Sums of at most fifty positive ninth powers: the exceptional value 5 360 377 770

Abstract

Vaughan and Wooley proved that every sufficiently large positive integer is a sum of at most 50 positive ninth powers, without an effective threshold. Benfield and Lippard, who computed the corresponding largest exception for the exponents k=5,6,7,8, left the ninth-power case open, remarking that the computation was out of reach for the machines available to them. We prove that 5 360 377 770 is not a sum of at most 50 positive ninth powers, that it is a sum of 51 of them, so that its minimal ninth-power representation length is exactly 51, and that it is the only exception in a window of 20 001 consecutive integers centred on it. Consequently, if every integer above 5 360 377 770 is a sum of at most 50 positive ninth powers — the effective form of the Vaughan–Wooley bound, which the literature does not supply — then 5 360 377 770 is exactly the largest integer that is not, answering the question of Benfield and Lippard conditionally. The refutation is not a failed search but a proved one: the depth-first search we use carries a reachability prune, and the statement that the prune discards no representation is itself part of the formal development, so that a search returning false refutes representability rather than merely failing to find it. The same machinery reproduces, as theorems, the four values already computed by Benfield and Lippard. Everything is machine-checked in Lean 4.

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