Common primes of two binomial coefficients, index by index

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This lead's results that are not drawn from a cited source are Plasma AI's, credited to Plasma AI. All of them are author-recorded at tier 0. No independent review of the lead or of any of its results is retained, and no claim of novelty is made for them.

Target and route

The target is the affirmative answer to Problem 699: for every $1\le i<j\le n/2$, some prime $p\ge i$ divides both $\binom ni$ and $\binom nj$. A triple is bad when no such prime exists. The threshold includes $p=i$ when $i$ is prime; the second manuscript cited below works with the strict threshold $p>i$. The problem page's current assessment records the problem as open, and this lead does not change that.

The route excludes bad triples one index at a time. Write $\binom ni=U_i(n)V_i(n)$, where $U_i(n)$ is supported on the primes below $i$ and $V_i(n)$ on the primes at least $i$. For a bad triple, van Doorn and Rocca's localization places each prime power of $V_i(n)$ at a point of the triangle $\Delta_i=\{(r,s):r,s\ge0,\ r+s<i\}$, so an integer polynomial $F$ vanishing to order $m$ on $\Delta_i$ gives $V_i(n)^m\mid F(j,n-j)$. Their finiteness argument then uses an ineffective $S$-part theorem. The lead replaces that input. For each prime $p<i$, choose $r_p$ maximizing $v_p(n-r_p)$ and write $n-r_p=P_pu_p$ with $P_p$ a power of $p$. Padé approximation gives effective alternatives forcing some cofactors $u_p$ to be large. Polynomials with different multiplicities on different rows of $\Delta_i$ turn these alternatives into a strict exponent gain and a numerical height above which no bad triple exists. Interval certificates and exact binomial gcd checks then exclude every row below that height. Large indices use an orbit polynomial whose discriminant bounds the gcd of the two coefficients from below; comparing it with a small-prime upper bound confines a bad row, and a prime in $(n-i,n]$ then divides both coefficients.

Results

1. No bad triple at $i=7$ or at any $i\ge9$. For every admissible triple with $i=7$ or $i\ge9$, some prime $p\ge i$ divides both coefficients. At seven the analytic estimates hold for $n-6\ge10^{100000000}$ and the finite exclusion reaches $n\le10^{100000001}$; at nine they hold for $n-8\ge10^{241547627}$ and the finite exclusion reaches $n\le10^{241547629}$. The proof uses van Doorn and Rocca's localization, fat-point divisor, orbit polynomial and discriminant, and for $i\ge906$ Lemma 1 of their second manuscript.
2. Restrictions at $i=3$. Unconditionally, a bad triple with $i=3$ has $n=2^hu$ with $u$ odd and $3u^3<2^h$, $4\le j<n/2$, $n>10^{19}$, $v_2(n)\ge49$, and at least three distinct prime factors of $(n-2)/2$. The bounds on $n$ and $v_2(n)$ come from an exact search of 127,892 rows with certified primality; the other restrictions are uniform proofs. Neither nonexistence nor finiteness at $i=3$ is proved.
3. Finiteness at $i=3$ under abc. Assuming the abc conjecture over $\mathbb Q$, only finitely many bad triples have $i=3$. A single exponent $1+\epsilon=10/9$ suffices, and a valid constant $K\ge1$ for that exponent gives $n<10^{263}K^{69}$ for every bad row. With van Doorn and Rocca's fixed-index finiteness at $i=4,5,6,8$, rational abc therefore makes the set of all bad triples finite. This does not show that the set is empty, and abc supplies no value of $K$.

Van Doorn and Rocca's manuscripts settle $i=1$ and $i=2$, exclude every $i\ge1476$, and prove finiteness for each fixed $i\ge4$ without an effective bound. These are on their pending claim page, not results of this lead. Together with item 1, they leave every possible counterexample with $i\in\{3,4,5,6,8\}$.

Pages

The Problem 699 research guide is the reading route and tabulates what is known at each index. The proof guide explains the five steps of the argument and names the proof of each completed index in a table that partitions them. The cutoff-nine theorem states item 1, joining the complete proof at seven and the complete proof at nine to the earlier cutoff at ten. The discriminant and interval proof excludes $i=29$ and $35\le i\le905$ with 399,713 certified intervals and takes $i\ge906$ from the calculation retained beside the second manuscript's card. It also gives the first cubic reduction, and proves that a nonzero polynomial vanishing to order $m$ at every point of $\Delta_3$ has degree at least $9m/4$.

The remaining-indices guide records what is known at $i=4,5,6,8$ and the precise gaps. The index-three guide collects the case $i=3$. Within it, the cubic follow-up proves the bounds $n>10^{19}$ and $v_2(n)\ge49$, the residue-group theorem proves the three prime factors of $(n-2)/2$, and the conditional cubic argument proves item 3. The methods guide states the divisibility, approximation, nonvanishing and finite-range tools, and the computation guide gives the entry points of the retained certificates. Superseded bounds and approaches that did not finish are kept in the guide's archive.

Why the lead is blocked

At $i=4,5,6,8$ only finiteness is known, from van Doorn and Rocca's ineffective $S$-part argument; the available effective version saves too little there. Numerical improvements to the existing cofactor bounds are not enough by themselves. At specified cofactor profiles at five, six and eight, exact all-degree bounds show that polynomials with the given row multiplicities cannot give the strict degree saving the divisor argument needs. A new argument must supply stronger cofactor inequalities, restrictions from the actual column, or a relation putting all remaining bad pairs on finitely many algebraic curves, where an effective Runge criterion would apply.

At $i=3$ even global finiteness is open. The small-prime part satisfies $U_3(n)\le n$, and no uniform smaller power of $n$ bounds it, since $U_3(2^h)=2^h$. The least degree-to-multiplicity ratio on $\Delta_3$ is $9/4$, while the manuscript's exponent argument would need a ratio below $2$. The missing step is an upper bound on $v_2(n)$, or another argument excluding the family as $v_2(n)$ grows. Finiteness is proved on every fixed algebraic curve, and effective finiteness is proved when $j$ or $n-2j$ is at most a fixed multiple of $n^{3/4}$ or when one residue group of $(n-2)/2$ is bounded; none of these gives global finiteness.

Evidence

The proofs are computer-assisted, by exact integer and rational certificates and directed analytic estimates; no Lean formalization is claimed. The combined checkers at seven and nine verify the numerical source bounds, every retained initial interval inequality, exact polynomial coverage, the joint interval bindings and the overlap with the finite range. They reuse the retained interpolation ranks. By default they then regenerate every joint Padé witness sequence, 9,811,438 witnesses at seven and 11,599,987 at nine, in roughly an hour of processor time per index; the --quick option skips that regeneration and says so. With the pinned dependencies of the evidence folder installed in the repository's .venv/, run from the repository root:

.venv/bin/python wiki/research/erdos_699/computations/erdos_699_index_7_complete.py
.venv/bin/python wiki/research/erdos_699/computations/erdos_699_index_9_complete.py

The large-index interval calculation and the cubic searches have their own entry points, named on the pages above. These checks are author evidence and confer no tier.

Sources

The manuscripts are Wouter van Doorn and Stefano Rocca's Partial Progress on Erdős Problem #699 and Binomial coefficients sharing a large prime divisor, both unpublished. The first says that its specific arguments follow L. Price's Overleaf project Common Prime Divisor of Binomial Coefficients (2026). Their proofs have not been reviewed here: both cards stand at author-recorded reading, and their proofs count as not verified. The cutoff $i\ge906$ is the repository calculation of the second card's evidence leg, which supplies the cutoff $i\ge1000$ that the manuscript states without its calculation and lowers it to $906$. The cofactor estimates build on the Padé construction of Bennett, Filaseta and Trifonov, On the factorization of consecutive integers (2009), read in its 2007 author preprint. The proofs also use explicit prime estimates and the exhaustive prime-gap computation of Oliveira e Silva, Herzog and Pardi (Math. Comp. 83 (2014)), as cited on the pages that use them.