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Just curious, is there any results pertaining to:

For every positive integers $n$, there exists a prime $p$ such that $n$ divides $p+1$.

I would appreciate if you could share any literature here, if available. Also, is there any results if I were to replace $p+1$ with $p-1$ instead?

Bill Dubuque
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Chong Eu Meng
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    These are both special cases of Dirichlet's Theorem on Arithmetic Progressions. https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions – Darth Geek Oct 30 '24 at 19:43
  • Yea, this is a special case of Dirichlet's theorem, which says there is always inf8nitely many prime of the form $ak+b$ for $a,b$ integers with $a\neq0$ and $\gcd(a,b)=1.$ https://en.wikipedia.org/wiki/Dirichlet's_theorem_on_arithmetic_progressions?wprov=sfti1 – Thomas Andrews Oct 30 '24 at 19:44
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    In the case of $p-1$, this can be proven by elementary properties of cyclotomic polynomials, without needing the full strength of Dirichlet's theorem: https://math.stackexchange.com/questions/1347389/infinitude-of-the-primes-p-equiv1-operatornamemod-n – Erick Wong Oct 30 '24 at 19:53
  • I seem to remember something by Erdös on $p-1.$ – suckling pig Oct 31 '24 at 02:41

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