Infinitely many $n$ with all exponents in the factorisation of $n(n+1)$ distinct? (Erdős #913)
Statement
Are there infinitely many positive integers $n$ such that, writing the factorisation of $n(n+1)$ into distinct primes as $$n(n+1)=\prod_i p_i^{k_i},$$ all of the exponents $k_i$ are distinct?
Acceptance. FULLY RESOLVES (proof-shaped): a complete written or machine-checkable proof that infinitely many $n$ have $n(n+1)$ with all prime-exponents distinct, or a proof that only finitely many do (with the finite list). A proof conditional on a clearly stated prime-generation hypothesis (such as the infinitude of primes $p$ with $8p^2-1$ prime) is a genuine advance but must be flagged as conditional and does not by itself fully resolve the unconditional question. ADVANCES (checkable): an unconditional lower bound showing at least $\gg X^{c}$ (for some explicit $c>0$) integers $n\leq X$ have the distinct-exponent property, with proof; a proof of infinitude conditional on a named standard conjecture with the reduction spelled out; or a certified computation extending OEIS A359747 over a documented range with a reproducible program and a report of the observed counting function. Deliver the proof/formalisation or the reproducible search code with certified output.
Background
Posed by Erdős [Er82c, p.28]; listed as open on erdosproblems.com/913 (fetched 2026-07-21, status 'open'), OEIS A359747. Bloom records a conditional route: it is expected (though unproven) that there are infinitely many primes $p$ for which $8p^2-1$ is also prime; for any such $p$ the choice $n=8p^2-1$ works, since then $n=8p^2-1$ is prime (exponent $1$) and $n+1=8p^2=2^3p^2$ contributes exponents $3$ and $2$, so $n(n+1)$ has exponent multiset $\{1,2,3\}$, all distinct. Thus the statement follows modulo a standard but currently intractable prime-generation conjecture. The problem is formalised in Lean in the Google DeepMind Formal Conjectures project. This is related to but distinct from Erdős #912, which concerns the asymptotic count of distinct exponents in the factorisation of $n!$ rather than infinitude of the distinct-exponent property for $n(n+1)$. An attacker would extend OEIS A359747 by scanning $n$ and testing the exponent multiset of $n(n+1)$ for distinctness (the coprime factors $n$ and $n+1$ contribute their exponents without interaction), gather density statistics, and attempt an unconditional proof via sieve or approximation methods that sidestep full prime-tuple conjectures.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #913 (T. F. Bloom) | website |
| REF-02 | OEIS A359747 | website |
| REF-03 | Lean formalisation of Erdős #913 (DeepMind Formal Conjectures) | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.