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Willans' formula is a prime-generating formula due to Willan (1964) that is defined as follows. Let F(j) = |_cos^2[pi((j-1)!+1)/j]_| (1) = {1 for j=1 or j prime; 0 otherwise ...

A Gaussian sum is a sum of the form S(p,q)=sum_(r=0)^(q-1)e^(-piir^2p/q), (1) where p and q are relatively prime integers. The symbol phi is sometimes used instead of S. ...

The constant e^pi that Gelfond's theorem established to be transcendental seems to lack a generally accepted name. As a result, in this work, it will be dubbed Gelfond's ...

The integer sequence defined by the recurrence relation P(n)=P(n-2)+P(n-3) (1) with the initial conditions P(0)=P(1)=P(2)=1. This is the same recurrence relation as for the ...

A sequence which arises in the hypothetical reproduction of a population of rabbits. Let the substitution system map 0->1 correspond to young rabbits growing old, and 1->10 ...

Ramanujan's Dirichlet L-series is defined as f(s)=sum_(n=1)^infty(tau(n))/(n^s), (1) where tau(n) is the tau function. Note that the notation F(s) is sometimes used instead ...

A Wilson prime is a prime satisfying W(p)=0 (mod p), where W(p) is the Wilson quotient, or equivalently, (p-1)!=-1 (mod p^2). The first few Wilson primes are 5, 13, and 563 ...

The absolute difference of two numbers n_1 and n_2 is |n_1-n_2|, where the minus sign denotes subtraction and |x| denotes the absolute value.

A mathematical object S is said to be additively closed if a,b in S implies that a+b in S.

The first of several addends, or "the one to which the others are added," is sometimes called the augend. Therefore, while a, b, and c are addends in a+b+c, a is the augend.