Algebra Etc.

Every invertible complex matrix is the exponential of some complex matrix.

The number of twin prime pairs less than x is bounded by cx (log log x)² / (log x)² for some constant c and sufficiently large x.

Primitive Pythagorean triangles with equal area johndcook.com/blog/2025/07/3…

The only possible regular polygons with vertices at integer coordinates are squares.

Today's exponential sum

'One algorithmic idea in numerical linear algebra is more important than all the others: QR factorization.' -- Numerical Linear Algebra, Trefethen and Bau

Cramér’s conjecture: For consecutive primes p_n and p_(n+1), the gap satisfies p_(n+1) - p_n = O( (log p_n)² ).

Finding tangent circles johndcook.com/blog/2025/07/3…

Spanish mnemonic for π: Sol y Luna y Mundo proclaman al Eterno Autor del Cosmo.

Aligning one matrix with another The orthogonal Procrustes problem johndcook.com/blog/2026/02/1…

The quotient theorem for tensors: If inner products with T are always tensors, T is a tensor.

More here: johndcook.com/blog/2026/05/1…

It seems the inverse of a right shift should be a left shift. But shifts aren't invertible. But the pseudoinverse of a right sift is indeed a left shift.

Ring theory database ringtheory.herokuapp.com

Here is an example. With this A and B, (AB)^+ = 2 B^+ A^+. 2/2

The Moore-Penrose pseduoinverse A^+ of a matrix A acts a lot like an inverse. If A is invertible, the pseudoinverse equals the inverse. But the analog of the identity for the the inverse of a product doesn't hold in general. 1/2

Invertible binary matrices johndcook.com/blog/2026/05/1…

Bit twiddling operations correspond to multiplication by a matrix mod 2. For example, here's the matrix that corresponds to computing x ^= x >> 2 for an 8-bit number.

Reverse engineering the Mersenne Twister with Linear Algebra johndcook.com/blog/2026/05/1…

A local ring is a ring that has exactly one maximal ideal.