Erdős 699: complete exclusion for every index at least ten

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This is a retained earlier research note. The current reading guide gives the latest status and distinguishes completed proofs from remaining cases.

P699 research · Erdős problem 699

Theorem. For every admissible triple $1\le i<j\le n/2$ with $i\ge10$, a prime $p\ge i$ divides both $\binom ni$ and $\binom nj$. The cases $i=1,2$ and the new complete case $i=7$ are also settled.

The new complete cases are $i=12$ and $i=14$. The previous complete proof already covered $i=10,11,13$ and every $i\ge15$, so these two cases fill every remaining gap above nine. The prime threshold remains the original $p\ge i$.

Subsequent complete cutoff. The proof at nine excludes every $i\ge9$. Seven is also complete. The preceding proof and its source calculations are retained below.

The two new complete cases

Both bounds are fully numerical. Neither depends on an unevaluated effective constant. The finite ranges are also completely excluded, rather than left as possible exceptional sets.

At twelve, explicit integral and factorial estimates make the twenty unequal-block Padé sources numerical. Finite prime-interval certificates reduce their coefficient-gcd starting indices to at most $18795$. The degree-72 ramp then gives the numerical tail in the table.

At fourteen, the numerical tail proof uses 64 selected pair alternatives. Exact coverage of 543 minimal vectors and 13002 collision-capacity lists gives positive gain $7/625$. The previously verified $831\times841$ interpolation matrix is retained. A sharper coefficient estimate supplies the algebraic bound $10^{6255983}$; the largest source threshold, with the row shift included, gives $10^{8516269}$.

Both tails lie within the range $10^{15170264}$ of the existing six-prime strip certificate. Its exponent-$1/3$ pair estimates, combined with explicit weighted ramps, reduce the finite caps to those in the table. New exponent-$2/5$ certificates then strengthen every pair on these smaller ranges. Their exceptional solutions are kept individually and disable an estimate only on the exact row windows where it could fail.

The completed interval partitions have 550744 certified intervals in total. The fifteen residual rows at twelve are all at most 736, and every admissible column in them is excluded by exact binomial gcd arithmetic. Fourteen has no residual rows. Every part of each finite range is covered.

The complete proof records are:

• Twelve: proof, checker, and certificate.
• Fourteen: numerical tail, finite proof, and the combined checker with its certificate.

These are computer-assisted mathematical proofs, with exact integer and rational certificates and directed analytic bounds. They are not Lean formalizations. The linked notes give the reproduction commands and source provenance; the numerical dependencies are pinned in the repository.

Complete exclusion at seven

The index-seven proof now joins all five initial and 258 joint numerical source bounds at $h=n-6\ge10^{100000000}$ to the finite exclusion $n\le10^{100000001}$. No exceptional set remains at seven. The block adjustment lemma and exact interval covers supply the missing analytic bridge.

Further progress on the remaining indices

The new quintic cofactor bound shows that every sufficiently large bad index-five triple must have $\{r_2,r_3\}=\{0,1\}$ and

$$\boxed{(n-4)^{1/100}<u_2,u_3<(n-4)^{57/100}.}$$

The threshold is effective, but this does not exclude intermediate cofactor sizes or settle index five.

A new simultaneous-approximation construction has an exact three-factor determinant, additional prime divisibility of its minors, and extra powers of two for the seed $10,9,8$. These are proved auxiliary lemmas. They have not yet supplied cofactor inequalities strong enough to settle six or eight. The complete conjecture for all $i\ge4$, and the separate cubic finiteness question, remain open.