Erdős 699: an interpolation system for one index-13 case

P699 research · Erdős problem 699

Complete exclusion. Index $13$ is completely excluded. The data below record the initial one-pattern construction.

Superseded. The complete index-13 proof establishes the explicit bound $n<10^{56000}$. It uses improved nonincreasing row multiplicities and a surjective jet map, covering every position assignment and removing the nonvanishing gap described below.

This note records the initial exact algebraic reduction for one canonical small-prime position assignment. No binomial rows or columns were enumerated.

Input and exact data

Use the BFT lower vector, indexed by primes $(2,3,5,7,11)$,

and the canonical positions

Thus the weighted rough-part exponents of the rows $t=0,\ldots,12$ are

Take coordinate-line exponents

and target row multiplicities

At each grid point $(x,y)$ with nonnegative integer coordinates and $x+y\le12$, put

The exact totals are

The residual multiplicities, by fixed $x$ and increasing $y$, are:

0:  0  0  9  7  7 18 19 18 13  6  0  0  0
1:  0  9  7  6 16 14 14 16 12  6  0  0
2:  9  7  6 15 12  9 12 15 12  6  1
3:  7  6 15 11  7  7 11 15 12  7
4:  7 16 12  7  6  7 12 16 14
5: 18 14  9  7  7  9 14 19
6: 19 14 12 11 12 14 20
7: 18 16 15 15 16 19
8: 13 12 12 12 14
9:  6  6  6  7
10: 0  0  1
11: 0  0
12: 0

The fixed linear system

Let $V$ be the rational vector space of polynomials $G(X,Y)$ of total degree at most 111 that vanish to multiplicity at least $R_{xy}$ at every grid point. These are homogeneous rational linear conditions on coefficients.

There are 6328 monomials of total degree at most 111. The multiplicity conditions impose at most

linear equations. Consequently

For any nonzero $G$ in this space, clear its coefficient denominators and define

Then $\deg H=528$, $\deg F\le639$, and $F$ has multiplicity at least $m_t$ at every Kummer point on row $x+y=t$.

The weighted row sum is

so the degree gain is positive:

All these statements are exact integer or rational arithmetic. They do not depend on a floating-point optimization result.

Consequence for bad pairs

Write $c=n-j$. The Kummer valuation argument in adapted_certificates.md applies to every integer polynomial with these row multiplicities. Its lower estimate, combined with the stated BFT vector at the stated positions, gives

whenever $F(j,c)$ is nonzero, beyond effective BFT and elementary thresholds. On the other hand,

Therefore every such nonzero evaluation gives an effective upper bound for $n$.

Choose any rational basis of $V$ and clear denominators individually. There are finitely many basis polynomials, and all their coefficient bounds are effectively computable by exact linear algebra. Beyond the maximum of the associated effective row bounds, every bad pair in this canonical position case must lie in

Here $H(j,c)>0$ throughout the physical range $13<j\le c$, so multiplication by $H$ introduces no physical zero.

Precisely what remains

The missing step is to compute the common polynomial divisor of the degree-111 space $V$, or equivalently enough basis elements to determine its curve components.

• If the gcd is constant, the common zero set is finite. A resultant or a zero-dimensional Groebner basis gives an effective bound for all its integer coordinates. This would settle this canonical position case effectively.
• The same conclusion holds if all common curve factors have no points in the physical region $13<j\le c$ beyond an explicit bound; coordinate and bounded-sum grid lines would be harmless examples.
• A common curve component meeting the physical region remains an obstacle. A general finiteness statement for that component would not by itself supply an effective bound.

A single nonzero interpolating polynomial is insufficient: it may vanish at the putative bad pair. Dimension at least 108 does not imply that the common zero set is finite. No gcd or rank computation has been completed.