Problem 707

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claims/: The 2 claim pages of Problem 707, one per claimant's result; the problem's standing derives from them.

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Statement. Let $A\subset \mathbb{N}$ be a finite Sidon set. Is there some set $B$ with $A\subseteq B$ which is perfect difference set modulo $p^2+p+1$ for some prime $p$?

Status. DISPROVED (LEAN): Alexeev and Mixon's 2025 counterexamples (claim page), whose Lean proofs this corpus has not audited, and Hall's 1947 counterexample (claim page), which they recognized as the first disproof.

Source. erdosproblems.com/707, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #707, https://www.erdosproblems.com/707.

References.

• [AlMi25] B. Alexeev and D. G. Mixon, Forbidden Sidon subsets of perfect difference sets, featuring a human-assisted proof. arXiv:2510.19804 (2025); published in Proc. Natl. Acad. Sci. USA 123 (2026), no. 21, e2531760123, DOI.
• [Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.
• [Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.
• [Er97c] Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I, Algorithms Combin. 13, Springer (1997), 47--67; p. 54 states the completion conjecture modulo $p^2+p+1$ for a prime power $p=q^\alpha$, "I now feel this conjecture is perhaps too optimistic", and the weaker $(1+\epsilon)n^2$ conjecture; no prize is printed for it. Library home: erdos_1997_some_my_favorite_problems_results; paged at conjecture_p54.
• [Gu04] Guy, Richard K., Unsolved problems in number theory. 3rd ed., Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp. Section C9 "Packing sums of pairs", pp. 176--177: "Erdős also asks if a Sidon sequence $a_1<a_2<\cdots<a_k$ can be prolonged to a perfect difference set (see C10), i.e., $a_1<a_2<\cdots<a_k<a_{k+1}<\cdots<a_{p+1}=p^2+p+1$ with the differences $a_u-a_v$, $1\le u,v\le p+1$, $u\ne v$, representing every nonzero residue mod $p^2+p+1$ exactly once?", followed by the weaker question whether it can be prolonged with $a_n<(1+o(1))n^2$; no prize is printed for it. Section C10 "Modular difference sets and error correcting codes", p. 181, asks "Can a given finite sequence, which contains no repeated differences, always be extended to form a perfect difference set?" after Singer's existence theorem for prime-power $k$ and the conjecture that no perfect difference set exists otherwise. Library home: guy_2004_unsolved_problems_number_theory.
• [Ha47] Hall, Jr., Marshall, Cyclic projective planes. Duke Math. J. (1947), 1079-1090.

Formalization. Statement in formal-conjectures.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• alexeev_2025_forbidden_sidon_subsets_perfect_difference_sets
• singer_1938_theorem_finite_projective_geometry_some_applications_number_theory
• singer_1938_theorem_finite_projective_geometry_some_applications_number_theory / theorem_p380
• guy_1991_western_number_theory_problems
• guy_1991_western_number_theory_problems / problem_91_05
• guy_2004_unsolved_problems_number_theory
• erdos_1997_some_my_favorite_problems_results
• erdos_1997_some_my_favorite_problems_results / conjecture_p54