Fermat numbers and the infinitude of primes - MathOverflow

Fermat numbers and the infinitude of primes

2010-04-23T08:24:57Z

Wonder whether any of you guys know why it is that the proof of the infinitude of primes that is based on the coprimality of any pair of (distinct) Fermat numbers is commonly attributed to Pólya.

In the first paragraph of this letter from Golbach to Euler there is already an argument along those lines, but since documents crediting it to Professor Pólya are not rare out there, it seems like it's passed unnoticed by a nonzero number of persons.

So, what do you think about this? It's not like Fermat numbers are essential to the proof or that there are no other demonstrations of the result... It's just that I'd really like to know about the origins of this discrepancy between the sources.

Answer by John Stillwell for Fermat numbers and the infinitude of primes

2010-04-23T10:39:08Z

It's interesting that the coprimality of Fermat numbers was already known in Goldbach's time. The reason for attributing the proof to Polya is presumably that such a proof is indicated as an exercise in Polya and Szego (1924). Because of this, Ribenboim, in his Little Book of Big Primes calls it "Polya's proof." Maybe the rumor started there.

[Added later] In the light of the comments that have come in, it now looks to me as though 1. Goldbach could have observed that he had a proof of the infinitude of primes, but didn't care to mention it, and 2. that the attribution of this observation to Polya starts with Hardy.

Re 1. In the 18th century, were people interested in finding new proofs of the infinitude of primes? For example, when Euler proved that $\Sigma 1/p=\infty$ (paper E72 in the Euler Archive) he did not remark that this gives a new proof of the infinitude of primes. It could very well be that Goldbach did not consider it interesting to prove again that there are infinitely many primes.

Re 2. One should bear in mind that Hardy knew Polya well. Polya visited him in England just after the publication of Polya & Szego and collaborated with him on the book Inequalities, published in 1934 ( four years before H&W). So Hardy could well have learned the proof directly from Polya.

Answer by Álvaro Lozano-Robledo for Fermat numbers and the infinitude of primes

2010-05-07T22:28:35Z

Hello,

As far as I know, the problem began with Hardy and Wright's "An introduction to the theory of numbers", first published in 1938. Indeed, in Section 2.4, page 14, they write

Second proof of Euclid’s theorem. Our second proof of Theorem 4, which is due to Polya, depends upon a property of what are called ‘Fermat’s numbers’...

Since Hardy and Wright's book has always been so popular, I suspect that many have given credit to Pólya, following their words.

Notice, however, that Dickson's 1952 "History of the theory of numbers" correctly attributed the theorem back to Goldbach (see p. 375 of Volume I):

Chr. Goldbach called Euler's attention to Fermat's conjecture that $F_n$ is always prime, and remarked that no $F_n$ has a factor $<100$; no two $F_n$ have a common factor.

Answer by J. H. S. for Fermat numbers and the infinitude of primes

2010-05-08T03:35:19Z

@Álvaro:

Agreed that a proof of the coprimality of any pair of distinct Fermat numbers appears in the very first paragraph of the aforementioned missive from Goldbach to Euler. That is not under discussion here. Thing is that, as Professor Lemmermeyer noted above, Goldbach himself did not seem to notice that this result would (immediately) provide him with a proof of the infinitude of the primes. As I commented before, one of my initials beliefs on this matter was that the exclamation "at quantulum hoc est ad demonstrandum omnes illos numeros esse absolute primos?" in the July 20th letter was somehow implying that Golbach had actually found the connection between both facts. Yet, your knowledgeable comments have just made me change my mind on this wrong impression that I initially had.
You are absolutely right when you express that the proof given by Hardy and Wright passes through the argument given by Goldbach in his letter to Euler. That's the reason that I said it is kind of weird to see H & W adscribing the result to Pólya.

Answer by Andres Caicedo for Fermat numbers and the infinitude of primes

2010-05-29T05:47:23Z

I am quoting from the nice book "The development of Prime Number Theory" by W. Narkiewicz, Springer (2000), pg. 8.

Any infinite sequence of pairwise coprime positive integers leads to a proof of [the infinitude of primes]. Such a proof first appears in a letter of C.Goldbach to Euler dated July 20, 1730 [footnote: The original date is July 20/31, the double dating being a consequence of the use of the Julianic calendar in Russia before 1918. It seems that this was the first proof of the infinitude of primes which essentially differed from that of Euclid.] (see Fuss 1843, I, 32-34; Euler-Goldbach 1965) and is sometimes attributed to G.Pólya (e.g. in Hardy, Wright (1960), Chandrasekharan (1968). P.Ribenboim (Nombres premiers: mystères et records. 1994) wrote that this attribution appears in an unpublished list of exercises of A.Hurwitz preserved in ETH in Zürich.) This proof was published in the well-known collection of exercises of G.Pólya and G.Szegö (1925).

What is interesting here is that Hurwitz died in 1919, prior to Hardy & Wright, and to Pólya & Szegő, so it is likely that Pólya rediscovered the argument on his own, unaware of Goldbach's letter, presented it to colleagues, and they would naturally attribute it to him.

Answer by Franz Lemmermeyer for Fermat numbers and the infinitude of primes

2011-12-29T18:37:10Z

On p. 167 of Beiträge zur Zahlentheorie, insbesondere zur Kreis- und Kugeltheilung, mit einem Nachtrage zur Theorie der Gleichungen (1891), Scheffler deduces the infinitude of primes from the fact that Fermat numbers are pairwise coprime. I don't think that Scheffler's book was widely read, however.