Bertrand’s postulate is an early result on the distribution of prime numbers: For every positive integer n, there exists a prime number that lies strictly between n and 2n. The proof is ported from John Harrison’s formalisation in HOL Light [1]. It proceeds by first showing that the property is true for all n greater than or equal to 600 and then showing that it also holds for all n below 600 b...

Let N be a positive integer and let A and B be dense subsets of {1, . . . , N}. The purpose of this paper is to establish a good lower bound for the greatest prime factor of ab+ 1 as a and b run over the elements of A and B respectively. 1991 AMS Mathematics Subject Classification. Primary 11N30, Secondary 11L05, keywords: greatest prime factor, Selberg’s sieve, Kloosterman sums.

We prove that there exists an integer p0 such that X split (p)(Q) is made of cusps and CM-points for any prime p > p0. Equivalently, for any non-CM elliptic curve E over Q and any prime p > p0 the image of the Galois representation ρE,p is not contained in the normalizer of a split Cartan subgroup. This gives a partial answer to an old question of Serre.

We present a family of congruences which hold if and only if a natural number n is prime. 2000 Mathematics Subject Classification: 11A51, 11A07. The subject of primality testing has been in the mathematical and general news recently, with the announcement [1] that there exists a polynomial-time algorithm to determine whether an integer p is prime or not. There are older deterministic primality ...

For any positive integer n, we define the function P (n) as the smallest prime p such that n | p!. That is, P (n) = min{p : n |p!, where p be a prime}. This function is a generalization of the famous Smarandache function S(n). The main purpose of this paper is using the elementary and analytic methods to study the mean value properties of P (n), and give two interesting mean value formulas for it.

Let ts,n be the n-th positive integer number which can be written as a power p, t ≥ s, of a prime p (s ≥ 1 is fixed). Let πs(x) denote the number of prime powers p, t ≥ s, not exceeding x. We study the asymptotic behaviour of the sequence ts,n and of the function πs(x). We prove that the sequence ts,n has an asymptotic expansion comparable to that of pn (the Cipolla’s expansion).

We investigate conditions which ensure that systems of binomial polynomials with integer coefficients are simultaneously free of large prime factors. In particular, for each positive number ", we show that there are infinitely many strings of consecutive integers of size about n, free of prime factors exceeding n, with the length of the strings tending to infinity with speed log log log log n. ...

This paper discusses the additive prime divisor function A(n) := ∑ pα||n αp which was introduced by Alladi and Erdős in 1977. It is shown that A(n) is uniformly distributed (mod q) for any fixed integer q > 1.

In this paper, we show that the diophantine equation Fn = p ± p has only finitely many positive integer solutions (n, p, a, b), where p is a prime number and max{a, b} ≥ 2.

Recently, Bhaintwal and Wasan studied the Generalized Reed-Muller codes over the prime power integer residue ring. In this paper, we give a generalization of these codes to Generalized Reed-Muller codes over Galois rings.