Erdős 699: complete exclusion at index nine

P699 research · Erdős problem 699 · Complete index seven

Theorem. For every $9<j\le n/2$, a prime $p\ge9$ divides both $\binom n9$ and $\binom nj$.

There is no remaining exceptional set at nine. The combined checker and proof record join the complete numerical infinite tail to the finite exclusion. This is a computer-assisted proof using exact integer and rational certificates and directed analytic bounds.

The two ranges meet

Put $h=n-8$. Every ingredient of the infinite-tail proof now applies at the same height:

The first three rows exclude every bad triple with $n-8\ge10^{241547627}$. If $n>10^{241547629}$, this condition certainly holds. The fourth row handles every smaller row, so the ranges leave no gap.

The prime threshold is the original $p\ge i$. At index nine the selected smaller primes are exactly $2,3,5,7$.

Complete coverage of the analytic sources

The initial-source proof establishes eleven pair sources implying all nineteen weakened baseline alternatives, together with the four ordinary arithmetic-progression sources. Its complete interval chains contain 299324 witnesses. Every interval inequality and endpoint is checked again by the combined checker. The symmetric $(3,7)$ pair already has a sufficient uniform tail and needs no additional interval.

For the joint clauses, the new height-cover driver and certificate give the following partition:

• 440 clauses hold directly at the target height by adjusted uniform estimates. They include all 224 clauses whose earlier uniform bounds were already sufficient.
• 350 clauses are covered by 11599987 interval witnesses, grouped into 321 direction triples. Every chain reaches a verified uniform tail: 19 use the earlier bound and 331 use the adjusted bound.

The combined checker independently regenerates every direct-target numerical certificate and its analytic coefficient-gcd bound. It also regenerates the uniform bound selected at the end of each interval chain. It checks that the direct and interval-covered identifiers partition all 790 clauses, with exactly the individual threshold weakenings used by the polynomial forest.

The block adjustment lemma allows the source heights to be lowered without changing the cofactor clauses. A clause has the form

with nominal block parameter $t$. If $x=\log h$ and $m\le tx$, failure of the clause bounds the residual denominator cost by

Here $\kappa$ is a directed lower bound for the relevant seed-prime coefficients. At a fixed integer $m$, the logarithmic error estimates are maxima of finitely many affine functions of $x$. Strict endpoint inequalities therefore certify entire intervals of real heights. For initial sources, evaluating the residual denominator cost directly on the prescribed cofactor box also permits larger approximation indices.

The joint calculation uses a compact directed prime-log prefix through $10^{10}$ and proper prime powers through $10^{14}$. The prime-power gcd lemma supplies the common coefficient divisor for both adjacent Padé rectangles. Larger prime intervals use the retained explicit Chebyshev bounds. The retained joint estimates allow pair gaps through eight and product gaps through 64, with maximum shift eight. Actual gap zero is permitted by the nonzero-integer argument.

Every accepted joint interval was checked with exact arithmetic during construction. The joint certificate retains the first and last witnesses, the count, and a SHA-256 digest of each complete witness sequence. The combined checker regenerates every sequence by default, running the generator to a separate output, and compares each count and digest with the certificate; its --quick option checks only the source bindings and endpoints and reports that reduced coverage. The initial-source certificate, by contrast, retains every individual witness, so its complete interval inequalities are checked by the combined verifier. Its optional gcd replay separately reconstructs the pointwise prime sums.

Polynomial coverage and the finite range

The relaxed baseline uses all 128 retained polynomial certificates. It covers 104148 of the $9^4\cdot16=104976$ labeled position and minimal-cofactor cases, including collisions of selected prime-power positions. The relaxed forest closes the remaining 828 cases with 24322 exact rational Farkas leaves. The reductions for the central column are those of the original effective proof.

The Newton-basis coefficient bounds give the weighted leaf threshold

Arithmetic nonvanishing removes the two largest common-zero thresholds. The remaining common-zero bound has exponent 162400114, and the baseline coefficient bound is $151670000/17$. Both lie below the stated polynomial height. The interpolation ranks remain unchanged; the combined checker uses their retained certificates.

The finite proof excludes every $n\le10^{241547629}$. Prime-pair estimates with cofactor exponents $41/100$ and then $9/20$ reduce a hypothetical bad row to $n<10^{26}$. The final partition contains 607975 certified intervals and thirteen residual rows, all at most 5832. Exact binomial gcd calculations exclude all 6762 admissible columns in those rows. The terminal-prime lemma handles the earlier prefix.

Reproduction and scope

With the retained dependencies available, run:

.venv/bin/python wiki/research/erdos_699/computations/erdos_699_index_9_complete.py

The checker verifies the full source partition, the numerical source constants and selected uniform tails, every retained initial interval, the joint endpoint bindings, all exact forest leaves, the finite-proof bindings, the overlap of the two ranges, and, unless --quick is given, the full joint witness replay. The proof record names its inputs by basename as its own run record; the checker validates each input by its domain and coverage. The initial-source driver reconstructs the individual prime sums. No interpolation rank is recomputed by the combined checker.

Together with the earlier cutoff-ten result, this proves the original assertion for every $i\ge9$. The completed index-seven proof is separate. This argument does not establish global finiteness at index three.