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arxiv: 2504.19451 · v3 · submitted 2025-04-28 · 🧮 math.NT · cs.AI

Artificial Intelligence in Number Theory: LLMs for Algorithm Generation and Ensemble Methods for Conjecture Verification

Ali Saraeb This is my paper

Pith reviewed 2026-05-22 19:21 UTC · model grok-4.3

classification 🧮 math.NT cs.AI
keywords large language modelsDirichlet L-functionsconductor predictionalgorithm generationconjecture verificationmachine learningzeros of L-functionsnumber theory
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The pith

Large language models solve algorithmic number theory tasks at 95 percent accuracy or higher when given optimal non-spoiling hints, while a gradient-boosting classifier predicts the conductor of Dirichlet L-functions from statistical traits

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper demonstrates two practical uses of artificial intelligence in number theory. It first evaluates an open-source large language model on sixty tasks drawn from textbooks and online forums, consisting of thirty algorithmic problems and thirty computational questions. With hints that guide without revealing the answer, the model reaches at least 0.95 accuracy relative to the correct output on every task. The second part trains a LightGBM multiclass classifier to predict the conductor q from moments, finite-difference statistics, and FFT magnitudes computed on the initial nontrivial zeros of the corresponding L-function. On a test set drawn from 214 randomly selected Dirichlet L-functions, the classifier reaches at least 93.9 percent accuracy for small q, supplying empirical evidence that the zeros uniquely determine the modulus.

Core claim

The initial nontrivial zeros of Dirichlet L-functions encode sufficient statistical information to identify the conductor q of the underlying character, as shown by training a LightGBM classifier on moments, finite-difference statistics, and FFT magnitudes extracted from those zeros and obtaining at least 93.9 percent test accuracy. In a separate application, the Qwen2.5-Math-7B-Instruct model produces correct algorithmic and computational outputs in number theory at 0.95 accuracy or better when supplied with an optimal non-spoiling hint.

What carries the argument

LightGBM multiclass classifier that maps a vector of statistical features (moments, finite-difference statistics, FFT magnitudes) of the first nontrivial zeros to the conductor q of the Dirichlet L-function

If this is right

  • Algorithmic and computational problems in number theory can be solved to high accuracy by current large language models when modest non-revealing guidance is supplied.
  • The folklore conjecture that the initial zeros uniquely determine the conductor holds at least for small q, because the classifier recovers q from zero-derived statistics alone.
  • Statistical summaries of low-lying zeros can serve as a practical proxy for recovering arithmetic invariants such as the conductor without explicit knowledge of the character.
  • Ensemble methods supply a concrete, reproducible route for empirically checking conjectures in analytic number theory that link zeros to arithmetic data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same feature-based classification approach could be tested on conductors larger than those appearing in the current training set to probe how far the uniqueness extends.
  • Interactive hint systems might be combined with zero-based classifiers to create hybrid tools that both generate candidate algorithms and check consistency with known L-function data.
  • If the zero-to-conductor mapping proves robust, similar pipelines could be applied to other arithmetic objects whose L-functions are expected to carry identifying information in their low zeros.

Load-bearing premise

The chosen statistical features extracted from the initial zeros are sufficient to identify the conductor uniquely without the classifier overfitting to the 214 training examples.

What would settle it

Apply the trained classifier to the initial zeros of Dirichlet L-functions whose conductors lie outside the range seen in training and check whether accuracy remains above 90 percent or falls sharply.

read the original abstract

This paper presents two concrete applications of Artificial Intelligence to algorithmic and analytic number theory. Recent benchmarks of large language models have mainly focused on general mathematics problems and the currently infeasible objective of automated theorem proving. In the first part of this paper, we relax our ambition and focus on a more specialized domain: we evaluate the performance of the state-of-the-art open-source large language model Qwen2.5-Math-7B-Instruct on algorithmic and computational tasks in algorithmic number theory. On a benchmark of thirty algorithmic problems and thirty computational questions taken from classical number-theoretic textbooks and Math StackExchange, the model achieves at least 0.95 accuracy (relative to the true answer) on every problem or question when given an optimal non-spoiling hint. The second part of the paper empirically verifies a folklore conjecture in analytic number theory stating that the modulus \(q\) of a Dirichlet character \(\chi\) is uniquely determined by the initial nontrivial zeros \(\{\rho_1,\dots,\rho_k\}\) (for some \(k\in\mathbb{N}\)) of the corresponding Dirichlet \(L\)-function \(L(s,\chi)\). We train a LightGBM multiclass classifier to predict the conductor \(q\) for 214 randomly chosen Dirichlet \(L\)-functions from a vector of statistical features of their initial zeros (moments, finite-difference statistics, FFT magnitudes, etc.). The model empirically verifies the conjecture for small \(q\), achieving at least 93.9\% test accuracy when sufficient statistical properties of the zeros are incorporated. For the second part of the paper, code and dataset are available.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents two applications of AI to number theory. In the first part, it evaluates the Qwen2.5-Math-7B-Instruct LLM on a benchmark of 30 algorithmic problems and 30 computational questions drawn from classical textbooks and Math StackExchange, claiming at least 0.95 accuracy relative to the true answer when each task is supplied with an optimal non-spoiling hint. In the second part, it trains a LightGBM multiclass classifier on a vector of statistical features (moments, finite-difference statistics, FFT magnitudes, etc.) extracted from the initial nontrivial zeros of 214 randomly chosen Dirichlet L-functions to predict the conductor q, reporting at least 93.9% test accuracy and thereby empirically supporting the folklore conjecture that q is uniquely determined by these zeros for small q. Code and dataset are made available for the second part.

Significance. If the methodological details are clarified, the LLM benchmark illustrates a practical, domain-specific use of current models for algorithm generation in number theory, though the dependence on hints restricts broader claims. The classifier experiment supplies concrete empirical support for a long-standing conjecture in analytic number theory; confirmation that the feature set was fixed in advance would make this a useful data point for future theoretical investigations into the distribution of zeros. The public release of code and data for the ML component is a clear strength that facilitates independent checks.

major comments (2)
  1. [LLM benchmark description] In the section describing the LLM benchmark: the selection process and criteria for the 'optimal non-spoiling hint' supplied to each of the 60 tasks are not specified. Because the reported 0.95 accuracy depends on these hints, the absence of a reproducible protocol undermines the central performance claim.
  2. [LightGBM multiclass classifier] In the section on the LightGBM multiclass classifier: the paper does not state whether the feature set (moments, finite-difference statistics, FFT magnitudes, etc.) was fixed before any model training or was chosen or refined after inspecting performance on the full collection of 214 L-functions. It also omits the train/test split ratio, cross-validation procedure, and uncertainty estimates around the 93.9% test accuracy. These omissions are load-bearing for the claim that the accuracy constitutes independent empirical verification of the conjecture rather than a possible consequence of post-hoc tuning on the given examples.
minor comments (2)
  1. [Title and abstract] The title mentions 'ensemble methods' for conjecture verification, yet the body describes a single LightGBM classifier; a brief clarification of whether any ensembling step is employed would improve consistency.
  2. [Conjecture verification] Consider reporting the precise number k of initial zeros used to compute the feature vectors, as this parameter directly affects the strength of the uniqueness claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We are pleased that the referee recognizes the potential significance of both the LLM benchmark and the empirical support for the Dirichlet conductor conjecture. Below, we provide point-by-point responses to the major comments and outline the revisions we intend to make.

read point-by-point responses

  1. Referee: In the section describing the LLM benchmark: the selection process and criteria for the 'optimal non-spoiling hint' supplied to each of the 60 tasks are not specified. Because the reported 0.95 accuracy depends on these hints, the absence of a reproducible protocol undermines the central performance claim.

    Authors: We agree with the referee that the lack of a specified protocol for hint selection limits the reproducibility of the benchmark results. The hints were selected by the authors to be minimal interventions that provide necessary background or point to an appropriate method without revealing the answer or critical steps. To address this, we will revise the manuscript to include a clear description of the hint generation criteria and include illustrative examples for several problems from the benchmark. This will allow future researchers to understand and potentially apply similar hinting strategies. revision: yes

  2. Referee: In the section on the LightGBM multiclass classifier: the paper does not state whether the feature set (moments, finite-difference statistics, FFT magnitudes, etc.) was fixed before any model training or was chosen or refined after inspecting performance on the full collection of 214 L-functions. It also omits the train/test split ratio, cross-validation procedure, and uncertainty estimates around the 93.9% test accuracy. These omissions are load-bearing for the claim that the accuracy constitutes independent empirical verification of the conjecture rather than a possible consequence of post-hoc tuning on the given examples.

    Authors: We confirm that the feature set was predetermined based on common statistical features for analyzing sequences of complex numbers, prior to any training or performance evaluation. However, we recognize the importance of explicitly documenting the experimental setup. In the revised version, we will add details on the a priori feature selection, the train/test split ratio used, the cross-validation method employed, and uncertainty measures such as confidence intervals or standard deviations obtained from multiple runs. The public code repository will be updated with comments reflecting these choices to facilitate verification. revision: yes

Circularity Check

0 steps flagged

No significant circularity; classifier accuracy provides independent evidence

full rationale

The paper's second part trains LightGBM on independently computed statistical features (moments, finite-difference statistics, FFT magnitudes) extracted from the initial zeros of 214 L-functions to predict conductor q, reporting 93.9% test accuracy. This does not reduce by construction to a fitted parameter encoding q, as the features derive solely from the zeros and the test split evaluates generalization. The first part is an empirical LLM benchmark on algorithmic tasks with non-spoiling hints, not a derivation. No self-citations, self-definitional steps, or ansatzes are load-bearing. The chain is self-contained against the external benchmark of held-out classification performance.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard assumptions about LLM prompting and ML generalization plus the domain assumption that zero statistics suffice to determine conductors.

free parameters (1)
  • LightGBM hyperparameters and feature set
    Tuned to achieve the reported 93.9% test accuracy on the 214 examples.
axioms (1)
  • domain assumption Statistical summaries of the first nontrivial zeros contain enough information to distinguish different conductors q for small values.
    This is the folklore conjecture under empirical test in the second part.

pith-pipeline@v0.9.0 · 5823 in / 1378 out tokens · 71869 ms · 2026-05-22T19:21:30.274858+00:00 · methodology

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