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Schäffer's Conjecture

Schäffer's conjecture concerns perfect powers among the power sums

1^k+2^k+...+x^k=y^n,

where x,y,k,n are positive integers and n>=2. Apart from x=y=1, the conjecture predicts no solutions unless (k,n) is one of (1,2), (3,2), (3,4), and (5,2), or (x,y,k,n)=(24,70,2,2) (Chen et al. 2026). Each of the four exceptional pairs (k,n) admits infinitely many solutions.

Chen et al. (2026) report a proof for every even k with 2<=k<=42. The exception 1^2+...+24^2=70^2 must be retained: the unqualified no-solution wording of their Theorem 1.2 omits this exception, although it is listed in their statement of the conjecture.

See also

Perfect Power, Power Sum, Square Pyramidal Number

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References

Chen, I.; Choi, S.; and Houle, J. "Schäffer's Conjecture for Even k<=42." 20 Oct 2026. https://arxiv.org/abs/2610.28768.

Cite this as:

Weisstein, Eric W. "Schäffer's Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SchaffersConjecture.html

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