The main purpose of this paper is to prove a conjecture of the Euler numbers and its generalization by using the analytic methods. That is, for any prime p ≡ 1 (mod 4) and integer α 1 we proved E φ(p α )/2 ≡ 0 (mod p α ), where E 2n are the Euler numbers and φ(n) the Euler function.

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Let p be an odd prime and a be an integer coprime to p. Denote by N(a, p) the number of pairs of integers b, c with bc ≡ a (mod p), 1 b, c (p − 1)/2 and with b, c having different parity. The main purpose of this paper is to study the sum p−1 a=1 (N (a, p) − (p − 1)/8) 2 , and obtain a sharp asymptotic formula.

Given the sequence of Pell numbers {P n }, we evaluate the integral part of the reciprocal of the sum ∞ k=n 1 P 3 n explicitly in terms of the Pell numbers themselves. MSC: Primary 11B39

The main purpose of this paper is using the analytic method to study the mean value properties of the two-term exponential sums with Dirichlet characters, and give an explicit formula for its fourth power mean.