The fourth known primitive solution to $a^5+b^5+c^5+d^5=e^5$

Abstract

We report the fourth known primitive solution to the Diophantine equation $a^5+b^5+c^5+d^5=e^5$, extending the small set of solutions identified in 1966, 1996, and 2004. The new solution has target value $e=1,956,878$, more than an order of magnitude larger than those of all previously known primitive solutions. It was obtained by a large-scale computational search using an optimized meet-in-the-middle algorithm. To make this search feasible under fixed memory, we introduce a partitioning strategy that trades additional preprocessing for reduced memory usage, and show that this approach remains practical for search ranges well beyond those explored here. Finally, we compare structural properties of known fourth- and fifth-power counterexamples to Euler's sum-of-powers conjecture, observing that most known fifth-power solutions have their largest summand close to the target value.

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