In 1644, Mersenne made the following conjecture:
The Mersenne numbers, $M_n=2^n−1$, are prime for $n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257$, and no others.
Euler found that the Mersenne number $M_{61}$ is prime, refuting the conjecture.
For context, $M_{61} = 2 305 843 009 213 693 951$. I imagine that this would be incredibly large for most 18th-century number theorists.
Thus, a natural question is: Do we know how Euler proved this? From what I've read, he wasn't Ramanujan-like in his results. Indeed, he tended to have proofs for such things, even if he never published/mentioned them (unless to show colleagues that he had already derived their published results years before them). Yet, I also doubt that he checked primes up to $\sqrt{M_{61}}$.
(And if it was indeed a case of mathematical mysticism, how could one use non-Eulerian cleverness to offer an alternative disproof? )
Edit: As Daniel Fischer commented, it actually wasn't Euler! "$M_{61}$ was determined to be prime in 1883 by Ivan Mikheevich Pervushin, though Mersenne claimed it was composite, and for this reason it is sometimes called Pervushin's number," according to Wikipedia. It was disproven a century later, but I suppose it would still be useful to know how it was disproved.