Problem 222

Erdős Problem #222: declared status 'open'. Formalized: no. 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 $n_{k+1}-n_k$. Current best: The best known upper bound is due to Bambah and Chowla [BaCh47], who proved that$$n_{k+1}-n_k \ll n_k^{1/4}.$$The differences are listed at A256435 on the OEIS. Prize: no. OEIS: A256435. Tags: number theory, squares.