Algebra Etc.

A Toeplitz matrix is constant along each diagonal.

The direct product of a finite number of nilpotent groups is nilpotent.

If m and n are relatively prime, so are the Fibonacci numbers F_m and F_n.

Every free module over a ring with identity is projective.

Source: johndcook.com/expsum/

Today’s exponential sum

Trig mnemonic diagrams johndcook.com/blog/trig-mnem…

In 1693, Bernard de Bessy listed all 880 possible 4x4 magic squares.

New post: Powers don’t clear fractions johndcook.com/blog/2026/03/1…

Montgomery's trick for efficiently computing multiple modular inverses at once johndcook.com/blog/2026/01/1…

Group actions and tone rows. Do reversion, inversion, and rotation always produce distinct sequences of notes? johndcook.com/blog/2026/03/1…

最近, 下の関係式に気づいたが, もっと一般化された関係式がすでにあることを知った. #cite_note-17" target="_blank" rel="nofollow noopener">en.wikipedia.org/wiki/Faulhaber…

Continued fractions and the Euclidean algorithm albany.edu/~hammond/gellm… [pdf]

The first paper on category theory: 'Natural isomorphisms in group theory' by Eilenberg and Mac Lane pnas.org/content/pnas/2… [pdf]

Atonal music and group theory johndcook.com/blog/2026/03/1…

Answer here: johndcook.com/blog/2026/03/1…

Notice anything special about the following sequence? 8 6 10 3 1 11 1 3 6 8 12 9 7 10 4 2 5 11 2 4 7 9 5 12

Every periodic continued fraction is the continued fraction expansion of a real quadratic irrational number.

The continued fraction expansion of a real number is finite if and only if the number is rational.

Circulant matrices, eigenvectors, and the FFT johndcook.com/blog/2023/05/1…