Problem 222

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 $n_{k+1}-n_k$ .

Source: erdosproblems.com/222

Number theory
Squares

Wiki page

Status

Open

No claim settles this problem.

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.

Problem 222

Status

Proof claims

Discussion