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Math-2620 · mathematics · Combinatorics

Power saving for prime-intersective polynomial differences

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every h in Z[x] of degree at least two with positive leading coefficient and a unit root modulo every positive integer, do $c_h,C_h>0$ exist such that every A subset [N] avoiding the nonzero values h(p) for rational primes p as differences satisfies $|A|<=C_h$ $N^{1-c_h}$?

Short English statement

For every h in Z[x] of degree at least two with positive leading coefficient and a unit root modulo every positive integer, do $c_h,C_h>0$ exist such that every A subset [N] avoiding the nonzero values h(p) for rational primes p as differences satisfies $|A|<=C_h$ $N^{1-c_h}$?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2620-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2620

Polynomial differences at prime arguments: Prime-intersective means a root r coprime to q for every q, exactly the unit-root condition. The manuscript’s result quantifies over every such positive-leading h of degree>=2 and every positive N, with c_h>0,C_h and avoidance of nonzero h(p) differences at rational prime inputs. Its exponent may depend on h.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

Power saving for prime-intersective polynomial differences

OpenAI's claim

Polynomial differences at prime arguments: For a fixed integer polynomial with positive leading term and a unit root modulo every modulus, the paper claims a fixed power saving for sets avoiding its nonzero values at prime inputs as differences. Both the exponent and constant may depend on the polynomial.

Mathematical statement of the claimed result

Mathematical claim 1.
An integer polynomial $h$ is prime-intersective if each positive modulus $q$ has a root $r$ with $\gcd(r,q)=1$ and $h(r)\equiv0\pmod q$. If $\deg h\ge2$ and its leading coefficient is positive, then the claim supplies $c_h>0,C_h\ge1$ such that \[(A-A)\cap\{h(p):p\text{ prime}\}\subseteq\{0\}\quad\Longrightarrow\quad|A|\le C_hN^{1-c_h}\] for every integer $N\ge1$ and $A\subseteq\{1,\ldots,N\}$. Both constants may depend on the polynomial.

Outline of the argument

Polynomial differences at prime arguments: The progression-compatible polynomial-difference method is adapted to a distribution supported on prime arguments. Root normalizations retain the required coprimality through successive arithmetic progressions. Prime-distribution estimates from the companion Dirichlet zero-free result supply the local comparison needed to contract the density-dominating multilinear quantity.

Formal verification

Family 182 describes Lean coverage for the square-difference power saving; the general polynomial and prime-input estimates are separate.

Polynomial differences at prime arguments: No manuscript-specific Comparator is listed in the pinned inventory.

The documented selection does not cover the entire manuscript assertion relevant here.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources

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