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Let sigma(n) be the divisor function. Then lim sup_(n->infty)(sigma(n))/(nlnlnn)=e^gamma, where gamma is the Euler-Mascheroni constant. Ramanujan independently discovered a ...

Lehmer's totient problem asks if there exist any composite numbers n such that phi(n)|(n-1), where phi(n) is the totient function? No such numbers are known. However, any ...

An Euler-Jacobi pseudoprime to a base a is an odd composite number n such that (a,n)=1 and the Jacobi symbol (a/n) satisfies (a/n)=a^((n-1)/2) (mod n) (Guy 1994; but note ...

A prime number is said to be weakly prime if changing a single digit to every other possible digit produces a composite number when performed on each digit. The first few ...

Let U(P,Q) and V(P,Q) be Lucas sequences generated by P and Q, and define D=P^2-4Q. (1) Let n be an odd composite number with (n,D)=1, and n-(D/n)=2^sd with d odd and s>=0, ...

The twin composites may be defined by analogy with the twin primes as pairs of composite numbers (n,n+2). Since all even number are trivially twin composites, it is natural ...

A primality test that provides an efficient probabilistic algorithm for determining if a given number is prime. It is based on the properties of strong pseudoprimes. The ...

Van der Corput sequences are a means of generating sequences of points that are maximally self-avoiding (a.k.a. quasirandom sequences). In the one-dimensional case, the ...

A Fermat pseudoprime to a base a, written psp(a), is a composite number n such that a^(n-1)=1 (mod n), i.e., it satisfies Fermat's little theorem. Sometimes the requirement ...

A compositeness certificate is a piece of information which guarantees that a given number p is composite. Possible certificates consist of a factor of a number (which, in ...