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Given a matrix A, let |A| denote its determinant. Then |A||A_(rs,pq)|=|A_(r,p)||A_(s,q)|-|A_(r,q)||A_(s,p)|, (1) where A_(u,w) is the submatrix of A formed by the ...

A (-1,0,1)-matrix is a matrix whose elements consist only of the numbers -1, 0, or 1. The number of distinct (-1,0,1)-n×n matrices (counting row and column permutations, the ...

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 ...

Fermat's 4n+1 theorem, sometimes called Fermat's two-square theorem or simply "Fermat's theorem," states that a prime number p can be represented in an essentially unique ...

A number n for which the harmonic mean of the divisors of n, i.e., nd(n)/sigma(n), is an integer, where d(n)=sigma_0(n) is the number of positive integer divisors of n and ...

A positive integer which is divisible by the sum of its digits, also called a Niven number (Kennedy et al. 1980) or a multidigital number (Kaprekar 1955). The first few are ...

A number n is called k-hyperperfect if n = 1+ksum_(i)d_i (1) = 1+k[sigma(n)-n-1], (2) where sigma(n) is the divisor function and the summation is over the proper divisors ...

The reversal of a positive integer abc...z is z...cba. The reversal of a positive integer n is implemented in the Wolfram Language as IntegerReverse[n]. A positive integer ...

The square torus is the quotient of the plane by the integer lattice.

The abundancy of a number n is defined as the ratio sigma(n)/n, where sigma(n) is the divisor function. For n=1, 2, ..., the first few values are 1, 3/2, 4/3, 7/4, 6/5, 2, ...