Fermat's Other Theorems
Three riffs on Fermat: a trivially true 'first theorem,' his little theorem in tiny and huge sizes, and a will asking his son not to publish the famous unproved conjecture.
Content note: Mild language ('half-assed', 'friggin').
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Cartoon by Ben Orlin, Math with Bad Drawings. https://cartoons.mathwithbaddrawings.com/2025-08-07-fermat-s-other-theorems/ (CC BY-NC 4.0)Transcript
[1] FERMAT'S FIRST THEOREM: aⁿ + bⁿ = cⁿ has no solutions satisfying the condition 1 < c < a < b < n. Fermat: "I have a proof of this fact, but honestly, this margin is WAAAY too big for it."
[2] FERMAT'S LITTLE THEOREM: If p is prime, then aᵖ − a is a multiple of p. EX: 10¹³ − 10 is a multiple of 13. / FERMAT'S TINY THEOREM (same, in tiny print). EX: 2³ − 2 is a multiple of 3. / FERMAT'S HUGE THEOREM: Get this: if p is prime, then aᵖ − a is, no joke, a friggin' *exact multiple* of p. EX: 1000⁷⁹¹⁹ − 1000 is a multiple of 7919.
[3] FERMAT'S LAST WILL AND TESTAMENT: "I leave to my son Clément-Samuel all of my papers, which he should feel free to publish. Well… all except that one half-assed conjecture I couldn't actually prove. Definitely don't publish that one. (I'd specify further, but this margin is too narrow.)"