Greg Egan on Nostr: In 1909, Arthur Wieferich proved that if 𝑥ᵖ+𝑦ᵖ=𝑧ᵖ for an odd prime ...

In 1909, Arthur Wieferich proved that if

𝑥ᵖ+𝑦ᵖ=𝑧ᵖ

for an odd prime 𝑝 that does not divide positive integers 𝑥, 𝑦, or 𝑧, then

𝑝² must divide 2ᵖ⁻¹−1

Fermat’s little theorem says 𝑝 itself always divides 2ᵖ⁻¹−1.

Soon afterwards, Dmytro Grave checked every prime under 1000, and conjectured that no such “Wieferich Primes” exist!

Then in 1913, Waldemar Meissner checked 1093 …

Now that we have computers this is trivial, but in 1913 it was a slog. Meissner showed that 2³⁶⁴−1 was divisible by 1093², and since 1093 – 1 = 3 × 364, and 𝑡³−1 is divisible by 𝑡−1, that’s enough.

Later, Emil Haentzschel pointed out that 2¹⁸²+1 is divisible by 1093², which does the same job because 𝑡⁶−1 is divisible by 𝑡+1. Just how annoying this was to Meissner has not been recorded.