Automated Search for Conjectures on Mathematical Constants using
Analysis of Integer Sequences
- URL: http://arxiv.org/abs/2212.09470v2
- Date: Sun, 11 Jun 2023 20:18:52 GMT
- Title: Automated Search for Conjectures on Mathematical Constants using
Analysis of Integer Sequences
- Authors: Ofir Razon, Yoav Harris, Shahar Gottlieb, Dan Carmon, Ofir David and
Ido Kaminer
- Abstract summary: Formulas involving fundamental mathematical constants had a great impact on various fields of science and mathematics.
Recent efforts to automate the discovery of formulas for mathematical constants, such as the Ramanujan Machine project, relied on exhaustive search.
Here we propose a fundamentally different method to search for conjectures on mathematical constants: through analysis of integer sequences.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: Formulas involving fundamental mathematical constants had a great impact on
various fields of science and mathematics, for example aiding in proofs of
irrationality of constants. However, the discovery of such formulas has
historically remained scarce, often perceived as an act of mathematical genius
by great mathematicians such as Ramanujan, Euler, and Gauss. Recent efforts to
automate the discovery of formulas for mathematical constants, such as the
Ramanujan Machine project, relied on exhaustive search. Despite several
successful discoveries, exhaustive search remains limited by the space of
options that can be covered and by the need for vast amounts of computational
resources. Here we propose a fundamentally different method to search for
conjectures on mathematical constants: through analysis of integer sequences.
We introduce the Enumerated Signed-continued-fraction Massey Approve (ESMA)
algorithm, which builds on the Berlekamp-Massey algorithm to identify patterns
in integer sequences that represent mathematical constants. The ESMA algorithm
found various known formulas for $e, e^2, tan(1)$, and ratios of values of
Bessel functions. The algorithm further discovered a large number of new
conjectures for these constants, some providing simpler representations and
some providing faster numerical convergence than the corresponding simple
continued fractions. Along with the algorithm, we present mathematical tools
for manipulating continued fractions. These connections enable us to
characterize what space of constants can be found by ESMA and quantify its
algorithmic advantage in certain scenarios. Altogether, this work continues in
the development of augmenting mathematical intuition by computer algorithms, to
help reveal mathematical structures and accelerate mathematical research.
Related papers
- FrontierMath: A Benchmark for Evaluating Advanced Mathematical Reasoning in AI [2.1061205911958876]
FrontierMath is a benchmark of hundreds of original, exceptionally challenging mathematics problems crafted and vetted by expert mathematicians.
Current state-of-the-art AI models solve under 2% of problems, revealing a vast gap between AI capabilities and the prowess of the mathematical community.
As AI systems advance toward expert-level mathematical abilities, FrontierMath offers a rigorous testbed that quantifies their progress.
arXiv Detail & Related papers (2024-11-07T17:07:35Z) - LeanAgent: Lifelong Learning for Formal Theorem Proving [85.39415834798385]
We present LeanAgent, a novel lifelong learning framework for theorem proving.
LeanAgent continuously generalizes to and improves on ever-expanding mathematical knowledge.
It successfully proves 162 theorems previously unproved by humans across 23 diverse Lean repositories.
arXiv Detail & Related papers (2024-10-08T17:11:24Z) - Unconditional correctness of recent quantum algorithms for factoring and computing discrete logarithms [0.0]
In 2023, Regev proposed a multi-dimensional version of Shor's algorithm that requires far fewer quantum gates.
We prove a version of this conjecture using tools from analytic number theory.
As a result, we obtain an unconditional proof of correctness of this improved quantum algorithm.
arXiv Detail & Related papers (2024-04-25T09:30:19Z) - MathScale: Scaling Instruction Tuning for Mathematical Reasoning [70.89605383298331]
Large language models (LLMs) have demonstrated remarkable capabilities in problem-solving.
However, their proficiency in solving mathematical problems remains inadequate.
We propose MathScale, a simple and scalable method to create high-quality mathematical reasoning data.
arXiv Detail & Related papers (2024-03-05T11:42:59Z) - When can you trust feature selection? -- I: A condition-based analysis
of LASSO and generalised hardness of approximation [49.1574468325115]
We show how no (randomised) algorithm can determine the correct support sets (with probability $> 1/2$) of minimisers of LASSO when reading approximate input.
For ill-posed inputs, the algorithm runs forever, hence, it will never produce a wrong answer.
For any algorithm defined on an open set containing a point with infinite condition number, there is an input for which the algorithm will either run forever or produce a wrong answer.
arXiv Detail & Related papers (2023-12-18T18:29:01Z) - Algorithm-assisted discovery of an intrinsic order among mathematical
constants [3.7689882895317037]
We develop a computer algorithm that discovers an unprecedented number of continued fraction formulas for fundamental mathematical constants.
The sheer number of formulas unveils a novel mathematical structure that we call the conservative matrix field.
Such matrix fields unify thousands of existing formulas, generate infinitely many new formulas, and lead to unexpected relations between different mathematical constants.
arXiv Detail & Related papers (2023-08-22T23:27:47Z) - A streamlined quantum algorithm for topological data analysis with
exponentially fewer qubits [5.478764356647437]
We present an improved quantum algorithm for computing persistent Betti numbers.
We discuss whether quantum algorithms can achieve an exponential speedup for tasks of practical interest.
arXiv Detail & Related papers (2022-09-26T17:56:11Z) - NaturalProofs: Mathematical Theorem Proving in Natural Language [132.99913141409968]
We develop NaturalProofs, a multi-domain corpus of mathematical statements and their proofs.
NaturalProofs unifies broad coverage, deep coverage, and low-resource mathematical sources.
We benchmark strong neural methods on mathematical reference retrieval and generation tasks.
arXiv Detail & Related papers (2021-03-24T03:14:48Z) - Bayesian Quadrature on Riemannian Data Manifolds [79.71142807798284]
A principled way to model nonlinear geometric structure inherent in data is provided.
However, these operations are typically computationally demanding.
In particular, we focus on Bayesian quadrature (BQ) to numerically compute integrals over normal laws.
We show that by leveraging both prior knowledge and an active exploration scheme, BQ significantly reduces the number of required evaluations.
arXiv Detail & Related papers (2021-02-12T17:38:04Z) - Quantum-Inspired Algorithms from Randomized Numerical Linear Algebra [53.46106569419296]
We create classical (non-quantum) dynamic data structures supporting queries for recommender systems and least-squares regression.
We argue that the previous quantum-inspired algorithms for these problems are doing leverage or ridge-leverage score sampling in disguise.
arXiv Detail & Related papers (2020-11-09T01:13:07Z) - Linear algebra and quantum algorithm [0.0]
Quantum algorithm is expressed by linear algebra on a finite dimensional complex inner product space.
The mathematical formulations of quantum mechanics had been established in around 1930, by von Neumann.
arXiv Detail & Related papers (2020-08-14T08:09:12Z)
This list is automatically generated from the titles and abstracts of the papers in this site.
This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.