Erdős 699: why the completed cases are proved

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details/: Individual proof notes and the finite and infinite-range arguments they use.

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Erdős 699 research guide

The completed conclusion is that every admissible triple with $i=1,2,7$ or $i\ge9$ has a common prime divisor $p\ge i$. This page explains the proof by its mathematical ingredients. The case references below cover all these indices without requiring the sequence of earlier cutoff announcements.

From a bad triple to incompatible size bounds

Write $c=n-j$, $S_i=\{p:p<i\}$, and

$$\binom ni=U_i(n)V_i(n),$$

where $U_i$ is supported on $S_i$ and every prime dividing $V_i$ is at least $i$. The following steps are the common structure of the proofs.

1. Local divisibility records a finite triangular grid. Under badness, each $q^a\Vert V_i(n)$ divides both $j-r$ and $c-s$ for some nonnegative $r,s$ with $r+s<i$. Consequently an integer polynomial vanishing to order $m$ on $\Delta_i=\{(r,s):r,s\ge0,\ r+s<i\}$ satisfies

$$V_i(n)^m\mid F(j,c).$$

This is the central reduction in van Doorn and Rocca's detailed manuscript. Their explicit nonvanishing polynomials have degree-to-multiplicity ratio less than $i-1$ for every $i\ge4$. Combining this with the sharp $S_i$-part theorem proves finiteness. Its constant is ineffective, so that argument alone does not give a stopping bound for computation.

2. Explicit cofactor estimates replace the ineffective input. For each $p<i$, choose $r_p$ maximizing $v_p(n-r_p)$ and write

$$n-r_p=P_pu_p,\qquad P_p=p^{v_p(n-r_p)}.$$

The numerator factors differ by at most $i-1$. Padé approximation therefore gives effective alternatives forcing some of their cofactors $u_p$ to be large. The proof keeps these alternatives simultaneously, with the actual prime labels and positions. Coincident positions are included. At seven and nine, the adjustment of Padé blocks also makes the estimates valid low enough to meet the finite proof.

3. Position-dependent polynomials give a strict exponent gain. Different multiplicities on the rows $r+s=t$ use more of the cofactor information than one uniform multiplicity. The weighted polynomial argument is a concrete model. Harder cases use exact interpolation and rational certificates showing that every permitted cofactor configuration gives a positive gain. Nonvanishing must also be established: the interpolation lemma constructs a family whose members cannot all vanish in the physical range, while some of the smallest indices use separate arithmetic arguments. Coefficient bounds then turn the gain into a numerical height above which bad triples are impossible.

4. The remaining finite range is covered by inequalities on whole intervals. Local valuation bounds control $U_i(n)$ throughout an interval of rows. On bounded height ranges, prime-power strip estimates give stronger cofactor bounds by solving congruences between powers of small primes. Their exceptional solutions are retained and applied only to the row windows they affect. An interval is excluded when its small-prime upper bound contradicts the polynomial or discriminant lower bound. Any residual individual rows are checked with exact binomial gcd arithmetic. This proves coverage of the entire finite range, even when its endpoint has millions of digits.

5. The uniform large-index argument closes the infinite range of indices. An orbit polynomial has every coefficient divisible by $\binom ni/\gcd(\binom ni,\binom nj)$. Its explicit discriminant gives a lower bound on that gcd. Comparing it with a small-prime upper bound confines the row; a prime in $(n-i,n]$ then divides both binomial coefficients. The symmetric discriminant and interval proof supplies the complete range $i\ge35$ and the additional case $i=29$, including its dependence on the retained large-index argument and explicit prime-gap results.

Direct references for the completed cases

The rows of this table partition the completed indices. Each reference includes its analytic inputs and links to the required finite records.

For the two most recently completed indices, the overlap is explicit:

Thus neither seven nor nine has an outstanding finite exceptional set. The large displayed numbers are thresholds at which complementary proofs meet, not unexamined search ranges.

What the computational records establish

The small-index proofs combine exact integer and rational calculations with directed bounds for analytic quantities. Modular ranks establish rational interpolation claims; exact rational inequalities cover the permitted cases; interval certificates and residual gcd checks exhaust the finite ranges. The associated source theorems remain mathematical inputs.

At seven and nine, the combined checkers bind the selected source data, polynomial coverage, and finite certificates and verify their overlap. By default they then regenerate every joint Padé witness sequence and compare each direction's witness count and sequence digest with the retained joint record; --quick skips that regeneration and says so. The construction checked each interval, the individual replay drivers regenerate the sequences on their own, and the initial-source drivers can also reconstruct every pointwise coefficient-gcd bound. The proof notes distinguish these longer replays from the combined checks. An exact finite certificate is not, by itself, a completed Lean formalization of the whole result.

The computation guide gives the entry points for reproducing the proofs. For the remaining mathematical questions, return to the research guide.