@AlgebraFact

Tweets about algebra, number theory, and miscellaneous math by @JohnDCook.

JohnDCook.com
Joined January 2010
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A matrix Q is orthogonal iff it has orthonormal columns (or rows).
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The NIST elliptic curves have been in the news twice lately. johndcook.com/blog/2026/10/2…

Backdoors, Incompetence, and Elliptic Curves

The NIST curve P-384 has been in the news twice lately, once for a result from Anthropic and once for incredibly bad key choices.

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For n >= 5, the alternating group A_n is simple.
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Greco-Latin squares and magic squares johndcook.com/blog/2022/01/1…

Greco-Latin squares | Mutually orthogonal Latin squares

Greco-Roman squares, orthogonal Latin squares, when they exist, and how they relate to magic squares.

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Volume to Area ratio for Regular Solids johndcook.com/blog/2026/07/2…
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If matrices A and B commute, then exp(A + B) = exp(A) exp(B).
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How NASA used Hadamard matrices to transmit photos from Mars johndcook.com/blog/2026/08/1…
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An n x n matrix A is non-singular iff the system Ax = b is consistent for all b.
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A subgroup N of G is normal if g^{-1}Ng = N for all g in G.
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New post: Refactoring Factorials johndcook.com/blog/2026/10/1…

Refactoring factorials

Introduction to the recent paper "Decomposing a factorial into large factors".

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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.
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Every positive integer can be written as a sum of distinct Fibonacci numbers.
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Geometric equations written as determinants with a column of 1s johndcook.com/blog/2022/12/1…

Geometric determinant formulas with projective coordinates

Why are equations in geometry often expressed as determinants with a column of 1s?

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If 2^p - 1 is prime, 2^{p-1}(2^p - 1) is an even perfect number. Euclid, Prob IX.36.
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An n x n matrix A is non-singular iff dim(range(A)) = n.
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Every permutation can be written as a product of disjoint cycles.
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Pascal's triangle with odd numbers highlighted
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LaTeX package for displaying Pascal-like triangles, such as the illustration above. johndcook.com/blog/2026/10/1…

LaTeX package for Pascal-like triangles

LaTeX package to make it easier to create illustrations of Pascal's triangle and similar triangles such as Bell's triangle, Leibniz' triangle, etc.

johndcook.com
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An n x n matrix A is non-singular iff the rows (columns) of A are linearly independent.
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The regions with the same color have the same area. johndcook.com/blog/2026/10/1…
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