulo 5 and 25 for overpartitions

We present two new Ramanujan-type congruences modulo 5 for overpartitions. We also give an affirmative answer to a conjecture of Dou and Lin, which includes four congruences modulo 25 for overpartition.     

On congruences of Euler numbers modulo powers of two

In this paper, we establish some identities involving the Euler numbers, the Euler numbers of order 2 and the central factorial numbers, and give a new proof of a classical result due to M.A. Stern.

Video abstract

For a video summary of this paper, please visit http://www.youtube.com/watch?v=kdNsdTDA-FE.