By some extremely simple arguments, we point out the following: (i) If n is the least positive kth power non-residue modulo a positive integer m, then the greatest number of consecutive kth power residues mod m is smaller than m/n. (ii) Let OK be the ring of algebraic integers in a quadratic field K = Q( √ d) with d ∈ {−1,−2,−3,−7,−11}. Then, for any irreducible π ∈ OK and positive integer k no...

In this article we classify actions of the groups Z pk (p is a prime integer) on compact oriented surfaces.

In this article we classify actions of the groups Z pk (p is a prime integer) on compact oriented surfaces.

Let [Formula: see text] be a Lucas sequence and, for every prime number text], let the rank of appearance in that is, smallest positive integer such divides whenever it exists. Furthermore, an odd integer. Under some mild hypotheses, we prove asymptotic formula primes as text].

We now establish a pleasantly simple description of Carmichael numbers, due to Korselt. First, we need the following notion. Let a and p be coprime (usually, p will be prime, but this is not essential). The order of a modulo p, denoted by ordp(a), is the smallest positive integer m such that a ≡ 1 mod p. Recall [NT4.5]: If ordp(a) = m and r is any integer such that a ≡ 1 mod p, then r is a mult...

In a binary decision diagram (BDD), a non-terminal node representing a function f = xf0_xf1 has two edges for f0 and f1. In the arithmetic ternary decision diagram (Arith TDD), each non-terminal node has three edges, where the third edge denotes f2 = f0 + f1, and + is an integer addition. The Arith TDD represents the extended weight function, an integer function showing the numbers of true mint...

In this paper, we show that if p is a prime and ifA = {a1, a2, . . . , am} is a set of positive integers with the property that aiaj +p is a perfect square for all 1 ≤ i < j ≤ m, then m < 3 · 2168. More generally, when p is replaced by a squarefree integer n, the inequality m ≤ f(ω(n)) holds with some function f , where ω(n) is the number of prime divisors of n. We also give upper bounds for m ...

Extended digit multiplication can be an effective benchmark for comparing contemporary CPUs to other architectures and devices. The Great Internet Mersenne Prime Search (GIMPS), a distributed computing effort to find large prime numbers, has produced highly optimized code for multiplying large, multimillion digit numbers on Pentium processors. This paper presents a hardware large integer multip...

There exist many results about the Diophantine equation (qn − 1)/(q − 1) = ym, where m ≥ 2 and n ≥ 3. In this paper, we suppose that m = 1, n is an odd integer and q a power of a prime number. Also let y be an integer such that the number of prime divisors of y − 1 is less than or equal to 3. Then we solve completely the Diophantine equation (qn − 1)/(q − 1) = y for infinitely many values of y....

Let p be a fixed odd prime. Using certain results of exponential Diophantine equations, we prove that (i) if p ≡ ± 3(mod  8), then the equation 8 (x) + p (y) = z (2) has no positive integer solutions (x, y, z); (ii) if p ≡ 7(mod  8), then the equation has only the solutions (p, x, y, z) = (2 (q) - 1, (1/3)(q + 2), 2, 2 (q) + 1), where q is an odd prime with q ≡ 1(mod  3); (iii) if p ≡ 1(mod  8)...