Erdős problem 649 · problems.science

Let $P(m)$ denote the greatest prime factor of $m$ . Is it true that, for any two primes $p,q$ , there exists some integer $n$ such that $P(n)=p$ and $P(n+1)=q$ ?

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FormalConjectures/ErdosProblems/

649.lean

Retained formal statement4 of 6

Sampaio independently observed that the answer to Erdős' original problem is no if one of the primes can be $2$ - for example this is false with $p=19$ and $q=2$ , since if $n+1=2^k$ and $19\mid n$ then (since $2$ is a primitive root modulo $19$ ) we must have $18\mid k$ , and hence $73\mid 2^{18}-1\mid n$ .

FormalConjectures/ErdosProblems/649.leanErdos649.erdos_649.variants.sampaio1 lineExact file

1¬∃ n, n.maxPrimeFac = 19 ∧ (n + 1).maxPrimeFac = 2

TextbookStatement only, no proof

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