Existence of three consecutive powerful numbers

Determine whether there exist three consecutive powerful numbers; equivalently, prove or disprove the conjecture that no three consecutive powerful numbers exist.

Background

A positive integer is powerful if every prime divisor occurs in its prime factorization with exponent at least two. The paper notes that consecutive powerful pairs exist, including 8 and 9, and that infinitely many such pairs can be constructed from suitable Pell-equation solutions.

The unresolved problem concerns whether a triple of consecutive integers can all be powerful. The paper presents the nonexistence assertion as a conjecture attributed to Erdős, Mollin, and Walsh, and explicitly states that it remains open. The paper’s main theorem addresses only a restricted family of candidate triples, so it does not resolve the general conjecture.

References

In this topic, Erd"os , Mollin and Walsh present the following conjecture. There are no three consecutive powerful numbers. This conjecture is very useful. For example, it implies that there are infinitely many non-Wieferich primes (a Wieferich prime is a prime $p$ which is a solution to the congruence equation $2{p-1}\equiv 1$ mod $p2$, see , and ). It appears to be very hard and remains open.

— An elementary note on three consecutive powerful numbers  (2608.23418 - Ma, 24 Aug 2026) in Section 1, Introduction and Statement of the result