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Math-533-A2 · revision r1 · on Math-533

OpenAI claim summary for Math-533

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Polynomial differences at prime arguments: The polynomial h(x)=(x-1)^2 is prime-intersective (the root one is coprime to every modulus). A set avoiding all square differences avoids h(p) for primes, so the theorem gives the recorded power saving, with positive exponent depending only on this fixed polynomial.

Degree-uniform intersective differences: Taking the intersective polynomial h(x)=x^2 gives the required power saving for square-difference-free sets; the theorem treats every intersective polynomial and permits a degree-only exponent.

Square-difference-free power saving: The square paper gives |A|<=C N^(1-c) when no nonzero square difference occurs. The general paper specializes to h(t)=t^2; the prime companion specializes to h(t)=(t-1)^2, which has the unit root 1 modulo every q and forbids only the squares (p-1)^2, hence also applies to square-difference-free A. In each case constants can be absorbed by decreasing the positive exponent for sufficiently large N, meeting the on-site fixed power-size threshold.

The claimed result implies the stated question and also supplies the additional strength described above.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Difference Sets Containing Squares
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Mathematical claim 2.
  6. Mathematical claim 3.
  7. Outline of the argument
  8. Formal verification
  9. Status caveat
  10. Sources

Difference Sets Containing Squares

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.

Degree-uniform intersective differences: For an intersective polynomial with a positive leading term and degree $k\ge2$, the claimed bound for difference-avoiding subsets is $O_h(N^{1-c_k})$; the exponent depends only on $k$, although the implicit constant depends on the polynomial.

Square-difference-free power saving: The manuscript claims absolute positive constants $c,C$ with $|A|\le C N^{1-c}$ whenever $A\subseteq[1,N]$ has no nonzero square difference; the improvement is a fixed power of $N$.

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.
Mathematical claim 2.
For each degree $k\ge2$ there is $0<c_k<1$ such that every intersective $h\in\mathbb Z[x]$ of degree $k$ with positive leading coefficient has constants $C_h\ge1,N_h$ satisfying \[(A-A)\cap h(\mathbb N_{>0})\subseteq\{0\},\quad A\subseteq\{1,\ldots,N\},\quad N\ge N_h\quad\Longrightarrow\quad|A|\le C_hN^{1-c_k}.\] Intersective means having a root modulo every positive integer. The saving exponent depends only on the degree; the prefactor and threshold may depend on all coefficients.
Mathematical claim 3.
The claimed square-difference bound gives absolute $c>0,C<\infty$ with \[|A|\le CN^{1-c}\] for every $N\ge1$ and $A\subseteq\{1,\ldots,N\}$ having no distinct $a,b$ with $|a-b|=m^2$ for any positive integer $m$. Equivalently every larger subset contains a nonzero square difference.

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.

Degree-uniform intersective differences: Local roots are normalized until their first nonzero Taylor coefficients are units, without changing their multiplicities. Auxiliary polynomials are chosen to remain compatible under passage to progressions. A multilinear density lower bound and a distribution-comparison argument then produce contraction with numerical choices controlled by the degree rather than the coefficients.

Square-difference-free power saving: A probability law on tuples of residues over many primes defines a nonnegative multilinear quantity dominating density. A rational Fourier lift places the indicator into this residue model. Comparison with shorter intervals works for every residue choice, and a positive fraction of negligible choices makes the quantity contract by a fixed power.

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.

Degree-uniform intersective differences: No manuscript-specific Comparator is listed in the pinned inventory.

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

Square-difference-free power saving: The manuscript links SquareDifference.lean.

The documented selection includes this manuscript’s main assertion; that assertion retains the restrictions stated above.

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