#222: next record gap between sums of two squares is 112
The target (erdosproblems.com/222, OPEN). Write n1<n2<··· for the integers representable as sums of two squares. Problem 222 asks for good upper and lower bounds on the consecutive differences nk+1−nk. We do not improve those asymptotic bounds, but we extend the exact record-gap frontier.
The new record. The consecutive represented integers
133,858,454,292 and 133,858,454,404
differ by 112. Therefore the 111 integers strictly between them are all nonrepresentable. This is the first record after the published difference-107 gap at 99,951,600,293..99,951,600,400, and difference 112 remains the global maximum through the represented endpoint 200,000,000,000. The only other gap of difference at least 108 before that endpoint is a later difference-108 gap at 190,766,104,961..190,766,105,069.
A direct certificate for all 111 misses. The endpoints have the explicit representations
133858454292 = 49986^2 + 362436^2
133858454404 = 35298^2 + 364160^2.For every interior integer, the public certificate supplies a prime p≡3 (mod 4) occurring to an odd exact exponent. Such an integer cannot be a sum of two squares. Dependency-free 64-bit primality and valuation code and PARI/GP independently verify every row, so this particular gap does not depend on trusting the large scan.
Why it is a record. The primary scanner enumerates every pair 0≤a≤b with a2+b2 in each closed chunk and marks the resulting integers in a packed bitmap. A complete baseline from zero recovers the published unique difference-107 maximum. Two post-anchor runs with different chunk boundaries agree on 15,137,795,993 represented integers, the new maximum, and the full list of gaps at least 108. A separately written scanner reverses the coordinate loops, uses a byte array, static scheduling, and a third partition; it produces exactly the same count and gaps.
Hostile audit. The auditor ran a new single scan of the entire interval 0..200,000,000,000 with a fourth chunk width. It counted 30,700,929,089 represented integers and recovered the same two large gaps. That total also equals the two author counts after subtracting their two represented overlap endpoints. Both scanners were separately checked against the prime-factor characterization on 35 represented-boundary cases and random partitions, and PARI regenerated the 111-row obstruction byte for byte.
Scope. This is a bounded exact computational result. It makes no assertion about the next record beyond 200 billion and does not move the asymptotic lower or upper bounds in Problem 222. The live page had no claimed proof or current worker, the relevant OEIS record tables still ended at the old gap, and exact-endpoint searches found no indexed prior occurrence on 2026-08-11. That is a dated search result, not an absolute priority claim.