In [a video](https://youtu.be/W1OkVkq2vFM) I watched last night on [nuking](https://math.stackexchange.com/questions/798215/) [mathematical mosquitos](https://mathoverflow.net/questions/42512), Matt Parker gave the following proof of the infinitude of primes: suppose there are finitely many primes. The [Green-Tao theorem](https://en.wikipedia.org/wiki/Green%E2%80%93Tao_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?



Use of infinitude of primes in the Green-Tao theorem
This was in fact answered by [Thomas Bloom](https://mathoverflow.net/users/385/thomas-bloom) in [this comment](https://mathoverflow.net/questions/42512/awfully-sophisticated-proof-for-simple-facts?page=3&tab=active#comment100823_42513) 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!


