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Updated Oct 7

Problem 1103

claims/: The 3 claim pages of Problem 1103, one per claimant's result; the problem's standing derives from them.

Statement. Let $A$ be an infinite sequence of integers such that every $n\in A+A$ is squarefree. How fast must $A$ grow?

Status. Open. The site labels the problem OPEN (page last edited 3 December 2025). Its commentary credits two bounds. Konyagin's 2004 bound on the finite analogue, Problem 1109, gives $a_j\ge j^{15/11-o(1)}$ for every infinite set with squarefree sums, recorded on Konyagin 2004. Van Doorn and Tao's first arXiv version (30 November 2025) proves $a_j>0.24\,j^{4/3}$ for all $j$ and constructs a squarefree such set with $a_j\le\exp(Cj/\log j)$ , recorded on van Doorn and Tao 2025. The site's proof-claims tab carries a partial proof claim by Xiyu Hu, submitted on 23 July 2026 under the username hxypqr with GPT-5.6 Sol named as assistance: an infinite set with squarefree pairwise sums, doubles included, and $|A\cap[1,x]|\gg(\log x)^2$ , so that $a_j\le\exp(C\sqrt j)$ , built from van Doorn and Tao's extension method and Konyagin's quadratic Brun sieve, with a partial Lean 4 development; it is recorded on its claim page. The claim concerns the construction side only and has no acceptance evidence.

Source. erdosproblems.com/1103, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1103, https://www.erdosproblems.com/1103.

References.

[Er81h] Erdős, P., Some problems and results on additive and multiplicative number theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182.
[Ko04] Konyagin, S. V., Problems of the set of square-free numbers. Izv. Ross. Akad. Nauk Ser. Mat. (2004), 63-90.
[vDTa25] W. van Doorn and T. Tao, Growth rates of sequences governed by the squarefree properties of its translates. arXiv:2512.01087 (2025). Published as Acta Arith. 224 (2026), 173-195, DOI 10.4064/aa251207-28-5.

Formalization. None recorded.

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.

Linked from (14)

Konyagin's lower bound on the growth of sequences with squarefree sums Van Doorn and Tao's growth bounds for squarefree sums Hu's logarithmic-square construction for squarefree pairwise sums problems/integer_sequences/E1103/claims Sequences and Densities of Integers additive_bases/erdos_1981_problems_results_additive_multiplicative_number_theory Question (p. 179): must an infinite sequence all of whose sums a_i + a_j are squarefree grow fast?Sequences and Densities of Integers integer_sequences/doorn_2025_growth_rates_sequences_governed_squarefree_properties Remark 7 (p. 5): no sequence has every residue modulo p^2 below p^2/2 Unnumbered bound (p. 4): a sequence with squarefree sums has a_j >> j^{15/11} exp(-O(log j/sqrt(log log j)))integer_sequences/erdos_1987_divisibility_properties_integers_form Theorem 2 (p. 117): a subset of {1,...,N} with all pairwise sums squarefree has fewer than 3N^{3/4} log N elements integer_sequences/konyagin_2004_problems_set_square_free_numbers

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