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In a video I watched last night on nuking mathematical mosquitos, Matt Parker gave the following proof of the infinitude of primes: suppose there are finitely many primes. The Green-Tao theorem says there are arbitrarily long arithmetic progressions in the primes, hence there cannot be finitely many primes. Contradiction.

Leaving aside the slightly dubious and unnecessary use of proof by contradiction, it made me wonder whether or not this proof was circular (and Parker himself remarks: "Green and Tao took it as a given that there are infinitely many prime numbers and my pithy proof may very well be circular!"). Namely, is there some fact about the infinitude of primes that is used deep in the proof of the Green-Tao theorem? For instance, in some density arguments or similar?

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David Roberts
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    I think this proof is not circular. On the other hand, as I recall, Green and Tao do use basic facts like $\zeta(s)$ has a pole at $s=1$ which implies readily that there are infinitely many primes. – GH from MO Feb 11 '16 at 23:01
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    https://mathoverflow.net/questions/42512/awfully-sophisticated-proof-for-simple-facts#comment100823_42513 – Peter Humphries Feb 11 '16 at 23:12
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    What they use about primes (Goldston-Yildirim result) is of course much much more than infinitude. – Fedor Petrov Feb 11 '16 at 23:14
  • @PeterHumphries I already linked to that. – David Roberts Feb 12 '16 at 01:59
  • @FedorPetrov can you clarify? – David Roberts Feb 12 '16 at 02:00
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    Clearly this question is not doing well! I don't mind migrating it to M.SE, though clearly the people best placed to answer it would be here. – David Roberts Feb 12 '16 at 02:02
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    In a way the question appears to be answered in the question you link to as pointed out by @PeterHumphries Why is that answer not sufficient? –  Feb 12 '16 at 02:10
  • @quid ah, it would help if the link went to the specific comment that points out the relevant usage: https://mathoverflow.net/questions/42512/awfully-sophisticated-proof-for-simple-facts?page=3&tab=active#comment100823_42513 I did in fact scan all the answers at that question to check if they already mentioned Green-Tao, but didn't have the patience to check all the comments too! – David Roberts Feb 12 '16 at 05:44
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    I'm voting to close this question as off-topic because it has been answered in a different question (see comments). – András Bátkai Feb 12 '16 at 07:36
  • @AndrásBátkai answered in a comment on an answer to a different question, which I've now extracted for ease of finding. – David Roberts Feb 12 '16 at 07:56
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    The majority of the proof comes from ergodic-theortical ideas which usually does not distinguish the primes(and actually in a later paper they've proved the Mobius-Nilsequences conj. which shows Mobius randomness in the settings of the dynamical systems they are considering).The primes come into play only on sect.$9$ in their paper,where they construct a psuedo-random measure.The infinitide of the primes can be seen implicitly in the pre-sieving step(W-trick),as you want $W$ to grow on $N$ (at-least morally) and not bounded a-priori,and more explicitly in Prop.9.1 they use Dirichelt's thm. – Asaf Feb 12 '16 at 08:23
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    There is an elementary simple proof of the infinitude of primes by Euclid: Let $p_1,...,p_n$ be primes. Hence there is a prime divisor $p_{n+1}$ of $1+p_1\cdots p_n$ diffrent from the $p_i,i\le n$. So there is no need to use Green-Tao, Prime Number Theorem or the like. – Todd Leason Feb 12 '16 at 10:03
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    @Todd I am a mathematician, and if you check the links, you see that the point is that this is total overkill and purposefully so. – David Roberts Feb 12 '16 at 11:45
  • @Asaf that would have been a great answer! – David Roberts Feb 12 '16 at 11:46
  • These might be silly questions, but here goes nothing. Doesn't a circular argument require assuming a priori the result to be proved? Or does "circular" here really just mean "forgetful"? If so, how do we measure "forgetfulness"? (E.g. it seems that most here agree that proving that primes get arbitrarily large means that the argument required the infinitude of primes, but why is that exactly?) – Pace Nielsen Feb 12 '16 at 16:35

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This was in fact answered by Thomas Bloom in this comment in response to exactly my question above (posed by Qiaochu Yuan):

[Green and Tao] need to embed $[1,N]$ in $Z_p$ for some prime bigger than $N$ to get a nice field structure for some arguments to work.

At least this is a more visible answer to this question if people are searching for it!

David Roberts
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