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Updated Oct 7

Problem 222

Statement. Let $n_1<n_2<\cdots$ be the sequence of integers which are the sum of two squares. Explore the behaviour of (i.e. find good upper and lower bounds for) the consecutive differences $n_{k+1}-n_k$ .

Status. Open.

Source. erdosproblems.com/222, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #222, https://www.erdosproblems.com/222.

References.

[BaCh47] Bambah, R. P. and Chowla, S., On numbers which can be expressed as a sum of two squares. Proc. Nat. Inst. Sci. India 13 (1947), no. 2, 101-103.
[DEKKM22] Dietmann, R. and Elsholtz, C. and Kalmynin, A. and Konyagin, S. and Maynard, J., Longer Gaps Between Values of Binary Quadratic Forms. International Mathematics Research Notices 2023, no. 12, 10313–10349.
[Er51] Erdős, P., Some problems and results in elementary number theory. Publ. Math. Debrecen (1951), 103-109.
[Ri82] Richards, Ian, On the gaps between numbers which are sums of two squares. Adv. in Math. (1982), 1-2.

Formalization. None recorded.

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.

Linked from (10)

Sequences and Densities of Integers integer_sequences/bambah_1947_numbers_which_can_be_expressed_as Theorem (p. 103) and (3): a sum of two squares lies between x and x + 2 sqrt(2+eps) x^{1/4}integer_sequences/dietmann_2023_longer_gaps_between_values_binary_quadratic Theorem 1 of arXiv:1810.03203v2 (p. 2): the gaps between sums of two squares have limsup of (s_{n+1}-s_n)/log s_n at least 195/449 Theorem 2 of arXiv:1810.03203v2 (p. 2): gaps between integers represented by forms of a fundamental discriminant D have limsup at least phi(|D|)/(2|D|(1+log phi(|D|)))Theorem 3 of arXiv:1810.03203v2 (p. 3): runs y_k + j^d (1 <= j <= k) avoiding sums of two squares, with limsup k/log y_k at least 1/(4d')integer_sequences/erdos_1951_problems_results_elementary_number_theory Inequality (2) (p. 103): infinitely many gaps between sums of two squares exceed c log u / (log log u)^{1/2}Theorem 1 (p. 103): long gaps in a sequence sifted by a divergent set of primes

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