Apéry was a French mathematician https://www.wikiwand.com/en/Roger...
For a pretty robust list of mathematical constants, see this Wikipe...
The Golden ratio is an important number in fields like art, archite...
Because of the importance of this problem, the Clay Institute a mil...
The Basel problem is asking to calculate the value of the Riemann Z...
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The Catalan constant $G$ is defined as $G=\frac{1}{1^2}-\frac{1}{3^...
The Euler-Mascheroni constant $\gamma$ is $\gamma = \int_1 ^\inft...
The irrationality measure, also called 'Liouville number' is given ...
Continued fractions have a rich mathematical history, with ancient ...
Polynomial continued fractions have a remarkable property of encomp...
The (standard) Bessel functions, denoted $J_\alpha, Y_\alpha$, are ...
The gamma function $\Gamma(x)$ generalizes the factorial operation ...
The hypergeometric function $_2 F_1$ generalizes many functions, su...
The Ramanujan Machine project, aside from being a tribute to Ramanu...
"Experimental mathematics is that branch of mathematics that concer...
Traditionally, mathematics is approached from a 'top-down' perspect...
In the original Ramanujan Machine paper, we aspired to have a futur...
This was very unexpected to the team who worked on the Ramanujan Ma...
The plot presented here takes inspiration from a real-world analog,...
BOINC is a platform that allows volunteers' computers to contribute...
The PSLQ algorithm was selected as one of the top 10 algorithms of ...
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The process performed here is analogous to defining a potential on ...
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The analogy between conservative matrix fields and conservative vec...
This observation is truly remarkable. We are essentially working wi...
The [Liouville–Roth](https://en.wikipedia.org/wiki/Liouville_number...
Factorial reduction can play a crucial part in achieving a substant...
Even if we were to discover a continued fraction within the ZigZagZ...
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Algorithm-assisted discovery of an intrinsic order among
mathematical constants
Rotem Elimelech, Ofir David, Carlos De la Cruz Mengual, Rotem Kalisch, Wolfgang
Berndt, Michael Shalyt, Mark Silberstein, Yaron Hadad, and Ido Kaminer
Technion - Israel Institute of Technology, Haifa 3200003, Israel
August 24, 2023
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
[1–7]. 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 [8–13]. 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 Apéry for the irrationality of ζ(3) [14, 15]. Utilizing
thousands of personal computers worldwide [16], 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.
1 Mathematical constants
Mathematical constants emerge naturally in various fields of science and branches of mathematics, such
as geometry, combinatorics, number theory, and probability theory [17]. Well-known examples of constants
are the ratio between the circumference and the diameter of a circle π, the base of the natural logarithm e,
the golden ratio φ, and the values of the infinite sum for integers n 2:
+ ···
The latter are the integer values of the Riemann zeta function ζ, a cornerstone of mathematical exploration
in the field of number theory, due to the remarkable connection between its complex zeros and the asymp-
totic distribution of prime numbers. The precise distribution of these zeros of ζ constitutes the Riemann
hypothesis, arguably the most important unsolved question in pure mathematics to this day.
The occurrence of a constant in diverse mathematical contexts reflects its significance and often leads to
the discovery of fundamental connections. A remarkable example is given by the so-called Basel problem,
posed in 1650, yet only solved in the mid-18th century by Euler [18] (see also [19]). His solution established
the relation ζ(2) = π
/6 and, in general, expressed all positive even values ζ(2n) as rational multiples of
even powers of π. Euler’s discovery also hinted for the first time at the profound, yet a priori not obvious,
relevance of π in questions related to the distribution of prime numbers. Notably, no analogous relation has
been found for the odd values ζ(2n + 1) to date, three centuries after Euler’s work.
arXiv:2308.11829v1 [cs.AI] 22 Aug 2023
Despite their tremendous impact, discoveries of new relations between mathematical constants and other
mathematical structures are sporadic events that stem from strong intuitive leaps or great strokes of creativ-
ity. In light of this, a captivating possibility is that a deeper underlying concept exists, encompassing all or a
significant group of constants and providing a framework for classifying and ordering them. Here we propose
such a mathematical structure—called conservative matrix field—and show that it not only recreates known
relations between constants, but also reveals new ones. Through these relations, the matrix fields help to
order constants within a cohesive framework that may shed new light on their interrelationships, intrinsic
properties, and complexity.
The complexity of mathematical constants
Central to understanding the intrinsic nature of a mathematical constant is the question of its irrationality,
which can be understood as a measure of its complexity. A rational number can be regarded as possessing
the most simple representation, being expressible as a quotient of two integers. In turn, an irrational number
exhibits a higher level of complexity, for it cannot be represented by such a quotient.
Even though the question of whether a number is rational or irrational may appear innocent at first
glance, for numerous fundamental constants, determining their rationality can be a highly non-trivial task.
The situation for the odd values of the Riemann zeta function serves to illustrate this claim. Indeed, the
proof of the irrationality of ζ(3) by Apéry in 1978 [14, 15] stands as the only definitive accomplishment in
this regard, while the matter remains unsettled for all of the remaining odd values ζ(2n + 1); see [20–22] for
partial results in this direction. The odd values of the zeta function, along with the Catalan, Gompertz, and
Euler–Mascheroni constants, represent a large collection of prominent mathematical constants for which the
question of irrationality is still open.
The distinction rational vs. irrational offers a first kind of hierarchy that orders numbers according to their
complexity. An attempt to refine this binary hierarchy arises from the concept of irrationality measure [23], a
numerical value that quantifies how fast a constant may be approximated by an infinite sequence of distinct
rational numbers. In this sense, any number that admits a fast enough rational approximation must be
irrational. Rational numbers are characterized by an irrationality measure of zero, while irrational numbers
are characterized by an irrationality measure of at least one. Although there are numbers with arbitrary
irrationality measures greater than one, it is noteworthy that most real numbers possess an irrationality
measure of one [24]. Consequently, there is more to be desired from a further-refined dichotomy between
rational and irrational numbers.
Our endeavor to establish a finer hierarchy revolves around quantifying the complexity of the actual
formulas that may represent each number, and clustering them accordingly. We now explain our notion
of formulas and elaborate on why this specific class of formulas is highly suitable for computer-supported
Algorithm-driven discovery of formulas for mathematical constants
Apéry’s celebrated proof of the irrationality of ζ(3) relies on producing an efficient approximation of this
number via the continued fraction formula
= 5
+ (n + 1)
) 12(2n + 1)
Truncating such an infinite continued fraction at a finite number of steps yields a sequence of rational
numbers approximating the target constant. In this way, the exploration of continued fractions offers an
approach for estimating irrationality measures.
Within the class of continued fractions, formulas like Eq. 1 constitute the distinguished subclass of
polynomial continued fractions, for which the partial numerators and denominators are defined by polyno-
mials with integer coefficients as a function of the depth n. The study of polynomial continued fractions is
particularly attractive due to their broad mathematical scope and computational simplicity [25, §VI] (see
also [26–29]). Remarkably, these continued fractions encompass a broad range of mathematical functions
including trigonometric, Bessel, gamma, and hypergeometric functions; they represent a large collection
of constants [30]; they generalize a wide class of infinite sums [31]; and they represent linear recurrence
formulas with polynomial coefficients [25]. The computational simplicity of polynomial continued fractions
stems from two key aspects. Firstly, the space of polynomial continued fractions is countable, which enables
its systematic exploration and analysis. Secondly, the rational approximation by a polynomial continued
fraction can be evaluated very efficiently [32, §5.2]. Those two facts make this space a fitting candidate for
computer-supported research.
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. The most successful algorithm employed there relied on a brute-force
search over the space of polynomial continued fractions.
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.
Our novel algorithm is thus designed to search for formulas based on factorial reduction, discovering
hundreds of polynomial continued fraction formulas for well-known mathematical constants. This output
significantly exceeds what has been manually discovered in the past (e.g., [28]). The algorithm is heuristic
in nature, as it prunes the search space based on the conjecture that continued fractions representing known
constants tend to have factorial reduction. At any rate, the actual formulas that the algorithm discovers can
be and are independently verified. A search utilizing the factorial reduction property makes the algorithm
conducive to distributed computing, enabling us to implement a massively parallel search, with which we
covered a significantly larger space of formulas than what was possible before.
Our algorithm-driven approach exemplifies the vision of experimental mathematics [11,34,35]—leveraging
algorithms at scale to generate new insights in mathematics. Through the generation of large amounts of
data, algorithms offer valuable hints about patterns and behaviors of mathematical structures that would be
otherwise imperceptible within a reasonable timeframe. These hints guide mathematicians in formulating
stronger conjectures and even provide a blueprint for rigorous proofs and generalizations. This process
can be regarded as a cycle: the results of algorithmic experiments create new conjectures that themselves
inspire new algorithmic experiments, leading to further discoveries and a deeper understanding of the original
problem. Our work executes two cycles of an algorithmic-driven scientific method (Fig. 1), leading to the
discovery of an underlying model—a novel mathematical structure.
We see a parallel between the relationship of algorithms to mathematics and that of experiments to the-
oretical physics. Computational frameworks can serve as virtual laboratories for mathematicians, enabling
systematic exploration of mathematical hypotheses on a large scale. The greater computational power avail-
able, the more precise and comprehensive the experiments, providing potential breakthrough opportunities.
This collaborative approach between algorithms and experts facilitates the exploration and testing of hy-
potheses, leading to an accelerated understanding of unsolved problems and augments the serendipitous
insights of brilliant mathematicians.
Figure 1: Algorithmic-driven scientific method in experimental mathematics and its realization in our work. Our work
encompasses two cycles of the scientific method, leading to the discovery of a novel mathematical concept.
Discovery of a novel mathematical structure: conservative matrix field
The insight that led us to find the concept of conservative matrix fields resulted exclusively from having
the large dataset of polynomial continued fraction formulas discovered by our novel algorithm [36]. When
examining these formulas, certain recurring symmetries and patterns arise. As an example, note the resem-
blance between the following three formulas for π:
The first one was discovered by Brouncker and proven by Euler (see [37]), the second one by the Ramanujan
Machine [6], and the third one by Pickett et al. [38]. Such striking similarities were found in many more
examples in our dataset of formulas, sparking the question of whether there exists a governing structure
within the space of formulas for each given constant.
The massive number of formulas produced by our algorithm allowed us to identify several infinite para-
metric families of formulas representing constants such as e, ln(2), π and ζ(3) (see Section 3). A study of
the relationships between the formulas revealed a mathematical structure, coined conservative matrix field
(Section 3), which unifies the infinite families and even derives new formulas with faster rates of convergence.
This structure was identified by observing a kind of “conservation law”, reminiscent of the defining property
of conservative vector fields in physics.
A remarkable application of conservative matrix fields is identifying promising paths for proving the
irrationality of constants, as we exemplify for ζ(3) [39]. This approach now directly applies to more cases
(Appendix I).
We put forward the conjecture that these matrix fields have the potential to unify all polynomial continued
fraction formulas associated with a specific constant. Interestingly, matrix fields also unveil connections be-
tween different constants, for instance, between π
and Catalan’s constant, π
and ζ(3), or e and Gompertz’s
constant. Based on these findings, we hypothesize that many constants, including values of the Riemann
zeta function, can be interconnected through the framework of conservative matrix fields, supporting the
idea that they give rise to a new hierarchy of mathematical constants.
2 The Distributed Factorial Reduction algorithm
Our algorithm aims to discover formulas that equate linear fractional transformations of mathematical
constants and polynomial continued fractions, both with integer coefficients. Due to the nature of this prob-
lem, the search space is intrinsically infinite. Even if we limit our search to a small list of constants, the search
space encompasses all their combinations with arbitrary coefficients, and the number of combinations grows
exponentially with the maximal coefficients allowed. A greater difficulty lies in the fact that an attempt to
explore the search space in parallel by distributing it across different workers leads to redundant computa-
tions of the same expressions. Specifically, division of the search space between workers over the space of
constants requires repeated computing of the same continued fraction by different workers, each equating
the result to different constants or different transformations of constants. Moreover, sharing the already
computed results among workers to avoid such redundancy leads to a prohibitively high communication
overhead between them.
Our algorithm overcomes these challenges by identifying continued fractions that possess a novel property—
which we call factorial reduction. This identification enables us to pinpoint which continued fractions relate
to (well-known) mathematical constants without having to equate them to the (infinite) space of possible
constants and transformations of constants. Our algorithm thus avoids the need to scan the space of con-
stants, which substantially reduces its complexity, and most importantly, makes it amenable to large-scale
distributed computing. This way, we scan the search space across non-communicating workers without
causing redundant calculations and avoid the communication bottleneck.
We conjecture that the property of factorial reduction is a signature of recurrence sequences that converge
to mathematical constants. Consequently, our algorithm not only searches for conjectured formulas but is
itself based on a conjecture. Importantly, every formula generated by the algorithm is independently verified,
thereby liberating it from the necessity of justifying the factorial reduction algorithm, making it a stand-alone
conjecture awaiting formal proof.
This conjecture-based algorithm proved to be extremely effective, leading to the discovery of hundreds
of new formulas for mathematical constants, many of which would have been hard or impossible to discover
by other means. Identifying this conjectured property was enabled by the abundance of formulas discovered
by the older algorithms of the Ramanujan Machine project [36]. In retrospect, the substantial number of
independently verified formulas exhibiting factorial reduction reinforces the belief that factorial reduction is
a distinctive feature of formulas converging to constants of interest.
Factorial reduction: an observation from the ‘laboratory’ of experimental math-
At the core of our algorithm lies a novel observation about the greatest common divisors (GCDs) g
of the
numerators p
and denominators q
of the convergents of polynomial continued fractions. The convergents
are the rational numbers p
found by truncating an infinite continued fraction. More precisely, if a
are the partial denominators and numerators of a continued fraction respectively (as in the rightmost
term of Eq. 2), then
= a
. (3)
These p
and q
are defined by the recursion u
= a
+ b
with initial conditions p
= 1, p
= a
and q
= 0, q
= 1. The growth rate of the resulting convergent numerator p
and denominator q
is a
power of a factorial, i.e. p
, q
for some positive integer d. See [6,25,40,41] for additional background
on continued fractions.
Our algorithm relies on the observation that polynomial continued fractions that converge to mathemat-
ical constants exhibit a peculiar behavior: Denoting g
= GCD(p
, q
) as the greatest common divisor of
the convergents, the reduced numerator and denominator grow exponentially at most, i.e.
, (4)
instead of the much faster factorial growth rate of p
and q
before reduction. We name this observed
property factorial reduction. See our complementary work [42] for additional details about this concept.
Factorial reduction is found to be extremely rare: a wide search hints that the space of continued fractions
with factorial reduction is a “lower-dimensional” subspace inside the space of all polynomial continued frac-
tions. However, surprisingly, we observed factorial reduction in every polynomial continued fraction formula
found by the Ramanujan Machine using previous algorithms (see Appendix A) for a wide range of constants,
including π, ζ(3), and Catalan’s constant. In fact, we examined formulas converging to these constants in
literature spanning over hundreds of years (e.g., [18, 25–28,31, 33,41,43, 44]). Interestingly, all the continued
fractions and infinite sums [45] we checked displayed the property of factorial reduction.
The algorithm presented in this work relies on this unexpected, albeit simple property. Concretely, the
algorithm numerically tests the growth rate of reduced convergents of polynomial continued fractions, saving
the ones that have exponential rates (rather than factorial rates). This approach alleviates the need to
identify a limit constant in advance. Thus, the property of factorial reduction enables efficient identification
of continued fractions that converge to mathematical constants. Fig. 2 displays the test for several polynomial
continued fractions, showcasing the efficacy of our algorithm.
Figure 2: Observation of the factorial reduction (FR) property - its connection to different constants and its sparsity
in the wide space of continued fractions. (a) Calculation of the growth rate s =
| of the reduced convergents
for different continued fractions, detailed in the table. We plot a negative s value to denote a negative limit to p
Even very similar continued fractions can have vastly different growth rates. We observe exponential growth whenever
the continued fraction converges to a mathematical constant. This approach identifies formulas for a wide range of
mathematical constants. (b) Comparison of the growth rate of continued fractions of the same form shows that the only
one having factorial reduction is precisely the one found by Apéry [14, 15] to converge to ζ(3). The factorial reduction
property is extremely rare, shown by its sensitivity to slight changes in the parameters of the continued fraction. (c) Table
of the polynomial continued fractions calculated in panel (a), denoting which ones have or do not have factorial reduction,
and which ones converge to a known constant or to a constant with no known closed form. Here, G is Catalan’s constant,
ζ(5, 1) = ζ(5) ζ(4) + ζ(3) ζ(2) + 1 is the continued fraction from Eq. 8 with s = 5, R = 1.