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Defining p_0=2, p_n as the nth odd prime, and the nth prime gap as g_n=p_(n+1)-p_n, then the Cramér-Granville conjecture states that g_n<M(lnp_n)^2 for some constant M>1.

The totient function phi(n), also called Euler's totient function, is defined as the number of positive integers <=n that are relatively prime to (i.e., do not contain any ...

A composite number n is a positive integer n>1 which is not prime (i.e., which has factors other than 1 and itself). The first few composite numbers (sometimes called ...

A generalized Vandermonde matrix of two sequences a and b where a is an increasing sequence of positive integers and b is an increasing sequence of nonnegative integers of ...

A Münchhausen number (sometimes spelled Münchausen number, with a single 'h') is a number equal to the sum of its digits raised to each digit's power. Münchhausen numbers ...

Almost all processes that are not obviously simple can be viewed as computations of equivalent sophistication (Wolfram 2002, pp. 5 and 716-717). More specifically, the ...

The Fermat number F_n is prime iff 3^(2^(2^n-1))=-1 (mod F_n).

Chebyshev noticed that the remainder upon dividing the primes by 4 gives 3 more often than 1, as plotted above in the left figure. Similarly, dividing the primes by 3 gives 2 ...

In 1891, Chebyshev and Sylvester showed that for sufficiently large x, there exists at least one prime number p satisfying x<p<(1+alpha)x, where alpha=0.092.... Since the ...

The decimal expansion of a number is its representation in base-10 (i.e., in the decimal system). In this system, each "decimal place" consists of a digit 0-9 arranged such ...