Alexeev and Mixon [arxiv/2510.19804] have disproved this conjecture, proving that $\{1,2,4,8\}$
cannot be extended to a perfect difference set modulo $p^2+p+1$
for any prime $p$.
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theorem erdos_707.counterexample_prime (A : Set ℕ) (hA : A = {1, 2, 4, 8}) :
Finite A ∧ IsSidon A ∧
∀ (B : Set ℕ) (p : ℕ),
Prime p → A ⊆ B → ¬IsPerfectDifferenceSet B (p ^ 2 + p + 1)
