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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.

actinidia
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    Without the LLT I'd say it is strange, so did he really prove it ? Maybe he just checked $a^{M_p-1} \equiv 1 \bmod M_p$ for many $a$ and considered it as a proof. – reuns Dec 18 '17 at 15:07
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    It wasn't Euler, according to wikipedia: "$M_{61}$ 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." – Daniel Fischer Dec 18 '17 at 15:12
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    For two primes, $p,q$ we have $2^q-1\equiv 0 \pmod p\implies q,|, p-1$ so you only need to check primes $p$ which are $1\pmod {61}$. That greatly reduces your search. – lulu Dec 18 '17 at 15:15
  • @DanielFischer I'm not sure why I was thinking it was Euler! Thank you for that, I've edited my answer to reflect this. – actinidia Dec 18 '17 at 15:19
  • Here is an excerpt from a letter from Euler to Bernoulli on the primality of $2^{31}-1$. In it, he describes a way to strengthen the congruence requirements on a potential prime divisor, but if I am reading correctly he declares that the method relies on some unproven conjectures. – lulu Dec 18 '17 at 15:21
  • @lulu "Ces règles sont fondées sur un principe dont la démonstration n'est encore connue." I'm very surprised that he'd just list out unsubstantiated cases like that, especially considering how deceptive number patterns can be. Ironically, he even mentions the deceptive polynomial $x^2-x+41$! But I suppose being Euler makes it easier to pull that off. Or perhaps (hopefully) his conjecture wasn't based in patterns, but rather in underlying theory. – actinidia Dec 18 '17 at 15:26
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    Oh, I'd say this style of writing is perfectly modern. It's not unusual, today, to see papers in number theory (or other topics) in which a "proof" is constructed by assuming the truth of a number of outstanding conjectures. So long as you are clear about which conjectures you are assuming, this is entirely acceptable. In contrast, Fermat and others of his time were comfortable with asserting that a small number of examples proved a result (as witness the supposed primality of the Fermat numbers...a conjecture which Fermat himself had the tools to disprove). – lulu Dec 18 '17 at 15:30
  • Euler was 18th century, not 17th. For a long time M(127) was the largest explicitly known prime. – DanielWainfleet Dec 18 '17 at 23:43
  • @DanielWainfleet Fixed. But that doesn't mean that Euler would have known all the primes below $M_{127}$ (or would it)? – actinidia Dec 18 '17 at 23:51
  • This might belong in hsm.se. – J.G. Dec 19 '17 at 00:38
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    The Wikipedia article on Mersenne primes has historical details. Mersenne did not claim that $M(p$) is composite for all $p>257.$ ($That $ would have raised some eyebrows!) . Euler showed that $M(31)$ is composite. There are about $150$ to $160$ primes less than $\sqrt {M(31)}$ that are congruent to $1$ mod $62,$ so $M(31)$ can be tested manually.... In "Mathematical Recreations and Essays" by Rousse-Ball & Coxeter, it says that Mersenne's statement "is not as impressive as it seems, for it contains 5 mistakes". – DanielWainfleet Dec 19 '17 at 01:58
  • Even the set of primes below$ \sqrt {M(127)}$ that are congruent to $1$ mod $62$ has on the order of $2^{52}\approx 4\times 10^{15}$ members. Not even Ramanujan could know them all. – DanielWainfleet Dec 19 '17 at 02:05
  • @DanielWainfleet Euler showed that $M(31)$ is composite ? Barely because it is prime. – Peter Mar 13 '18 at 21:18
  • @DanielWainfleet I don't know when the Lucas-Lehmer test was found. With this, it would be possible to show that $M_{127}$ is prime with a table calculator or even by hand, although it would be an enormous task. – Peter Mar 13 '18 at 21:22
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    More impressive than the primality-tests is that someone (I don't remember his name) has factored $2^{67}-1$ by hand and needed the Sundays of three years to find the $9$ digit-factor! – Peter Mar 13 '18 at 21:25
  • @Peter . See Lucas-Lehmer Prime Test in Wikipedia – DanielWainfleet Mar 14 '18 at 06:53

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As others have pointed out, it wasn't Euler who first proved $M61$ to be prime, but the Russian mathematician-priest L.M. Pervushin (or as it appeared in contemporary printed literature, Pervouchine).

He sent his "proof" to the Imperial Academy Of Sciences of Saint-Petersbourg in 1883, however this paper is unavailable to the general public at this time (it is housed in the Academy archives and no scan of the document has been made public). On the other hand, a short note in the 1887 edition of the Bulletin de l'Académie Impériale des Sciences de Saint-Pétersbourg, tome XXXI, columns 532-533, entitled "Sur un nouveau nombre premier, annoncé par le père Pervouchine" reveals the following interesting points (in original French and corresponding English translation):

En novembre de l'année 1883, dans la correspondance de notre Académie se trouve une communication qui contient l'assertion que le nombre

$$ 2^{61}-1 = 2305843009213693951 $$

est un nombre premier. Ce résultat nous a été annoncé par son auteur, L.M. Pervouchine, prêtre à Chadriusk (gouvernement de Perm), qui nous avait envoyé auparavant des communications intéressantes sur quelques propositions arithmétiques analogues.

In English:

In November of the year 1883, within the communications of our Academy there is a letter containing the assertion that the number

$$ 2 ^ {61} -1 = 2305843009213693951 $$

is a prime number. This result was announced to us by its author, L.M. Pervouchine, priest at Chadriusk (Perm state), who had previously sent us interesting papers on some similar arithmetic propositions.

Disappointingly, the Academy then goes on to state the following:

(...) laissant à la charge de l'auteur la responsabilité pour l'exactitude du résultat qu'il a obtenu au bout de ses longs et fatiguants calculs.

In English:

(...) leaving the responsibility of the result's accuracy to the author, which he obtained after a series of long and tiresome calculations.

Finally, a note on the letter itself:

Le manuscrit du père Pervouchine contenant sa communication de l'année 1883 (...) est accompagné de de quelques tables, calculées par l'auteur, et destinées à faciliter la vérification du résultat qu'il a obtenu.

In English:

The manuscript of father Pervouchine, containing his 1883 letter (...) is accompanied by some tables, calculated by the author, and intended to facilitate the verification of the result he obtained.

So it would appear that the method of calculation was indeed a heavily manual task, parts of which were recorded in tables and sent in a letter to the Academy. It looks like there was no peer review of these calculations, at least not up to 1887.

The note "Sur un nouveau nombre premier, annoncé par le père Pervouchine", Bulletin de l'Académie Impériale des Sciences de Saint-Pétersbourg, tome XXXI, columns 532-533, can be found here.

Also noteworthy is this interesting 1987 paper by Guy Haworth that details the history of the discovery of Mersenne primes.

Klangen
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    Knuth in Volume 2 of TAOCP says “see Istoriko-Mat. Issledovanii͡a 6 (1953), 559” — maybe that goes into more details and is a good place to look? – ShreevatsaR Mar 27 '19 at 02:13