The popular Mersenne Twister random number generator is so named because its period is a Mersenne prime.

New post: The number of points on an elliptic curve johndcook.com/blog/2026/10/2…

A matrix Q is orthogonal iff it has orthonormal columns (or rows).

The NIST elliptic curves have been in the news twice lately. johndcook.com/blog/2026/10/2…

For n >= 5, the alternating group A_n is simple.

Greco-Latin squares and magic squares johndcook.com/blog/2022/01/1…

Volume to Area ratio for Regular Solids johndcook.com/blog/2026/07/2…

If matrices A and B commute, then exp(A + B) = exp(A) exp(B).

How NASA used Hadamard matrices to transmit photos from Mars johndcook.com/blog/2026/08/1…

An n x n matrix A is non-singular iff the system Ax = b is consistent for all b.

A subgroup N of G is normal if g^{-1}Ng = N for all g in G.

New post: Refactoring Factorials johndcook.com/blog/2026/10/1…

An n × n real matrix B has a real logarithm if and only if B is nonsingular and for each negative eigenvalue λ and each positive integer k the number of k × k Jordan λ-blocks occurring in the Jordan form of B is even.

Every positive integer can be written as a sum of distinct Fibonacci numbers.

Geometric equations written as determinants with a column of 1s johndcook.com/blog/2022/12/1…

If 2^p - 1 is prime, 2^{p-1}(2^p - 1) is an even perfect number. Euclid, Prob IX.36.

An n x n matrix A is non-singular iff dim(range(A)) = n.

Every permutation can be written as a product of disjoint cycles.