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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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Đào Thanh Oai
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  • 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 $a<1, c$.) – Yaakov Baruch Oct 03 '19 at 18:38
  • 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... – Yaakov Baruch Oct 03 '19 at 19:54
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    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 $k$. – Greg Martin Oct 04 '19 at 03:00
  • @YaakovBaruch From your work, can you publish the good conjecture on ArXvi or any journal? – Đào Thanh Oai Oct 07 '19 at 02:14
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    @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. – Yaakov Baruch Oct 24 '19 at 11:06
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    The conjecture is a generalization of the Fermat last theorem, Beal conjecture, Fermat-Catalan conjecture and I think maybe equivalent to ABC conjecture – Đào Thanh Oai Oct 25 '19 at 08:17
  • @Yaakov Baruch: could there be a heuristics related to bifurcation theory suggesting that your $4.67$ actually is the first Feigenbaum constant? – Sylvain JULIEN May 25 '20 at 16:47
  • @SylvainJULIEN: the corresponding density for squarefull numbers is $\zeta(3/2)/\zeta(3) N^{1/2}\approx 2.173254 N^{1/2}$. I would imagine that for cubefull numbers the constant similarly relates to values of $\zeta$. – Yaakov Baruch May 26 '20 at 07:54
  • I wonder whether one can prove at least that there exists an absolute constant $k$ such that the set of all $k$-ful numbers does not have any Schur triples. – Seva Jul 04 '20 at 06:27

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