On consecutive happy numbers

Let e?1 and b?2 be integers. For a positive integer with 0?aj<b, define

     

On sums of binomial coefficients and their applications

In this paper we study recurrences concerning the combinatorial sum and the alternate sum , where m>0, n?0 and r are integers. For example, we show that if n?m-1 then

     

Congruences involving Bernoulli and Euler numbers

Let [x] be the integral part of x. Let p>5 be a prime. In the paper we mainly determine , , and in terms of Euler and Bernoulli numbers. For example, we have

     

Note on some congruences of Lehmer

In the paper, we generalize some congruences of Lehmer and prove that for any positive integer n with (n,6)=1

     

A combinatorial identity with application to Catalan numbers

By a very simple argument, we prove that if l,m,n∈{0,1,2,…} then

     

Notes sur la fonction ζ de Riemann, 4

Opération fondamentale de l'arithmétique, familière depuis des millénaires, la division euclidienne n'a pas livré tous ses secrets. Ainsi, notons pour k et a entiers positifs, le reste de la division euclidienne de k par a, et imaginons un instant que, par un choix convenable d'un entier n et de réels c₂,…,c_n, nous sachions rendre arbitrairement petite la quantité

     

On the index of circular units in the full group of units of a compositum of quadratic fields

For a compositum of quadratic fields , where d₁,…,ds are square-free odd integers and , we study the group C of circular units of k. We construct a basis of C, compute the index of C in the full group of units of k and derive a lower bound for the divisibility of this index by a power of 2. These results give a lower bound for the divisibility of the class number of the maximal real subfield of k by a power of 2.     

On character sums over flat numbers

Let q?2 be an integer, χ be any non-principal character mod q, and H=H(q)?q. In this paper the authors prove some estimates for character sums of the form

     

Cubic residues and binary quadratic forms

Let p>3 be a prime, u,v,d∈Z, gcd(u,v)=1, p?u²−dv² and , where is the Legendre symbol. In the paper we mainly determine the value of by expressing p in terms of appropriate binary quadratic forms. As applications, for we obtain a general criterion for and a criterion for εd to be a cubic residue of p, where εd is the fundamental unit of the quadratic field . We also give a general criterion for , where {Un} is the Lucas sequence defined by U₀=0, U₁=1 and Un+1=PUn−QUn−1 (n?1). Furthermore, we establish a general result to illustrate the connections between cubic congruences and binary quadratic forms.     

A restricted sum formula among multiple zeta values

For positive integers α₁,α₂,…,αr with αr?2, the multiple zeta value or r-fold Euler sum is defined as

     

On Delannoy numbers and Schröder numbers

The nth Delannoy number and the nth Schröder number given by

     

Exceptional congruences for the coefficients of certain eta-product newforms

Let F(z)=∑_n=1^∞a(n)qⁿ denote the unique weight 16 normalized cuspidal eigenform on . In the early 1970s, Serre and Swinnerton-Dyer conjectured that

     

On the parity of exponents in the standard factorization of n!

Let p₁,p₂,… be the sequence of all primes in ascending order. The following result is proved: for any given positive integer k and any given , there exist infinitely many positive integers n with

     

Some identities involving theta functions

Let (|q|<1). For k∈N it is shown that there exist k rational numbers A(k,0),…,A(k,k−1) such that

     

On the residue classes of integer sequences satisfying a linear three term recurrence formula

In this paper we investigate linear three-term recurrence formulae with sequences of integers (T(n))_n?0 and (U(n))_n?0, which are ultimately periodic modulo m, e.g.

     

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:

     

On the product of two primitive elements of maximal subfields of a finite field

Let denote a finite field with r elements. Let q be a power of a prime, and p₁,p₂, p₃ be distinct primes. Put

     

On the quadratic character of quadratic units

Let be a prime. Let a,b∈Z with p?a(a²+b²). In the paper we mainly determine by assuming p=c²+d² or p=Ax²+2Bxy+Cy² with AC−B²=a²+b². As an application we obtain simple criteria for εD to be a quadratic residue , where D>1 is a squarefree integer such that D is a quadratic residue of p, εD is the fundamental unit of the quadratic field with negative norm. We also establish the congruences for and obtain a general criterion for p|U(p−1)/4, where {Un} is the Lucas sequence defined by U₀=0, U₁=1 and Un+1=bUn+k²Un−1(n?1).