Tools used in the P699 proofs

../

details/: The divisibility, approximation, prime estimates, and nonvanishing lemmas behind the P699 arguments.

---

The proof guide explains how these ingredients fit together. This page is for a reader who wants the general mathematical statements before the individual index certificates.

1. Divisibility and the actual column. The localization lemma turns vanishing on the triangular grid into an exact divisor of a polynomial evaluated at $(j,n-j)$. It retains the common small-prime valuations of the two binomial coefficients. The weighted polynomial argument explains why different multiplicities on different rows help.
2. Effective cofactor bounds. The unequal Padé block construction and block adjustment lemma provide the approximation inequalities used at seven and nine. The coefficient-gcd lemma supplies their prime-power divisibility; each construction links the explicit analytic estimates needed to make it numerical.
3. Nonvanishing. The interpolation lemma gives a family of polynomials that cannot all vanish at an admissible pair. The coprime-polynomial argument is another effective way to control common zeros. The smallest completed cases also use the arithmetic arguments linked in their proofs.
4. Covering a bounded range. The prime-power strip argument bounds cofactors throughout whole regions of exponents and retains the exceptional solutions. Those estimates feed the interval proofs.
5. A possible tool for the remaining cases. The Runge criterion gives effective finiteness on a fixed curve when enough finite fibers supply poles. It does not put all bad pairs on finitely many curves.

The supporting statements retain the additional estimates and exact constants used by these arguments. They are reference material, rather than a second required reading list.