OBJECTIVE AND SOURCE REVIEW
Erdős #222; independent attempt, 2026-09-23. The problem asks for good bounds on gaps between successive sums of two integer squares. I read the full
222.tex. Its quarter-power construction is correct and classical; calling it sharp does not establish an optimal exponent. I examine the complementary constructive lower-bound mechanism explicitly.
WORK: A CERTIFIABLE GAP OF ANY FIXED LENGTH
Choose distinct primes $q_1,\ldots,q_H\equiv3\pmod4$. The Chinese remainder theorem supplies a residue $X$ modulo $Q=\prod_{j=1}^Hq_j^2$ satisfying \[
X+j\equiv q_j\pmod {q_j^2}\quad(1\le j\le H).
\] Every positive representative of this residue gives a block $X+1,\ldots,X+H$ with no sum of two squares. Indeed, if $q\equiv3\pmod4$ divides $a^2+b^2$, then $q\mid a,b$: otherwise $-1$ would be a square modulo $q$, contradicting Fermat's theorem. Thus such a prime cannot divide a sum of squares to exact exponent one, whereas our congruence forces precisely that exponent. Taking $0\le X<Q$, a gap between consecutive representable integers surrounding this block has length at least $H+1$ and begins below $Q$. Thus, with the maximal-gap convention of the source, \[
g(Q)\ge H+1.
\] There are arbitrarily many primes $3\pmod4$: given a finite list, a prime factor $3\pmod4$ of $4\prod q_j-1$ is outside it. So this is a completely constructive unbounded-gap proof, including the location bound in terms of its selected primes.
ADVERSARIAL VERIFICATION AND REMAINING GAP
One must impose residues modulo $q_j^2$, not merely modulo $q_j$, to exclude an even valuation. The block endpoint need not itself be a sum of squares; surrounding it by the nearest two representable integers only increases the gap. The large product $Q$ makes this weaker than the known logarithmic lower bounds recorded in the official entry. It is a verified certificate framework, not a new record. Neither it nor the supplied quarter-power upper bound determines the true order of the largest gaps.
SOURCES CHECKED
Complete local source:
gpt_pro_5.4/222.tex. Official indexed remarks checked at
https://www.erdosproblems.com/search_bib/Er57?sources_only=1, entry 222, 2026-09-23. No advanced external theorem is used in the CRT argument.
FINAL
UNRESOLVED. The constructive lower obstruction is quantified, with no verified improvement to the best general bounds.
COMPLETION ESTIMATE
Subjective progress toward determining gap growth.
COMPLETION: 10%