On powerful integers expressible as sums of two coprime fourth powers

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) <middle dot> 73993169(2) <middle dot> 338837713(2)= 427511122(4) + 1322049209(4),and that in fact this is the only solution up to 3.6125 <middle dot> 10(37). We also conjecture that1061853595348370798528584585707993395597624934311961270177857= 17(3) <middle dot> 38401618921(2) <middle dot> 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) approximate to 3.289 <middle dot> 10(173) and propose a candidate for the smallest solution with b not equal 17. Finally, we suggest several approaches that might allow our result to be extended past these ranges.