Everyone knows that there is an infinitude of primes. I know the Euclide, the Euler and the Erdos proofs. But are they the only known proofs ? I will try, here, to present a fourth one : Let the Fermat numbers $F_n=2^{2^n}+1$. They are not all primes, $F_5$ for example is not, Euler has proved it ! There are two possibilities, of course : if there is an infinitude of prime Fermat numbers, the primes are infinite. If there is an infinitude of compound Fermat numbers, the Goldbach theorem states that $\gcd(F_n,F_m)=1$ if $n\neq m$. So $F_n=p_1^{n_1}\cdots p_i^{n_i}$ and $F_m=q_1^{m_1}\cdots q_j^{m_j}$ and $p_k\neq{q_j},\forall{j,k}$. Thus $p_1=F_n^{1/n_1}p_2^{-n_2/n_1}\cdots p_i^{-n_i/n_1}$ are infinite and there is always an infinitude of primes! Do you know another proof mentioned in the literature, please? Thank you.
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We don't need to break into cases. For each Fermat number $F_m$, let $p_m$ be its smallest prime factor. The $p_m$ are all distinct. – André Nicolas Aug 21 '14 at 16:50
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I've always liked this one: http://www.cut-the-knot.org/proofs/InjectivePrimes.shtml – forallepsilon Aug 21 '14 at 16:58
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There are hundreds of proofs of the infinitude of primes. I'm certain someone has compiled a large list somewhere. Here is a smaller list that I found on the first page of a google search. – davidlowryduda Aug 21 '14 at 17:14
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"Infinitely many" or "an infinite set of", not "an infinity of". – Aug 21 '14 at 17:33
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You can simplify your proof a bit: since all $F_n$ are coprime and every number has a unique prime decomposition, there are necessarily an infinity of primes, whether or not the $F_n$ are themselves primes (just pick one prime factor of $F_n$, for each $n$). – Jean-Claude Arbaut Aug 21 '14 at 17:42
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1@Hurkyl The first dictionary I checked gives, as one meaning of "infinity", "an infinite or very great number or amount". So I think "an infinity of primes" is quite correct. – Andreas Blass Aug 21 '14 at 17:44
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@Andreas: I don't doubt that that misuse is sufficiently common to make it into a dictionary -- but it is still a misuse that is one source of confusion people have about the topic, and worth correcting. (in particular, no mathematical usage of 'infinity' shares this meaning) – Aug 21 '14 at 17:53
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2@Hurkyl Googling "an infinity of" I found lots of usesof this construction. Then, restricting the search by adding "math", I still found lots of uses and, on the few Google pages that I actually looked into, no claims that it's a misuse except for your comment here. – Andreas Blass Aug 21 '14 at 18:17
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1My hunch is that any rigorous book about prime numbers probably has at least one other proof besides the classic one. Somewhere I read a proof that I thought was inelegant because it requires proving the fundamental theorem of arithmetic first, and I don't know, the infiniteness of primes seems to me like a more elementary thing than unique factorization. – Mr. Brooks Aug 21 '14 at 21:25
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Related https://math.stackexchange.com/questions/50006/different-ways-to-prove-there-are-infinitely-many-primes – Henry Aug 04 '20 at 00:04
3 Answers
This page at MIT purports to be an attempt to collect as many proofs as possible. But I suspect it has been left unfinished because it's not very long and there are lots of proofs of this theorem.
This page presents several proofs and could probably be expanded.
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The key idea of Euclid's classical proof is that we can construct an infinite sequence of primes from any infinite sequence of coprimes, e.g. give an increasing sequence of naturals $\,f_n > 1\,$ that are pair-coprime, i.e. $\,(f_i,f_j) = 1\,$ for $\,i\ne j,\,$ then choosing $\,p_i\,$ to be a prime factor of $\,f_i\,$ yields an infinite sequence of primes, since the $\,p_i\,$ are distinct: $\,p_i\ne p_j,\,$ being factors of coprimes $\,f_i,\, f_j\,$.
Thus, since the Fermat numbers enjoy such coprimality, they yield an infinite sequence of primes. Ribenboim says the proof using the sequence of Fermat numbers is due to Goldbach (1730), and the idea of using coprimes was used in an exercise by Hurwitz (1891).
You can find these references, and many other proofs that there are infinitely many primes in the first chapter of Ribenboim's The new Book of Prime Number Records.
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There are six nice proofs presented in the first chapter of Proofs from THE BOOK, including one similar to yours using Fermat numbers.
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