Prime numbers are one of the most basic objects in mathematics and one of the most basic questions is to decide which numbers are prime (a clearly related problem is to find the prime factorisation of a number). Given a number n one would like a quick way to decide if n is prime, say using a computer. Quick can be given a precise meaning. First one measures the complexity by the number of digit...

Let Ω(n) and ω(n) denote respectively the total number of prime factors and the number of distinct prime factors of the integer n. Euler proved that an odd perfect number N is of the form N = pem2 where p ≡ e ≡ 1 (mod 4), p is prime, and p ∤ m. This implies that Ω(N) ≥ 2ω(N) − 1. We prove that Ω(N) ≥ (18ω(N) − 31)/7 and Ω(N) ≥ 2ω(N) + 51.

It is sufficient to prove that there is an excess of prime factors in the product of repunits with odd prime bases defined by the sum of divisors of the integer N = (4k+1) ∏l i=1 q 2αi i to establish that there do not exist any odd integers with equality between σ(N) and 2N. The existence of distinct prime divisors in the repunits in σ(N) follows from a theorem on the primitive divisors of the ...

It is sufficient to prove that there is an excess of prime factors in the product of repunits with odd prime bases defined by the sum of divisors of the integer N = (4k + 1) ∏l i=1 q 2αi i to establish that there do not exist any odd integers with equality between σ(N) and 2N . The existence of distinct prime factors in the repunits in σ(N) follows from a theorem on the primitive divisors of th...

Given a graph G, a subset M of V (G) is a module of G if for each v ∈ V (G) ∖M , v is adjacent to all the elements of M or adjacent to none of them. For instance, V (G), ∅ and {v} (v ∈ V (G)) are modules of G called trivial. Given a graph G, ωM(G) (respectively αM(G)) denotes the largest integer m such that there is a module M of G which is a clique (respectively a stable) set in G with ∣M ∣ = ...

What is the first prime? It seems that the number two should be the obvious answer, and today it is, but it was not always so. There were times when and mathematicians for whom the numbers one and three were acceptable answers. To find the first prime, we must also know what the first positive integer is. Surprisingly, with the definitions used at various times throughout history, one was often...

Zaremba conjectured that given any integer m > 1 , there exists an integer a < m with a relatively prime to m such that the simple continued fraction [0, cx.cr] for a/m has c¡ < B for i = 1, 2,..., r, where B is a small absolute constant (say 5 = 5). Zaremba was only able to prove an estimate of the form c, < C log m for an absolute constant C . His first proof only applied to the case where m ...

We consider what one can prove about the distribution of prime numbers in arithmetic progressions, using only Selberg's formula. In particular, for any given positive integer q, we prove that either the Prime Number Theorem for arithmetic progressions, modulo q, does hold, or that there exists a subgroup H of the reduced residue system, modulo q, which contains the squares, such that (x; q; a) ...

We extend results of Videla and Fukuzaki to define algebraic integers in large classes of infinite algebraic extensions of Q and use these definitions for some of the fields to show first-order undecidability. In particular, we show that the following propositions hold. (1) For any rational prime q and any positive rational integer m, algebraic integers are definable in any Galois extension of ...

In 1857 Bouniakowsky [6] made a conjecture concerning prime values of polynomials that would, for instance, imply that x + 1 is prime for infinitely many integers x. Let ƒ (x) be a polynomial with integer coefficients and define the fixed divisor of ƒ, written d(ƒ), as the largest integer d such that d divides f(x) for all integers x. Bouniakowsky conjectured that if f(x) is nonconstant and irr...