OBJECTIVE AND SOURCE REVIEW
Erdős #376; independent attempt, 2026-09-23. Are there infinitely many positive $n$ with $\gcd(\binom{2n}{n},105)=1$? I read all of
376.tex. It carefully labels the proposed very small exponent conditional, but its broader assertion that no unconditional power-saving upper bound is available is false literally: one base already provides one. I make that distinction precise and independently enumerate a bounded range.
WORK: AN UNCONDITIONAL BASELINE AND EXACT CHECK
By the digit characterization, every admissible $n<3^k$ has only digits zero or one in its $k$ base-three positions. There are exactly $2^k$ such nonnegative integers. Consequently \[
G_{3,5,7}(N)\le2(N+1)^{\log_3 2}.
\] To justify the digit condition directly, Legendre's factorial valuation gives $v_p(\binom{2n}{n})=\sum_{j\ge1}(\lfloor2n/p^j\rfloor-2\lfloor n/p^j\rfloor)$, the number of carries on doubling $n$ in base $p$. There are no carries exactly when every digit is at most $(p-1)/2$. The saved program
verification/batch_3_digits_records.py generates all $2^{20}$ zero-one base-three strings and tests their base-five and base-seven digits. It finds exactly 14 positive solutions below $3^{20}=3486784401$: \[
1,10,756,757,3160,3186,3187,3250,7560,7561,7651,
20007,59548377,59548401.
\] The list and bounds are saved in its JSON output. This is an exact bounded reproduction check, not a claimed record search.
ADVERSARIAL VERIFICATION AND REMAINING GAP
Leading zero digits are permitted, ensuring every integer below the cutoff is considered. Zero is explicitly excluded from the displayed list. The unconditional exponent $\log_3 2$ is much larger than the source's conditional $0.026$; it does not validate that stronger assertion. Neither any power-saving upper bound nor a finite list proves finiteness or infinitude. Constructing arbitrarily large simultaneous digit-restricted integers remains the exact gap.
SOURCES CHECKED
Complete local source
gpt_pro_5.4/376.tex; official primary statement and digit equivalence checked at
https://www.erdosproblems.com/latex/376, 2026-09-23.
FINAL
UNRESOLVED. A literal overstatement about unconditional bounds is corrected and a bounded exhaustive check reproduced.
COMPLETION ESTIMATE
Conservative subjective progress toward infinitude.
COMPLETION: 8%