On powerful integers expressible as sums of two coprime fourth powers

• Published in: Research in number theory 2023-12, Vol.9 (4), Article 78
• Main Authors: Elkies, Noam D., Goel, Gaurav
• Format: Article
• Language: English
• Subjects: Algorithms; Curves; Fifteenth Algorithmic Number Theory Symposium (ANTS XV); Mathematics; Mathematics and Statistics; Number Theory; Sums
• ISSN: 2522-0160; 2363-9555
• DOI: 10.1007/s40993-022-00415-9

We confirm the conjecture, made on MathOverflow (Question 191889) by the first-named author, that the smallest powerful integer N = a 2 b 3 > 1 expressible as a sum of two coprime fourth powers is 3088257489493360278725196965477359217 = 17 3 · 73993169 2 · 338837713 2 = 427511122 4 + 1322049209 4 , and that in fact this is the only solution up to 3.6125 · 10 37 . We also conjecture that 1061853595348370798528584585707993395597624934311961270177857 = 17 3 · 38401618921 2 · 382833034044850177 2 = 572132418369898 4 + 988478679472373 4 is the second-smallest solution. Further, we give an algorithm using the arithmetic of elliptic curves that, given generators of a certain Mordell–Weil group, can be used to quickly generate all such numbers up to any given bound. Using this algorithm, we report on finding all solutions for small b values up to 2 - 2 / 3 exp ( 400 ) ≈ 3.289 · 10 173 and propose a candidate for the smallest solution with b ≠ 17 . Finally, we suggest several approaches that might allow our result to be extended past these ranges.