We confirm the conjecture, made on MathOverflow (Question 191889) by the first-named author, that the smallest powerful integer N=a2b3>1\documentclass[12pt]{minimal}
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\begin{document}$$N = a^2 b^3 > 1$$\end{document} expressible as a sum of two coprime fourth powers is 3088257489493360278725196965477359217=173·739931692·3388377132=4275111224+13220492094,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} 3088257489493360278725196965477359217&= 17^3 \cdot 73993169^2 \cdot 338837713^2 \\&= 427511122^4 + 1322049209^4, \end{aligned}$$\end{document}and that in fact this is the only solution up to 3.6125·1037\documentclass[12pt]{minimal}
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\begin{document}$$3.6125 \cdot 10^{37}$$\end{document}. We also conjecture that 1061853595348370798528584585707993395597624934311961270177857=173·384016189212·3828330340448501772=5721324183698984+9884786794723734\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned}&1061853595348370798528584585707993395597624934311961270177857 \\&\quad = 17^3 \cdot 38401618921^2\cdot 382833034044850177^2 \\&\quad = 572132418369898^4 + 988478679472373^4 \end{aligned}$$\end{document}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 2-2/3exp(400)≈3.289·10173\documentclass[12pt]{minimal}
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\begin{document}$$2^{-2/3}\exp (400)\approx 3.289\cdot 10^{173}$$\end{document} and propose a candidate for the smallest solution with b≠17\documentclass[12pt]{minimal}
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\begin{document}$$b\ne 17$$\end{document}. Finally, we suggest several approaches that might allow our result to be extended past these ranges.
