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

     

Quartic residues and binary quadratic forms

Let be a prime, m∈Z and . In this paper we obtain a general criterion for m to be a quartic residue in terms of appropriate binary quadratic forms. Let d>1 be a squarefree integer such that , where is the Legendre symbol, and let ε_d be the fundamental unit of the quadratic field . Since 1942 many mathematicians tried to characterize those primes p so that ε_d is a quadratic or quartic residue . In this paper we will completely solve these open problems by determining the value of , where p is an odd prime, and . As an application we also obtain a general criterion for , where {u_n(a,b)} is the Lucas sequence defined by and .     

When subset-sums do not cover all the residues modulo p

Let . We prove that a subset of , where p is a prime number, with cardinality larger than such that its subset sums do not cover has an automorphic image which is rather concentrated; more precisely, there exists s prime to p such that

     

The distribution of patterns of inverses modulo a prime

We investigate the distribution of the numbers x∈[1,p] for which all lie in a subset of the set of multiplicative inverses modulo a prime p. Here the a_i are integers coprime to p and the numbers are distinct .     

Quartic, octic residues and Lucas sequences

Let be a prime and a,b∈Z with a²+b²≠p. Suppose p=x²+(a²+b²)y² for some integers x and y. In the paper we develop the calculation technique of quartic Jacobi symbols and use it to determine . As applications we obtain the congruences for modulo p and the criteria for (if ), where {Un} is the Lucas sequence given by U₀=0, U₁=1 and Un+1=bUn+k²Un−1(n?1). We also pose many conjectures concerning , or .     

Volumes and diophantine inequalities associated with decomposable forms

For homogeneous decomposable forms in n variables with real coefficients, we consider the associated volume of all real solutions to the inequality . We relate this to the number of integral solutions to the Diophantine inequality in the case where F has rational coefficients. We find quantities which bound the volume and which yield good upper bounds on the number of solutions to the Diophantine inequality in the rational case.     

The structure of analytic τ-sheaves

Let R be a complete discrete valuation -algebra whose residue field is algebraic over , and let K denote its fraction field. In this paper, we study the structure of τ-sheaves M without good reduction on the curve , seen as a rigid analytic space. One motivation is the Tate uniformization theorem for t-motives of Drinfeld modules, which we want to extend to general τ-sheaves. On the other hand, we are interested in the action of inertia on a generic Tate module T_?(M) of M.For a given τ-sheaf M on , we prove the existence of a maximal model for M on , an R-model of , and, over a finite separable extension R′ of R, of nondegenerate models for M.We prove the following ‘semistability’ theorem: there exists a finite extension K′ of K, a nonempty open subscheme C′⊂C, and a filtration

     

Complete decomposition of Dickson-type polynomials and related Diophantine equations

We characterize decomposition over C of polynomials defined by the generalized Dickson-type recursive relation (n?1)

     

Enumeration of isomorphism classes of extensions of p-adic fields

Let Ω be an algebraic closure of and let F be a finite extension of contained in Ω. Given positive integers f and e, the number of extensions K/F contained in Ω with residue degree f and ramification index e was computed by Krasner. This paper is concerned with the number of F-isomorphism classes of such extensions. We determine completely when p2?e and get partial results when p2||e. When s is large, is equal to the number of isomorphism classes of finite commutative chain rings with residue field , ramification index e, and length s.     

Arithmetical properties of wendt's determinant

Wendt's determinant of order n is the circulant determinant W_n whose (i,j)-th entry is the binomial coefficient , for 1?i,j?n, where n is a positive integer. We establish some congruence relations satisfied by these rational integers. Thus, if p is a prime number and k a positive integer, then and . If q is another prime, distinct from p, and h any positive integer, then . Furthermore, if p is odd, then . In particular, if p?5, then . Also, if m and n are relatively prime positive integers, then W_mW_n divides W_mn.     

Realizable classes of tetrahedral extensions

Let k be a number field and O_k its ring of integers. Let Γ be the alternating group A₄. Let be a maximal O_k-order in k[Γ] containing O_k[Γ] and its class group. We denote by the set of realizable classes, that is the set of classes such that there exists a Galois extension N/k at most tamely ramified, with Galois group isomorphic to Γ, for which the class of is equal to c, where O_N is the ring of integers of N. In this article we determine and we prove that it is a subgroup of provided that k and the 3rd cyclotomic field of are linearly disjoint, and the class number of k is odd.     

Kloosterman's uniformly distributed sequence

Applying the theory of uniform distribution, especially the Erdös-Turán-Koksma inequality and the Koksma-Hlawka inequality, to the two-dimensional Kloosterman sequence , j=1,2,…,?(n) (where , and ?(n) is the Euler function) we find an estimation for the discrepancy of this sequence and an error term for the Kth moment, K=1,2,…, of the sequence of distances as

     

The distribution of prime ideals in a real quadratic field with units having a given index in the residue class field

Let k be a real quadratic number field and the ring of integers and the group of units in k. Denote by the subgroup represented by elements of E of for a prime ideal in k. We show that for a given positive rational integer a, the set of prime numbers p for which the residual index of for lying above p is equal to a has a natural density c under the Generalized Riemann Hypothesis. Moreover, we give the explicit formula of c and conditions to c=0.     

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

     

The density of 0's in recurrence double sequences

Let be a double sequence over a finite field satisfying a linear recurrence with constant coefficients, with at most finitely many nonzero elements on each row. Given a nonzero element g of , we show how to obtain an explicit formula for the number of g's in the first qⁿ rows of A. We also characterize the cases when the density of 0's is 1.