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Convergence rates of the PCFs The plots present the absolute difference between the PCF value and the corresponding fundamental constant (that is, the error) versus the number of terms calculated in the PCF. On the left are PCFs with exponential/super-exponential convergence rates, and on the right are PCFs that converge polynomially. The majority of previously known PCFs for π converge polynomially, whereas all of our newly found results converge exponentially.
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Fundamental mathematical constants such as e and π are ubiquitous in diverse fields of science, from abstract mathematics and geometry to physics, biology and chemistry1,2. Nevertheless, for centuries new mathematical formulas relating fundamental constants have been scarce and usually discovered sporadically3–6. Such discoveries are often consider...
Citations
... Apéry then showed that this specific sequence p n /q n proved the irrationality of the number to which it converges. It is also demonstrated [2] that the linear recursion is equivalent to the following polynomial continued fraction (PCF): Apéry's paper inspired other researchers to apply related strategies to other problems in Diophantine approximations, to study irrationality measures of other constants, and to find applications in other fields [3][4][5][6][7][8][9][10]. ...
... Many of the PCF formulas that led us to the conjectures and proofs in this paper were originally found by the Ramanujan Machine project [10,30], which employs computer algorithms to find conjectured formulas for fundamental constants. Various algorithms are being developed as part of that project, and so far, they all focus on formulas in the form of PCFs. ...
... Since the algorithms check candidate formulas by their numerical match to target constants, the results are always in the form of conjectures rather than proven theorems. The first algorithms succeeded in finding conjectured PCF formulas for π , e, values of the Riemann zeta function ζ , and the Catalan constant [10]. These latter formulas led to a new world record for the irrationality bound of the Catalan constant. ...
Linear recursions with integer coefficients, such as the recursion that generates the Fibonacci sequence $${F}_{n}={F}_{n-1}+{F}_{n-2}$$ F n = F n - 1 + F n - 2 , have been intensely studied over millennia and yet still hide interesting undiscovered mathematics. Such a recursion was used by Apéry in his proof of the irrationality of $$\zeta \left(3\right)$$ ζ 3 , which was later named the Apéry constant. Apéry’s proof used a specific linear recursion that contained integer polynomials (polynomially recursive) and formed a continued fraction; such formulas are called polynomial continued fractions (PCFs). Similar polynomial recursions can be used to prove the irrationality of other fundamental constants such as $$\pi$$ π and $$e$$ e . More generally, the sequences generated by polynomial recursions form Diophantine approximations, which are ubiquitous in different areas of mathematics such as number theory and combinatorics. However, in general it is not known which polynomial recursions create useful Diophantine approximations and under what conditions they can be used to prove irrationality. Here, we present general conclusions and conjectures about Diophantine approximations created from polynomial recursions. Specifically, we generalize Apéry’s work from his particular choice of PCF to any general PCF, finding the conditions under which a PCF can be used to prove irrationality or to provide an efficient Diophantine approximation. To provide concrete examples, we apply our findings to PCFs found by the Ramanujan Machine algorithms to represent fundamental constants such as $$\pi$$ π , $$e$$ e , $$\zeta \left(3\right)$$ ζ 3 , and the Catalan constant. For each such PCF, we demonstrate the extraction of its convergence rate and efficiency, as well as the bound it provides for the irrationality measure of the fundamental constant. We further propose new conjectures about Diophantine approximations based on PCFs. Looking forward, our findings could motivate a search for a wider theory on sequences created by any linear recursions with integer coefficients. Such results can help the development of systematic algorithms for finding Diophantine approximations of fundamental constants. Consequently, our study may contribute to ongoing efforts to answer open questions such as the proof of the irrationality of the Catalan constant or of values of the Riemann zeta function (e.g., $$\zeta \left(5\right)$$ ζ 5 ).
... Which patterns found from mathematical data lead to interesting as opposed to trivial mathematics? More recently, this AI-guided "Conjecture Formulation" has been given much systematic thought [131,67,35,51,111,102,11,36]. In a six-month workshop in Cambridge in 2023 which I helped to co-organize [23], together with Professor K. Buzzard et al., we wanted to give some criteria on AI-driven theoretical discovery, and in particular on AI-assisted conjectures. ...
... The knot invariant relations found by saliency analyses [35] and the Reidemeister moves untangled for extremely complicated knots [61], though novel, interesting, and precise, were either readily proven or have not become sufficiently influential in the field; thus they fail Birch Test (N). Likewise, the continued fraction identities found by the Ramanujan machine [111], or the physical conservation laws found by AI-Feynman [125] also belong to this category. Even the faster matrix multiplication algorithm found by DeepMind [47] was shortly thereafter beaten by human researchers [85]. ...
Recent years have seen the dramatic rise of the usage of AI algorithms in pure mathematics and fundamental sciences such as theoretical physics. This is perhaps counter-intuitive since mathematical sciences require the rigorous definitions, derivations, and proofs, in contrast to the experimental sciences which rely on the modelling of data with error-bars. In this Perspective, we categorize the approaches to mathematical discovery as "top-down", "bottom-up" and "meta-mathematics", as inspired by historical examples. We review some of the progress over the last few years, comparing and contrasting both the advances and the short-comings in each approach. We argue that while the theorist is in no way in danger of being replaced by AI in the near future, the hybrid of human expertise and AI algorithms will become an integral part of theoretical discovery.
... Machine-learning specialists have recently tried to treat datasets of mathematical experiments and datasets of mathematical proofs and other texts as raw material for machine learning. It is argued that, in this case, the machine finds patterns that are significant for a person, offers correct texts for solving problems, etc. [45,46]. ...
... Computer identification of the coincidence of the values of two differently specified numerical constants calculated with high accuracy leads to the hypothesis that this coincidence is not accidental, but the exact values of the constants are equal [46]. As a final example, we point to the experimental discovery in 1995 by the Bailey-Borwein-Pluff formulas to calculate binary expansion digit π according to its numbering [47]. ...
This paper considers an approach to mathematical education adequate to the task of developing mathematics and its applications in the 21th century. This approach is based on improving the efficiency of the educational process by maintaining the motivation of students of various categories. The basis for the formation of motivation is, on the one hand, independent design; invention of mathematical objects, methods of action, and models of the world around us; and the discovery of facts of mathematical reality and, on the other hand, solving of new, unexpected, and feasible tasks for the student. The student’s work is similar to the work of a mathematician–researcher and programmer. The possibilities of research work in educational mathematics are significantly expanded due to computer-based intramathematic experiments. Debugging a computer program is a special kind of mathematical experiment.
... In recent decades, a growing number of discoveries in fields of mathematics have been assisted by computer algorithms, primarily for exploring large parameter spaces that humans would take too long to investigate [1][2][3][4][5][6][7]. As computers and algorithms become more powerful, an intriguing possibility arises-the interplay between human intuition and computer algorithms can lead to discoveries of novel mathematical concepts that would otherwise remain elusive [8][9][10][11][12][13]. ...
... The Ramanujan Machine project [6], named to honor Srinivasa Ramanujan's unique contributions to mathematics [33], proposed to automate the process of discovery of formulas via an algorithmic approach. The project developed algorithms to find formulas for constants numerically, yielding many new ones such as notable formulas for Catalan's constant. ...
... To broaden the exploration beyond the space that can be covered by brute-force searches, one is confronted with the necessity to develop a more sophisticated exploration strategy. It was through the analysis of the database of formulas obtained in [6] that we were able to identify an intriguing numerical property in the space of formulas. This property, called factorial reduction (defined in Section 2), guided the development of a more efficient algorithm for formula discovery. ...
In recent decades, a growing number of discoveries in fields of mathematics have been assisted by computer algorithms, primarily for exploring large parameter spaces that humans would take too long to investigate. As computers and algorithms become more powerful, an intriguing possibility arises - the interplay between human intuition and computer algorithms can lead to discoveries of novel mathematical concepts that would otherwise remain elusive. To realize this perspective, we have developed a massively parallel computer algorithm that discovers an unprecedented number of continued fraction formulas for fundamental mathematical constants. The sheer number of formulas discovered by the algorithm unveils a novel mathematical structure that we call the conservative matrix field. Such matrix fields (1) unify thousands of existing formulas, (2) generate infinitely many new formulas, and most importantly, (3) lead to unexpected relations between different mathematical constants, including multiple integer values of the Riemann zeta function. Conservative matrix fields also enable new mathematical proofs of irrationality. In particular, we can use them to generalize the celebrated proof by Ap\'ery for the irrationality of $\zeta(3)$. Utilizing thousands of personal computers worldwide, our computer-supported research strategy demonstrates the power of experimental mathematics, highlighting the prospects of large-scale computational approaches to tackle longstanding open problems and discover unexpected connections across diverse fields of science.
... A better understanding of AI's impact on science may not only help guide AI development, bridging AI advances more closely with scientific research, but also hold implications for science and innovation policy. This is especially the case given AI's recent remarkable success in advancing research frontiers across several fields [25][26][27][28][29][30][31][32] , from predicting the structure of proteins in biology [33][34][35] to designing new drug candidates in medicine [36][37][38] , from discovering natural laws in physics 39,40 to solving complicated equations and discovering new conjectures in mathematics [41][42][43] , from controlling nuclear fusion 44 to predicting new material properties [45][46][47] , from designing taxation policy 48 to suggesting democratic social mechanism 49 , and more [50][51][52] . These advances raise the possibility that, as AI continues to improve in accuracy, robustness, and reach, it may bring meaningful benefits to science, propelling scientific progress across a range of research areas while significantly augmenting researchers' innovative capacities. ...
The ongoing artificial intelligence (AI) revolution has the potential to change almost every line of work. As AI capabilities continue to improve in accuracy, robustness, and reach, AI may outperform and even replace human experts across many valuable tasks. Despite enormous efforts devoted to understanding AI's impact on labor and the economy and its recent success in accelerating scientific discovery and progress, we lack a systematic understanding of how advances in AI may benefit scientific research across disciplines and fields. Here we develop a measurement framework to estimate both the direct use of AI and the potential benefit of AI in scientific research by applying natural language processing techniques to 87.6 million publications and 7.1 million patents. We find that the use of AI in research appears widespread throughout the sciences, growing especially rapidly since 2015, and papers that use AI exhibit an impact premium, more likely to be highly cited both within and outside their disciplines. While almost every discipline contains some subfields that benefit substantially from AI, analyzing 4.6 million course syllabi across various educational disciplines, we find a systematic misalignment between the education of AI and its impact on research, suggesting the supply of AI talents in scientific disciplines is not commensurate with AI research demands. Lastly, examining who benefits from AI within the scientific workforce, we find that disciplines with a higher proportion of women or black scientists tend to be associated with less benefit, suggesting that AI's growing impact on research may further exacerbate existing inequalities in science. As the connection between AI and scientific research deepens, our findings may have an increasing value, with important implications for the equity and sustainability of the research enterprise.
... Probably, it is natural that mathematics is constructed in [7] from the very beginning as a human-machine object. Of course, we should also mention machine generation of experimental mathematical data [8] and conjectures based on this data [9][10][11][12][13]. ...
... There are many places that study generalized continued fraction and in particular polynomial continued fractions (see for example [2,6,8,7]) . This paper originated in the Ramanujan machine project [10] which aimed to find simple polynomial continued fraction presentations to interesting mathematical constants using computer automation. With the goal of trying to prove many of the conjectures discovered by the computer, and along the way understand Apery's proof, this conservative matrix field structure was found. ...
We provide a more accessible approach to Ap\'ery's proof that the Riemann zeta function at 3 is irrational. To achieve this, we introduce a new structure called the conservative matrix field, which facilitates the proof and can be applied to other mathematical constants, such as e, {\pi}, ln(2), in order to study their properties. The results obtained in this paper not only offer a more accessible proof of Ap\'ery's theorem, but also pave the way for further research and discovery in this field and relates it to other fields in number theory.
... Generalized Expansions and two related tools GCF.TXT by Dougherty-Bliss and Zeilberger and [7] and the Ramanujan Machine by a team of nine authors [16] are intriguing, but adjacent to the topic of this article, so they are not described further here. ...
... • at least one of the general purpose web search engines on your computers or smart phones, • one of the tools that identify algebraic numbers, returning results such as Root [polynomial, n] and have numerous integer-relation models (AskConstants, Maple identify, SymPy identify and FindPoly, or WolframAlpha), 16 -This is also a good place to find high precision values and program fragments for efficiently computing more digits. ...
There are now several comprehensive web applications, stand-alone computer programs and computer algebra functions that, given a floating point number such as 6.518670730718491, can return concise nonfloat constants such as 3 arctan 2 + ln 9 + 1, that closely approximate the float. This is analogous to unscrambling an egg. Such software includes AskConstants, Inverse Symbolic Calculator, the Maple identify function, MESearch, OEIS, Ordner, RIES, and WolframAlpha. Usefully often such a result is the exact limit as the float is computed with increasing precision. Therefore these program results are candidates for proving an exact result that you could not derive or conjecture without the program. Moreover, candidates that are not the exact limit can be provable bounds, or convey qualitative insight, or suggest series that they truncate, or provide sufficiently close efficient approximations for subsequent computation. This article describes some of these programs, how they work, and how best to use each of them. Almost everyone who uses or should use mathematical software can benefit from acquaintance with several such programs, because these programs differ in the sets of constants that they can return.
... For instance, the Ramanujan Machine Project [2,4,5] re-discovered an already known relation involving e π . Namely, that the continued fraction defined by b n = n 2 + 4 and a n = 2n + 1 converges to: ...
We provide the results of pattern recognition experiments on mathematical expressions. We give a few examples of conjectured results. None of which was thoroughly checked for novelty. We did not attempt to prove all the relations found and focused on their generation.
... The Ramanujan Machine project [1][2][3] detects new expressions related to constants of interests, such as ζ function values, γ and various algebraic numbers (to name a few). ...
The Ramanujan Machine project detects new expressions related to constants of interest, such as $\zeta$ function values, $\gamma$ and algebraic numbers (to name a few). In particular the project lists a number of conjectures involving even and odd $\zeta$ function values, logarithms etc. We show that many relations detected by the Ramanujan Machine Project stem from a specific algebraic observation and show how to generate infinitely many. This provides an automated proof of many of the relations listed as conjectures by the project (although not all of them).
