Erdős 699: complete exclusion for every index at least nine

P699 research · Erdős problem 699

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Theorem. For every admissible triple $1\le i<j\le n/2$ with $i\ge9$, a prime $p\ge i$ divides both $\binom ni$ and $\binom nj$. The cases $i=1,2,7$ are also completely settled.

The previous cutoff-ten proof handles every $i\ge10$. The new complete proof at nine closes the remaining case. Independently, the complete proof at seven removes that exceptional index as well. Both use the original prime threshold.

How the two new complete proofs close

In each row the two ranges overlap after the shift from $h$ to $n$. Every admissible row and column is therefore excluded.

The block adjustment lemma accounts for the smaller residual denominator cost when the Padé index changes. Each selected approximant then certifies an interval of logarithmic heights. Complete chains reach the numerical uniform tails, while keeping the cofactor clauses used by the polynomial forest.

At seven, five initial alternatives and 258 joint clauses are covered. At nine, eleven pair sources imply nineteen initial alternatives, four ordinary product sources provide additional constraints, and all 790 joint clauses are covered. The combined checkers verify the source bindings and exact relaxed polynomial coverage, then join the analytic ranges to the retained finite certificates.

The proofs also use integer Newton bases for coefficient bounds and arithmetic nonvanishing to remove the largest common-zero bounds. The original interpolation ranks are retained. These are computer-assisted proofs with exact integer and rational certificates and directed analytic estimates.

The index-seven combined checker and index-nine combined checker have matching seven and nine proof records. Their notes give the individual source and finite-certificate replays.

What remains

Any counterexample must have $i\in\{3,4,5,6,8\}$.

The complete conjecture for $i\ge4$ is reduced to four fixed indices. New column restrictions at six and eight constrain difficult configurations. Exact all-degree polynomial obstructions at six and eight show that stronger arithmetic input is needed at the specified cofactor profiles.