Analytic number theory · prime values of polynomial families · sieve methods

Bateman–Horn Conjecture

Collaboration beta

How often does every polynomial in a fixed admissible family take a prime value at the same integer input?

# {n \leq x : f_{1} (n), \dots, f_{k} (n) all prime} \sim S_{f} \int_{2}^{x} \frac{dt}{\prod_{i} \log f_{i} (t)}

Known results and sources

An admissible polynomial family can avoid every local congruence obstruction and occasionally produce prime values simultaneously—the frequency predicted by Bateman–Horn remains conjectural.

Research problem

Exact mathematical statement

Let $f_1,…,f_k\in\mathbb Z[t]$ be primitive, pairwise nonassociate, irreducible, nonconstant polynomials with positive leading coefficients. Assume the family is admissible: no prime divides $\prod_i f_i(n)$ for every integer $n$ . For each prime $p$ , set

\nu(p)=\#\{a\operatorname{mod} p: \prod_i f_i(a)\equiv0\pmod p\},\qquad \mathfrak S_{\mathbf f}=\prod_p\frac{1-\nu(p)/p}{(1-1/p)^k}.

The Bateman–Horn conjecture predicts

\#\{n\le x:f_1(n),…,f_k(n)\text{ are all prime}\}\sim\mathfrak S_{\mathbf f}\int_2^x\frac{dt}{\prod_i\log f_i(t)}.

Equivalently, the leading scale is $\frac{\mathfrak S_{\mathbf f}}{\prod_i\deg f_i}\frac{x}{(\log x)^k}$ . The classical fixed-family conjecture remains open. The retained source develops one unfinished program centered first on $f_1(n)=n$ and $f_2(n)=n+2$ ; it contains no complete proof.

Problem infographic

Problem at a glance

Scientific problem explainer for the open Bateman–Horn conjecture, showing an admissible family of irreducible integer polynomials, modular root counts nu of p, the singular-series local correction, the predicted simultaneous prime-value count, and the twin-prime specialization n and n plus two. — Local residue obstructions determine the singular-series correction to the predicted simultaneous prime-value count, but the classical asymptotic for every fixed admissible polynomial family remains open.

Current mathematical picture

Where work on Bateman–Horn Conjecture stands

Open conjecture

This curated overview retains the current work's exact Bateman–Horn statement and singular-series normalization, its conditional reduction of the weighted twin-prime specialization to interconvertible parity residuals, exact prime-annihilating, smooth–rough, finite-kernel, and character identities, the source-presented combined large-sieve estimate, scoped core excisions, route failures, and six work orders. The final hybrid cancellation theorem remains open; solving it would address only the weighted twin-prime nucleus, while the general conjecture still requires nonlinear and proper-prime-power work. The source includes no reproducible attachment, formal evidence, independent review, complete proof, or accepted result, and this state has no proof, review, acceptance, or publication effect.

Leading routeProjector-first reduced-conductor route

Sum primitive conductors and inflations before any absolute square, recover the true reduced additive phase, and then combine the finite-kernel or smooth–rough organization with reduced-conductor or Ramanujan dispersion.

Route status · Active route

Useful failureConditional Kloosterman side strip

A quoted Bettin–Chandee-type input yields provisional 1/38 uniform and 1/33 balanced standard-gauge strips, but the exact published theorem and all hypotheses remain unaudited, so this is an optional narrowed side branch rather than the active endpoint.

Route status · Narrowed route

Main reductionDerivative–tail blocks expose the full signed diagonal

The intrinsic -L'T_R transform, combined large-sieve bound, and divisor-stratified primitive orthogonality isolate the structure that an admissible high-conductor argument must preserve.

Evidence posture · Reported reduction

Completed special caseFixed and subpolynomial nonunit core cusps are excised

Grouped two-dimensional sieve estimates make each fixed nonunit prime-power core and the subpolynomial nonunit cusp source-reported o(X), with the unit, transition, and central scopes explicitly excluded.

Evidence posture · Reported special case

Priority open bridgeDerive the exact Q_R Poisson/character formula

Starting from (8.M1)–(8.M4), derive the Q_R-gauge primitive-character formula with reduced conductors, every coprimality condition, zero and negative frequencies, finite derivative–tail blocks, local factors, its own low-conductor estimate, and exact projector recovery.

Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Bateman–Horn Conjecture in numbers

1.6kretained lines of mathematical investigation1,641 in the current working snapshot

Argument development: 1,449 · 88%
Explored or eliminated routes: 10 · 1%
Computational analysis: 3 · 0%
Open obligations: 96 · 6%
Definitions and setup: 83 · 5%

18selected mapped statements7routes investigated7reported milestones6open questions4contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byPaul T. Bateman, Roger A. HornProposed1962Status checkedAug 6, 2026

Source recordedAug 6, 2026

Recommended next task

Derive the exact Q_R Poisson/character formula

Suggested move: Fix one Fourier and Gauss-sum convention, differentiate the analytic Poisson family, reduce each phase to its true conductor, and verify exact recovery of the additive projector before estimating any block.

What would count as progress

The separated formula includes zero mode, negative frequencies, local Euler factors, conductor inflation, and every coprimality condition.
A Q_R-specific low-effective-conductor estimate is proved without importing (5.21).
Exact summation of primitive data recovers the reduced additive phase.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

26 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Active routeProjector-first reduced-conductor route

Route status · Active route

Active routeCharacter-first derivative–tail route

Retain primitive characters long enough to expose the derivative–tail blocks, carry every c|f divisor stratum after squaring, and require exact recovery of the projector-first formula after summing primitive data.

Route status · Active route

Active routeFour-term finite-kernel dispersion

Keep the depth-(1,1), mixed, and depth-(2,2) pieces in one signed dispersion identity and seek cancellation of the order-X diagonal before absolute values.

Route status · Active route

Active routeLevel-2 spectral parity-kernel route

Insert B_R into the level-2 slice before absolute values and test whether a Kuznetsov or Eisenstein decomposition exposes exact continuous-spectrum diagonal cancellation.

Route status · Active route

Active routeAnalytic-input audit

Resolve the exact literature statements, normalizations, ranges, exceptional-character posture, and small-range errors behind every conditional analytic reduction before retaining any numerical strip or final asymptotic.

Route status · Active route

Explored alternatives

Other routes

2 recorded

Narrowed routeConditional Kloosterman side strip

Route status · Narrowed route

Route held in reserveGeneral Bateman–Horn extension

General low-divisor, multi-coordinate, proper-subtuple, nonlinear-conductor, proper-power, and partial-summation work remains in the current research map but paused until the weighted twin-prime nucleus is solved.

Route status · Route held in reserve

Route statements and reductions

Statements the next route can inspect and build on

Route statementDeterminant and odd Poisson geometry

Writing n=ar and n+2=bs gives the exact determinant equation bs-ar=2. In the recomputed odd sector, solution-line Poisson summation has prefactor X/(2rs) and phase (-1)^k e(-k r-bar/s), producing a level-2 slice rather than a full Hecke correspondence.