Erdős 699: a fully numerical stopping bound at index 20

P699 research · Erdős problem 699 · Interpolation nonvanishing

Every bad triple with $i=20$ satisfies the fully numerical bound

Every source threshold used in this proof has been numerically instantiated. The bound holds for all assignments and collisions of the small-prime maximum positions, and there is no remaining exceptional curve. Excluding the rows below this endpoint is a separate finite verification.

Twelve numerical cofactor estimates

For positive integers $u,v$, nonnegative exponents $a,b$, and $|p^a u-q^b v|\le100$, put $x=\min(p^a u,q^b v)$. Each row of the table gives

The first six are retained in the asymmetric pair table. The last six are certified by the numerical parameter checker and its recorded bounds. They use a finite, explicit coefficient-gcd lower bound and the asymmetric BFT threshold formula. No effective-but-unevaluated threshold is invoked.

For each prime $p<20$, select $r_p\in\{0,\ldots,19\}$ maximizing $v_p(n-r_p)$, and write $n-r_p=P_pu_p$. Set $e_p=\log u_p/\log(n-19)$. The table supplies the indicated disjunctions for the $e_p$ once $n\ge10^{27027659}$, because every relevant smaller input is at least $n-19>10^{27027658}$.

Choosing an endpoint witnessing each disjunction and removing dominated vectors leaves 60 componentwise minimal vectors $v$, with some $v\le e$ for every feasible $e$. A singleton maximum has exponent capacity $1-v_p$; a position shared by at least two primes has capacity $1$. The collision-capacity construction from the weighted certificate covers every possible grouping. Monotone row weights let rearrangement place the largest capacities at the earliest positions.

A surjective interpolation system

Use the integer vectors

The vector $m$ is nonincreasing. On the grid $x,y\ge0$, $x+y<20$, prescribe residual multiplicities

There are 3599 independent symmetric jet conditions. More precisely, the integer jet matrix on the 3600 symmetric monomials through degree $D=118$ has rank 3599 modulo 65521, as checked exactly. It therefore has full row rank over $\mathbb Q$.

The nonvanishing lemma supplies integer polynomials of degree at most $119$ with these prescribed residual multiplicities and no common zero when $X+Y>19$. Multiplication by

raises their multiplicities to at least $m_t$ on every Kummer row $t$. Here $\deg H=440$, so the resulting degree bound is $559$. The factor $H$ is positive throughout $20<j\le c$.

Exact enumeration of the 8001 collision-capacity configurations gives the uniform gain

One worst vector, in prime order $(2,3,5,7,11,13,17,19)$, is

with sorted capacities

The numerical endpoint

The cofactor-minor estimate in the general interpolation proof applies with

It bounds the complete divisibility constant by $10^E$, where

For completeness, the rational interpolation sections can be cleared using cofactors of a full-rank minor, whose coefficients are bounded by $r!(2\cdot20^{118})^r$. The multiplier for the nonvanishing generators is bounded by $1+(210+24)\binom{120}{2}=1670761<10^7$. These are the inputs to the displayed elementary power-of-ten bound.

Consequently every bad row beyond the source thresholds satisfies

Since $4000000\cdot319/1250=1020800>874790$, the algebraic argument excludes $n\ge10^{4000000}$ whenever the source estimates hold. All twelve source estimates certainly hold for $n\ge10^{27027659}$, proving the stated bound.

The checker and certificate retain the twelve source references, integer interpolation data, exact gain, modular rank, and final exponent comparison. Run the checker with the pinned dependencies installed in the repository's .venv/:

.venv/bin/python wiki/research/erdos_699/computations/erdos_699_index_20_numerical_tail.py

Only finite cofactor vectors, collision patterns, and a fixed polynomial jet matrix are enumerated; no binomial rows are scanned.