Problem Statement
Are there infinitely many $n$ such that if\[n(n+1) = \prod_i p_i^{k_i}\]is the factorisation into distinct primes then all exponents $k_i$ are distinct?
Categories:
Number Theory
Progress
It is likely that there are infinitely many primes $p$ such that $8p^2-1$ is also prime, in which case this is true with exponents $\{1,2,3\}$, letting $n=8p^2-1$.This problem has been formalised in Lean as part of the Google DeepMind Formal Conjectures project.
Source: erdosproblems.com/913 | Last verified: January 19, 2026