It is curious how aggregates rather than single instances creep into our subject even when we aren't looking for statistical trouble. Here is an example. In the Erdös spirit, I'll offer a $5 prize for anyone who can manage to provide a proof of the fact that • every linear form aX+b with a, b ∈ Z relatively prime represents at least one prime number; and yet • the proof doesn't actually show th...

This paper will prove that essentially only the obvious recurrences have almost all primes as divisors. An integer n is a divisor of a recurrence if n divides some term of the recurrence. In this paper, "almost all primes" will be taken interchangeably to mean either all but finitely many primes or all but for a set of Dirichlet density zero in the set of primes. In the context of this paper, t...

Abstract: We present three remarks on Goldbach’s problem. First we suggest a refinement of Hardy and Littlewood’s conjecture for the number of representations of 2n as the sum of two primes positing an estimate with a very small error term. Next we show that if a strong form of Goldbach’s conjecture is true then every even integer is the sum of two primes from a rather sparse set of primes. Fin...

Abstract Let denote the maximal number of different primes that may occur in order a finite solvable group G , all elements which have orders divisible by at most n distinct primes. We show for . As an application, we improve on recent bound Hung and Yang arbitrary groups.