# When Five Isn’t Prime

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2023-09-18 18:30:07

How far would you go to save a theorem? Would you invent a new kind of number? That’s what the mid-19th century German mathematician Ernst Eduard Kummer did, and while he was partly driven by the hope of proving Fermat’s Last Theorem, that wasn’t actually the theorem he was trying to save.

Most readers of this essay will already be familiar with Fermat’s Last Theorem (“FLT” for short), but everyone has a first time learning about FLT, and this essay may be yours, so I’ll remind/inform you that Fermat’s Last Theorem is the infamous assertion that if n is some positive integer bigger than 2, then the relation an + bn = cn can’t have a solution in which a, b, and c are non-zero integers.1 Fermat probably proved the claim for n = 4, and later mathematicians proved it for other specific values of the exponent n (focussing on prime exponents, since if you can rule out all the primes bigger than 2 as possible exponents, the general assertion will follow2). FLT is indeed a theorem now (thanks to Andrew Wiles, Richard Taylor, Ken Ribet and many others) but it technically shouldn’t be called Fermat’s theorem because it’s unlikely that Fermat proved it (even though he claimed he’d found a proof and a marvelous one at that); see my essay The Curious Incident of the Boasting Frenchman.

What’s indisputably true is that in March of 1847, roughly two centuries after Fermat made his boast, the French mathematician Gabriel Lamé underwent the mathematician’s equivalent of finding oneself suddenly naked in public: he sketched a proposed proof of Fermat’s claim to the Paris Academy only to have his idea publicly shot down minutes later by a colleague he admired. The colleague was Joseph Liouville, whose work Lamé had acknowledged in his presentation as a source of inspiration. As if that wasn’t enough of a blow, Liouville suggested that Lamé’s approach wasn’t just wrong; it was also a rather obvious thing to try. Ouch! Liouville may have been unkind but he was right; Leonhard Euler had made his own version of Lamé’s mistake a century earlier in a failed proof of the special case n = 3 of FLT.3