In this paper we confirm a conjecture of Sun which states that each positive integer is a sum of a square, an odd square and a triangular number. Given any positive integer m, we show that p = 2m + 1 is a prime congruent to 3 modulo 4 if and only if Tm = m(m + 1)/2 cannot be expressed as a sum of two odd squares and a triangular number, i.e., p = x+8(y+z) for no odd integers x, y, z. We also sh...

Using a sum-product result due to Bourgain, Katz, and Tao, we show that for every 0 < 2 ≤ 1, and every integer k ≥ 1, there exists an integer N = N(2, k), such that for every prime p and every residue class a (mod p), there exist positive integers x1, ..., xN ≤ p satisfying a ≡ 1 x1 + · · ·+ 1 xN (mod p).

Let p be a prime and let a be a positive integer. In this paper we determine ∑pa−1 k=0 ( 2k k+d ) /mk and ∑p−1 k=1 ( 2k k+d ) /(kmk−1) modulo p for all d = 0, . . . , pa, where m is any integer not divisible by p. For example, we show that if p 6= 2, 5 then p−1 ∑

Given an integer b > 1 and a string s of decimal digits, one may ask whether there exists an integer n such that n (in decimal form) ends in s. This paper answers that question for the case where the exponent b is relatively prime to 10. It extends the earlier work [2], where the question was answered for cubes.

For every integer n ≥ 1 let an be the smallest positive integer such that n + an is prime. We investigate the behavior of the sequence (an)n≥1, and prove asymptotic results for the sums ∑ n≤x an, ∑ n≤x 1/an, and ∑ n≤x log an.

Let q be a power of a prime and n a positive integer. Let P (q) be a parabolic subgroup of the finite general linear group GLn(q). We show that the number of P (q)conjugacy classes in GLn(q) is, as a function of q, a polynomial in q with integer coefficients. This answers a question of J. Alperin in [1].

Let p be an odd prime and a be a positive integer. In this paper we investigate the sum P p

We investigate the density of integers that may be written as p + 2, where p is a prime and k a nonnegative integer.

If p is a prime and n a positive integer, let νp(n) denote the exponent of p in n, and up(n) = n/p νp(n) the unit part of n. If α is a positive integer not divisible by p, we show that the p-adic limit of (−1) up((αp)!) as e → ∞ is a well-defined p-adic integer, which we call zα,p. In terms of these, we then give a formula for the p-adic limit of ( ap+c bpe+d ) as e → ∞, which we call ( ap∞+c b...