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Algebra Etc.

@AlgebraFact

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

JohnDCook.com
Joined January 2010
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Parallel addition, relativistic addition, and points at infinity johndcook.com/blog/2020/12/3…

Whether to add one or two infinities to the real numbers

Sometimes you want to add one infinity to the real numbers and sometimes you want to add two. This post gives several examples of each.

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A linear operator T is idempotent if T² = T.
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An n x n matrix A is non-singular iff dim(kernel(A)) = 0.
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A category has finite products and equalizers iff it has pullbacks and a terminal object.
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I have a truly marvelous Lean certificate for this proposition which this context-window is too narrow to contain.
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If an odd perfect number exists, it must have more than 2000 digits.
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The unbalanced group theorem johndcook.com/blog/2026/10/2…

The Unbalanced Group Theorem

The unbalanced group theorem is a key theorem in the classification of finite simple groups. It's proof was wrong but the theorem is true.

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A group is called simple if it has no normal subgroups other than the identity and the group itself.
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Algebra Etc. retweeted
Axiom (@axiommathai) profile picture
Axiom
@axiommathaiOct 27
The largest math proof in human history is now computer-verified. Today, Axiom announces a complete Lean auto-formalization of the Classification of Finite Simple Groups, the "periodic table of symmetry." About 15,000 pages, built by over 100 mathematicians across generations.
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Costas arrays johndcook.com/blog/2022/09/1…

Costas arrays

A Costas array is a variation on the n queens problem. This post generates and visualizes solutions.

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Hiding data in permutations johndcook.com/blog/2026/07/2…
Image from Algebra Etc. (@AlgebraFact)
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It is unknown whether odd perfect numbers exist.
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New post: Inversion in an ellipse johndcook.com/blog/2026/10/2…

Pole and polar with respect to an ellipse

Inversion in an ellipse. Pole and polar with respect to an ellipse. Duality between focus and directrix.

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A linear operator T is nilpotent if T^k = 0 for some positive integer k.
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New post: Poles and Polars johndcook.com/blog/2026/10/2…

Poles and polars in spherical, plane, and projective geometry

Duality in spherical and plane geometry. The dual of a line is called a pole, and the dual of a point is called a polar. Extension to projective geometry.

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Algebra Etc. retweeted
Mathelirium (@mathelirium) profile picture
Mathelirium
@matheliriumOct 24
For 130 years we only knew the zeros of the Riemann zeta function sit somewhere strictly between 0 and 1. Now, a proof has squeezed every one of them into the band from 1/8 to 7/8. Most of what the full Riemann hypothesis would tell us about the primes already follows from that, and the proof needed no new ideas, just old tools everyone thought were too weak. 
See below for link
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Computing polar triangles with linear algebra johndcook.com/blog/2026/10/2…
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Duality in spherical trig. Why, for example, there are two laws of cosines. johndcook.com/blog/2026/10/2…

Duality in spherical trigonometry

Duality in spherical trigonometry. Why where are two laws of cosines, for example, and why every identity has a dual counterpart.

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It is unknown whether the number of Mersenne primes is finite.
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Kan's paper introducing adjoint functors ams.org/journals/tran/1958-0… [pdf]
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