Problem 340

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Statement. Let $A=\{1,2,4,8,13,21,31,45,66,81,97,\ldots\}$ be the greedy Sidon sequence: we begin with $1$ and iteratively include the next smallest integer that preserves the Sidon property (i.e. there are no non-trivial solutions to $a+b=c+d$). What is the order of growth of $A$? Is it true that

$$\lvert A\cap \{1,\ldots,N\}\rvert \gg N^{1/2-\epsilon}$$

for all $\epsilon>0$ and large $N$?

Status. Open.

Source. erdosproblems.com/340, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #340, https://www.erdosproblems.com/340.

References.

• [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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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