Let n >1be an integer, and let Fpdenote a field of pelements for a prime p ≡1(modn). By 2015, the question of existence or nonexistence of n-th power residue difference sets in Fphad been settled for all n <24. We settle the case n =24by proving the nonexistence of 24-th power residue difference sets in Fp. We also prove the nonexistence of qualified24-th power residue difference sets in Fp. The proofs make use of a Mathematica program which computes formulas for the cyclotomic numbers of order 24 in terms of parameters occurring in quadratic partitions of p.
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