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Given an integer e>=2, the Payam number E_+/-(e) is the smallest positive odd integer k such that for every positive integer n, the number k·2^n+/-1 is not divisible by any ...

A general reciprocity theorem for all orders which covered all other known reciprocity theorems when proved by E. Artin in 1927. If R is a number field and R^' a finite ...

An algorithm that can be used to factor a polynomial f over the integers. The algorithm proceeds by first factoring f modulo a suitable prime p via Berlekamp's method and ...

The cototient of a positive number n is defined as n-phi(n), where n is the totient function. It is therefore the number of positive integers <=n that have at least one prime ...

Given binomial coefficient (N; k), write N-k+i=a_ib_i, for 1<=i<=k, where b_i contains only those prime factors >k. Then the number of i for which b_i=1 (i.e., for which all ...

Eisenstein's irreducibility criterion is a sufficient condition assuring that an integer polynomial p(x) is irreducible in the polynomial ring Q[x]. The polynomial ...

A number n is called equidigital if the number of digits in the prime factorization of n (including powers) uses the same number of digits as the number of digits in n. The ...

An object is unique if there is no other object satisfying its defining properties. An object is said to be essentially unique if uniqueness is only referred to the ...

Let U(P,Q) and V(P,Q) be Lucas sequences generated by P and Q, and define D=P^2-4Q. (1) Then {U_((n-(D/n))/2)=0 (mod n) when (Q/n)=1; V_((n-(D/n))/2)=D (mod n) when (Q/n)=-1, ...

A special case of the Artin L-function for the polynomial x^2+1. It is given by L(s)=product_(p odd prime)1/(1-chi^-(p)p^(-s)), (1) where chi^-(p) = {1 for p=1 (mod 4); -1 ...