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Let p be an odd prime, k be an integer such that pk and 1<=k<=2(p+1), and N=2kp+1. Then the following are equivalent 1. N is prime. 2. There exists an a such that ...

A generalization of the Gaussian sum. For p and q of opposite parity (i.e., one is even and the other is odd), Schaar's identity states ...

Let sigma_0(n) and sigma_1(n) denote the number and sum of the divisors of n, respectively (i.e., the zeroth- and first-order divisor functions). A number n is called sublime ...

Given a sequence of real numbers a_n, the supremum limit (also called the limit superior or upper limit), written lim sup and pronounced 'lim-soup,' is the limit of ...

Let N be an odd integer, and assume there exists a Lucas sequence {U_n} with associated Sylvester cyclotomic numbers {Q_n} such that there is an n>sqrt(N) (with n and N ...

Consider a symmetric triangle wave T(x) of period 2L. Since the function is odd, a_0 = 0 (1) a_n = 0, (2) and b_n = (3) = (32)/(pi^2n^2)cos(1/4npi)sin^3(1/4npi) (4) = ...

Given relatively prime integers p and q (i.e., (p,q)=1), the Dedekind sum is defined by s(p,q)=sum_(i=1)^q((i/q))(((pi)/q)), (1) where ((x))={x-|_x_|-1/2 x not in Z; 0 x in ...

A pseudoprime is a composite number that passes a test or sequence of tests that fail for most composite numbers. Unfortunately, some authors drop the "composite" ...

An automorphic graph is a distance-transitive graph Gamma for which the automorphism group Aut(Gamma) acts primitively on the vertices of Gamma and is not a complete graph or ...

A binomial number is a number of the form a^n+/-b^n, where a,b, and n are integers. Binomial numbers can be factored algebraically as ...