Are there infinitely many $n$ such that $(n 2 n)$ is coprime to $105$ ?

Open problem — our best result is machine-sealed: obstruction map, reproduced by an independent verifier. The conjecture itself is unsettled.

number theory · open · formalized (Lean) · 1 attempt

machinery: Kummer-carries,base-representations,digit-restriction-sets,exponential-sums,EGRS-two-prime-method,central-binomial-coprimality,Lucas-theorem

use this record

vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

evidence

obstruction map

machine-sealed

Erdős #376 (is C(2n,n) coprime to 105=3*5*7 for infinitely many n?): honest obstruction map, Opus-verified. Cheap verifier (every base-p digit of n <= (p-1)/2; for {3,5,7}: base-3 digits in {0,1}, base-5 in {0,1,2}, base-7 in {0,1,2,3}) agrees with direct gcd(C(2n,n),105) AND Kummer carry-count, 0 mismatches n<3000; reproduces OEIS A030979 exactly. The obstruction is the SIGN of the density exponent 1+r = 1 + sum_p log((p+1)/2p)/log p: for {3,5,7} it is +0.0260 (count ~ N^0.026, infinite but sub-polynomially sparse — matches the huge b-file gaps); for {3,5,7,11} it is -0.227 (finite — only {0,1,3160}, matching Graham's conjectured last term). #376 sits at +0.026, so close to 0 that infinitude is not robust to any uniform error term — exactly why digit-distribution/exponential-sum methods (which lose factors swamping a 0.026 exponent) cannot prove it. Honest obstruction map; no proof of finiteness or infinitude.

p | C(2n,n) iff adding n+n base p carries iff some base-p digit of n > (p-1)/2 (Kummer). Density exponent per prime from allowed-digit fraction f_p=(p+1)/(2p): 1+r=+0.0260 (3,5,7), -0.2268 (3,5,7,11) — Opus recompute matches the proposer's +0.026/-0.227 to 4 dp. Verifier 3-way agreement (digit==Kummer==gcd) 0 mismatches n<3000; A030979 terms <8000 = [0,1,10,756,757,3160,3186,3187,3250,7560,7561,7651]. {3,5,7,11} sparse base-3 {0,1} search to 3^20~3.5e9 yields exactly {0,1,3160} (finiteness side survives). Digit constraints across bases are positively correlated (deterministic low-order-digit coupling), which aids the conjecture but blocks any Borel-Cantelli/independence proof. Workflow C++ census to 1e14 = 16 solutions.

scripts/erdos376_coprime105.py selftest 0 mismatches; density exponents +0.0260/-0.2268 recomputed; A030979 + {0,1,3160} confirmed.

unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Using **Kummer’s theorem**, for any prime $p$, [ v_p\binom{2n}{n} ] equals the **number of carries** that occur when adding $n+(n)$ in base $p$. So (p\nmid \binom{2n}{n}) iff there are **no carries** in that base, which happens exactly when every base-$p$ digit of $n$ is at most $(p-1)/2$. [[nomath]](Because doubling a…

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.

candidate solution ↗

formal

AMS 11 · open (literature)

theorem erdos_376 : answer(sorry) ↔ { (n : ℕ) | n.centralBinom.Coprime 105 }.Infinite

formal-conjectures/376.lean ↗

oeis

A030979 — Numbers k such that binomial(2k,k) is not divisible by 3, 5 or 7.0,1,10,756,757,3160,3186,3187,3250,7560,7561,7651,20007,59548377,59548401,45773612811,45775397187,237617431723407,249919

links

Kummer's theorem · reference

related: #1093 · #1094

status