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Let pi_(m,n)(x) denote the number of primes <=x which are congruent to n modulo m (i.e., the modular prime counting function). Then one might expect that ...

There are two related conjectures, each called the twin prime conjecture. The first version states that there are an infinite number of pairs of twin primes (Guy 1994, p. ...

k+2 is prime iff the 14 Diophantine equations in 26 variables wz+h+j-q=0 (1) (gk+2g+k+1)(h+j)+h-z=0 (2) 16(k+1)^3(k+2)(n+1)^2+1-f^2=0 (3) 2n+p+q+z-e=0 (4) ...

Legendre showed that there is no rational algebraic function which always gives primes. In 1752, Goldbach showed that no polynomial with integer coefficients can give a prime ...

Call a number of the form n^2-k a "near-square number." Numbers of the form n^2-5 for n=1, 2, ... are -4, -1, 4, 11, 20, 31, 44, 59, 76, 95, ... (OEIS A028875). These are ...

Let F_n be the nth Fibonacci number, and let (p|5) be a Legendre symbol so that e_p=(p/5)={1 for p=1,4 (mod 5); -1 for p=2,3 (mod 5). (1) A prime p is called a Wall-Sun-Sun ...

Dickson states "In a letter to Tanner [L'intermediaire des math., 2, 1895, 317] Lucas stated that Mersenne (1644, 1647) implied that a necessary and sufficient condition that ...

By way of analogy with the prime counting function pi(x), the notation pi_(a,b)(x) denotes the number of primes of the form ak+b less than or equal to x (Shanks 1993, pp. ...

A power floor prime sequence is a sequence of prime numbers {|_theta^n_|}_n, where |_x_| is the floor function and theta>1 is real number. It is unknown if, though extremely ...

A pair of prime numbers (p,q) such that p^(q-1)=1 (mod q^2) and q^(p-1)=1 (mod p^2). The only known examples are (2, 1093), (3, 1006003), (5 , 1645333507), (83, 4871), (911, ...