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If A and B are coprime integers, it seems that at least one of the exponents of the prime factors of AB(A+B) must be one or two.

http://www.comparativetables.com/laeb.htm

Is this interesting?

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This generalization is false and it fails infinitely often.

Some counterexamples: $$ 271^3 + 2^3 3^5 73^3=919^3$$ $$ 3^4 29^3 89^3 + 7^3 11^3 167^3=2^7 5^4 353^3$$

In general consider the elliptic curve: $x^3 + y^3 = k$ where k is 3-full integer, i.e. every exponent is at least 3.

It may have infinitely many rational points $(x'/d,y'/d)$. Whenever $x',y'$ are coprime it is a counterexample $x'^3+y'^3= k d^3$ and there is explicit construction added ON A CONJECTURE OF ERDOS ON 3-POWERFUL NUMBERS, ABDERRAHMANE NITAJ.

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    $\begingroup$ To make this more explicit for the OP, take e.g. $A=271^3$ and $B=2^33^573^3$. Then $AB(A+B)=271^32^33^573^3(919^3)$, and no exponent below 3 appears. $\endgroup$ Jun 16, 2013 at 15:04

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