Problem 326

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claims/: The 1 claim page of Problem 326, one per claimant's result; the problem's standing derives from them.

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Statement. Does there exist $A=\{a_1<a_2<\cdots\}\subset \mathbb{N}$ which is a minimal basis of order $2$ (i.e. every large integer is the sum of $2$ elements from $A$, and no proper subset of $A$ has this property), such that

$$\lim_{k\to \infty}\frac{a_k}{k^2}=c$$

for some $c\neq 0$?

Formulation. Erdős first asked the question for any basis of order $2$, not necessarily minimal: "It was asked by Erdős whether there is an infinite sequence $\{a_k\}$ for which $n=a_i+a_j$ is solvable for every $n$ and which satisfies $a_k/k^2\to c$" [ErGr80, p. 47]. The answer to that question is yes: Cassels [Ca57] gave such a basis, with $a_k=ck^2+O(k)$, as [ErGr80], p. 47, and the site's commentary record. Erdős and Graham add that "there is a small amount of 'cheating' going on here", since Cassels starts from a basis $\{b_k\}$ for which $\limsup b_k/k^2$ differs from $\liminf b_k/k^2$ and then adds new terms. They call the minimal-basis question of the Statement "the 'correct' way of formulating the question" and conjecture that the answer is no [ErGr80, pp. 47–48]. They also give "another way of stating the problem" [ErGr80, p. 48]: does every basis of order $2$ have a subset $\{a_k\}$ which is also a basis and for which $\lim a_k/k^2$ does not exist? The site's earlier wording asked that question. A note posted on the thread on 2026-04-16, which the poster attributes to GPT-5.4, answered it no: the note's basis is the set of positive integers whose ternary digits are all $0$ or $1$, and in it every sub-basis has $b_k/k^2\to0$. The site then rewrote the problem as the present question, which the note does not address, so the note gets no claim page.

Status. Claimed: the site's label is OPEN (page last edited 2026-04-17), and the standing derives from one pending full claim: [[problems/additive_bases/E0326/claims/2026_05_20_bhalla|Bhalla's minimal basis with $a_k\sim ck^2$]], a manuscript of 2026-05-20 with a Lean formalization posted on 2026-06-14, not built or audited in this corpus.

Source. erdosproblems.com/326, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #326, https://www.erdosproblems.com/326.

References.

• [Ca57] Cassels, J. W. S., Über Basen der natürlichen Zahlenreihe. Abh. Math. Sem. Univ. Hamburg 21 (1957), 247-257.
• [ErGr80] Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève, Geneva, 1980; pp. 47–48. Library home: erdos_1980_old_new_problems_results_combinatorial_number_theory.

Formalization. Statement in formal-conjectures.

Progress

Not yet compiled.

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.

• bhalla_2026_regularly_thin_minimal_asymptotic_basis_order
• bhalla_2026_regularly_thin_minimal_asymptotic_basis_order / theorem_1_1
• nathanson_2014_paul_erdos_additive_bases
• nathanson_2014_paul_erdos_additive_bases / definition_p3