Problem 44

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Statement. Let $N\geq 1$ and $A\subset \{1,\ldots,N\}$ be a Sidon set. Is it true that, for any $\epsilon>0$, there exist $M$ and $B\subset \{N+1,\ldots,M\}$ (which may depend on $N,A,\epsilon$) such that $A\cup B\subset \{1,\ldots,M\}$ is a Sidon set of size at least $(1-\epsilon)M^{1/2}$?

Status. Open.

Source. erdosproblems.com/44, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #44, https://www.erdosproblems.com/44.

References.

• [Gu04] Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp.; section C9 "Packing sums of pairs", printed p. 177: whether a finite Sidon sequence can be prolonged to one with $a_n<(1+o(1))n^2$, "as dense as possible asymptotically". Library home: guy_2004_unsolved_problems_number_theory.

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.

• guy_1991_western_number_theory_problems
• guy_1991_western_number_theory_problems / problem_91_05
• guy_2004_unsolved_problems_number_theory