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A lucky number of Euler is a number p such that the prime-generating polynomial n^2-n+p is prime for n=1, 2, ..., p-1. Such numbers are related to the imaginary quadratic ...

Levy (1963) noted that 13 = 3+(2×5) (1) 19 = 5+(2×7), (2) and from this observation, conjectured that all odd numbers >=7 are the sum of a prime plus twice a prime. This ...

A Størmer number is a positive integer n for which the greatest prime factor p of n^2+1 is at least 2n. Every Gregory number t_x can be expressed uniquely as a sum of t_ns ...

A set S of positive integers is said to be Diophantine iff there exists a polynomial Q with integral coefficients in m>=1 indeterminates such that ...

An Artin L-function over the rationals Q encodes in a generating function information about how an irreducible monic polynomial over Z factors when reduced modulo each prime. ...

A p-adic number is an extension of the field of rationals such that congruences modulo powers of a fixed prime p are related to proximity in the so called "p-adic metric." ...

For an integer n>=2, let lpf(n) denote the least prime factor of n. A pair of integers (x,y) is called a twin peak if 1. x<y, 2. lpf(x)=lpf(y), 3. For all z, x<z<y implies ...

A primitive root of a prime p is an integer g such that g (mod p) has multiplicative order p-1 (Ribenboim 1996, p. 22). More generally, if GCD(g,n)=1 (g and n are relatively ...

Bertrand's postulate, also called the Bertrand-Chebyshev theorem or Chebyshev's theorem, states that if n>3, there is always at least one prime p between n and 2n-2. ...

The conjecture proposed by Catalan in 1888 and extended by E. Dickson that each aliquot sequence ends in a prime, a perfect number, or a set of sociable numbers. The ...