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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 ...

The square-triangle theorem states that any nonnegative integer can be represented as the sum of a square, an even square, and a triangular number (Sun 2005), i.e., ...

Vizing's theorem states that a graph can be edge-colored in either Delta or Delta+1 colors, where Delta is the maximum vertex degree of the graph. A graph with edge chromatic ...

Let s_1, s_2, ... be an infinite series of real numbers lying between 0 and 1. Then corresponding to any arbitrarily large K, there exists a positive integer n and two ...

Minkowski's question mark function is the function y=?(x) defined by Minkowski for the purpose of mapping the quadratic surds in the open interval (0,1) into the rational ...

A labeling phi of (the vertices) of a graph G with positive integers taken from the set {1,2,...,r} is said to be r-distinguishing if no graph automorphism of G preserves all ...

A doublecross graph is a graph with graph crossing number 2. The numbers of doublecross simple graphs on n=1 nodes are 0, 0, 0, 0, 0, 1, 39, ..., and the numbers of connected ...

Let E be an elliptic curve defined over the field of rationals Q(sqrt(-d)) having equation y^2=x^3+ax+b with a and b integers. Let P be a point on E with integer coordinates ...

Consider a Lucas sequence with P>0 and Q=+/-1. A Fibonacci pseudoprime is a composite number n such that V_n=P (mod n). There exist no even Fibonacci pseudoprimes with ...

A theorem due to Conway et al. (1997) which states that, if a positive definite quadratic form with integer matrix entries represents all natural numbers up to 15, then it ...