-2

I was wondering how Euclid showed that there are infinitely many primes by generating a prime number from finitely many primes, and if it could be used to answer if there are infinitely many pairs of primes whose difference is 2. I show my approach in my - short - article here (I know I should copy here the relevant bits, but the article is really short).

My question: why is it not proved like this already? Am I missing something?

Update:
It seems I was too lazy to check for counterexamples. I removed my article. Thank you for the lot of feedbacks.

Rushabh Mehta
  • 13,663
Dávid Laczkó
  • 865
  • 1
    Euclid never said that the product of the first $n$ prime numbers plus $1$ is prime. – WhatsUp Oct 05 '20 at 19:31
  • 3
    Yes, you're missing something. –  Oct 05 '20 at 19:32
  • 2
    You are missing a lot. Your "article" proves nothing, but I suspect it would take too much time to explain it all. – David G. Stork Oct 05 '20 at 19:32
  • 6
    This site is not an place to come to have articles reviewed and assessed. – amWhy Oct 05 '20 at 19:33
  • 3
    $2 \cdot 3 \cdot 5 \cdot 7 -1 = 209$ is not prime. – cosmo5 Oct 05 '20 at 19:35
  • 5
    Please read this answer https://mathoverflow.net/a/23521 under "Examples of common false beliefs in mathematics" – halrankard2 Oct 05 '20 at 19:41
  • 1
    Another one: https://math.stackexchange.com/q/1055365/42969 – Martin R Oct 05 '20 at 19:57
  • @WhatsUp Even this would not prove the twin prime conjecture. Only if $p$#$\pm1$ would be a pair of primes, we would get a prime-twin this way. – Peter Oct 06 '20 at 07:41
  • If the article is short, then there is no reason to not formulate it out here. But even then, amWhy's comment applies. – Peter Oct 06 '20 at 07:52
  • The primorials have been checked for primality upto very high limits. The only known twin prime pairs $p$#$\pm1$ occur for $p=3,5,11$. An interesting quesion would be whether there are infinite many such pairs, probably not. – Peter Oct 06 '20 at 07:57
  • Fermat was also too lazy when he conjectured that $2^{2^n}+1$ is always prime. Ironically, chances are very high that he already knew all primes of this form and that there are no more. Euler was the first to disrove this conjecture by finding $641\mid 2^{32}+1$ – Peter Oct 06 '20 at 08:04

1 Answers1

2

Euclid proof states that $\psi_n +1$ is itself prime or it contains new primes in its factorization, for example

$$2\cdot 3 \cdot 5\cdot 7 \cdot 11\cdot 13+1=30031=59 \cdot 509$$

user
  • 154,566
  • 1
    See the comment above. Even if $p$#$+1$ would always be prime , the twin prime conjecture would still not be proven. – Peter Oct 06 '20 at 07:44
  • 1
    @Peter My answer is related to the proof given by the asker. Thanks anyway for your additional comment. – user Oct 06 '20 at 12:44