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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?

Let three positive integers $a, b, c$ with $\gcd(a,b)=\gcd(b,c)=\gcd(c,a)=1$ then at least one of $a$, $b$, $c$, $a+b$, $b+c$, $c+a$ have h(P) $\le 3$

Stronger conjecture: Let two positive integers $a, b$ with $\gcd(a,b)=1$ then at least one of $a$, $b$, $a+b$ have h(P) $\le 3$ (proposed a year ago)

Note that the stronger conjecture is true up to $10^{18}$ (by Yaakov Baruch)

Relative question:

See also:

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  • $\begingroup$ What is the density of the set with $P\le N, h(P)\ge 3$ asymptotically? (I would expect an answer of the form $cN^a$ for specific positive reals $a<1, c$.) $\endgroup$
    – Yaakov Baruch
    Commented Oct 3, 2019 at 18:38
  • $\begingroup$ Empirically, with $N=10^n, n=1,2,\dots 21$, I suspect something like $\approx 4.67 N^{1/3}$. But some of the references at OEIS A036966 may be addressing this... $\endgroup$
    – Yaakov Baruch
    Commented Oct 3, 2019 at 19:54
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    $\begingroup$ Yes, I believe the counting function for numbers with $h(n)\ge k$ is $\sim c_k x^{1/k}$. This is trivially true for $k=1$ and well known for $k=2$, and I think the latter proof generalizes to any $k$. $\endgroup$
    – Greg Martin
    Commented Oct 4, 2019 at 3:00
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    $\begingroup$ @Đào Thanh Oai. There is nothing in my calculations that warrants publication. I also think the conjecture itself is one of many many that are very likely to be true (probabilistically, once checked for small values) but are in the same general order of difficulty to prove as an effective $abc$-conjecture. Most such conjectures seem to be stated as dead-end curiosities, not leading to some deeper or wider insight. $\endgroup$
    – Yaakov Baruch
    Commented Oct 24, 2019 at 11:06
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    $\begingroup$ The conjecture is a generalization of the Fermat last theorem, Beal conjecture, Fermat-Catalan conjecture and I think maybe equivalent to ABC conjecture $\endgroup$
    – Đào Thanh Oai
    Commented Oct 25, 2019 at 8:17

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