Is min exponents of three positive integers $n$, $n+1$ and $n+2$ $=1$ true or false?

Question

Given a positive integer $P>1$, let its prime factorization be written $$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}$$

Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,a_k)$

Is the follows property true or false?

The property: Let $n$ is a positive integer then $\min(h(n), h(n+1), h(n+2)) = 1$

PS: The property was checked up to $n=5.10^7$

This conjecture was mentioned at https://mathoverflow.net/questions/163444/integer-solutions-of-z3-y2-xx-1x1 — Gerry Myerson, Sep 22 '19 at 12:26

score 15 · Accepted Answer · edited Sep 22 '19 at 09:16

15

It is actually an old conjecture of Erdős, Mollin, and Walsh that the pattern you have noticed does indeed go on forever, i.e., there are no three consecutive powerful numbers.

edited Sep 22 '19 at 09:16

Wojowu

27,379

answered Sep 22 '19 at 06:57

Bjørn Kjos-Hanssen

24,629

Is min exponents of three positive integers $n$, $n+1$ and $n+2$ $=1$ true or false?

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