Erdős 699: a fully numerical tail at index 15

P699 research · Erdős problem 699 · The interpolation map

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Every bad triple with $i=15$ satisfies the fully numerical bound

$$\boxed{n<10^{15\,170\,264}.}$$

This replaces the unevaluated cofactor thresholds in the earlier interpolation argument. No new interpolation matrix is needed: its degree-257 symmetric jet map already has full row rank, and the nonvanishing polynomial family has total degree 962.

Explicit cofactor constraints

Use the original explicit BFT Theorem 2.1 table and the following seven additional asymmetric constraints. An entry $(a,b)$ for the pair $(p,q)$ means

$$u_p\ge(n-14)^a\quad\text{or}\quad u_q\ge(n-14)^b.$$

The first four are retained in the asymmetric prime-pair table. For the last three, the oriented seed parameters $(P,Q,a,b,c,d)$ are

$$(128,125,1,1,53,43),\quad (243,121,1,2,34,25),\quad (13,11,1,1,69,49).$$

The seed differences $aP-bQ$ are respectively $3,1,2$. The prime-interval GCD estimate is applied with $m_0=100000$ and 4000 interval terms. It supplies an exact rational lower bound $\ell$ with $G(c,d,dm-\delta)\ge\exp(dm\ell)$ for every $m\ge m_0$ and $\delta\in\{0,1\}$. This is a uniform analytic inequality; the parameter $m$ is not enumerated.

The numerical asymmetric BFT formula then certifies the displayed exponents and thresholds. For $(11,13)$ the oriented seed lists 13 first, so its targets are exchanged before evaluating that formula. All source differences are at most 100; the actual differences here are at most 14. The source derivation and asymmetric argument give the underlying inequalities.

Applying the existing polynomial family

Keep the line exponents and row multiplicities from the index-15 interpolation certificate. Enumerate the minimal cofactor vectors for the preceding disjunctions and the original explicit table. For each vector, consider every possible subset of primes at singleton maximal positions; the remaining primes are grouped as in the weighted-certificate argument. Rearrangement is applicable because the row multiplicities are nonincreasing.

The exact minimum degree gain is now

$$g=\frac{7993}{10000}>0.$$

The same polynomial coefficient estimate as before gives, for every bad row beyond the source thresholds,

$$n^g<10^{8725502}.$$

The row threshold $10^{15170264}$ exceeds every listed factor threshold by at least a full power of ten, so subtracting any selected position $0\le r_p\le14$ leaves each source estimate applicable. It also satisfies

$$15170264\cdot\frac{7993}{10000}>8725502.$$

This proves the stated complete numerical tail. The checker recomputes the three new GCD and asymmetric source bounds, performs the exact finite cofactor optimization, and verifies the final rational comparison. Its output records all parameters and certified intervals. The previously completed modular-rank calculation is an input and is not repeated.