Fundamental mathematical constants such as
, the golden ratio
, and many others play an instrumental
part in diverse ﬁelds such as geometry, number theory, calculus, fundamental physics, biology, and ecology
]. Throughout history simple formulas of fundamental constants symbolized simplicity, aesthetics, and
mathematical beauty. A couple of well-known examples include Euler’s identity
+ 1 = 0 or the
continued fraction representation of the Golden ratio:
The discovery of such Regular Formulas (RFs)
was often sporadic and considered an act of math-
ematical ingenuity or profound intuition. One prominent example is Gauss’ ability to see meaningful
patterns in numerical data that led to new ﬁelds of analysis such as elliptic and modular functions and
to the hypothesis of the Prime Number Theorem. He is even famous for saying “I have the result, but I
do not yet know how to get it” [
], which emphasizes the role of identifying patterns and RFs in data as
enabling acts of mathematical discovery.
In a diﬀerent ﬁeld but in a similar manner, Johannes Rydberg’s discovery of his formula of hydrogen
spectral lines [
], resulted from his data analysis of the spectral emission by chemical elements:
is the emission wavelength,
is the Rydberg constant,
upper and lower quantum energy levels respectively. This insight, emerging directly from identifying
patterns in the data, had profound implications on modern physics and quantum mechanics.
Unlike measurements in physics and all other sciences,
constants can be calculated
to an arbitrary precision (number of digits) with an appropriate formula, thus providing an
. In this sense, mathematical constants contain an unlimited amount of data (e.g. the
inﬁnite sequence of digits in an irrational number), which we propose to use as a ground truth for ﬁnding
new RFs. Since the fundamental constants are universal and ubiquitous in their applications, ﬁnding such
patterns can reveal new mathematical structures with broad implications, e.g. the Rogers-Ramanujan
continued fraction (which has implications on modular forms) and the Dedekind η and j functions [4, 5].
Consequently, having a
method to derive new RFs can help research in many ﬁelds of science.
In this paper, we establish a novel method to learn mathematical relations between constants and we
present a list of new conjectures found using this method. While the method can be leveraged for many
forms of RFs, we demonstrate its potential with equations of the form of generalized continued fractions
x = a
, . . .
are partial numerators and denominators respectively. GCFs in which
the partial numerators and denominators follow a closed-form expression like a polynomial have been of
interest to mathematicians for centuries and still are today, e.g. William Broucker’s π representation 
or [1, 7, 8].
We demonstrate our approach by ﬁnding identities between a GCF and the value of a rational function
at a fundamental constant. For simplicity, enumeration and expression aesthetics, we limit ourselves to
integer polynomials on both sides of the equality. We propose two search algorithms. The ﬁrst algorithm
uses the Meet-In-The-Middle (MITM) algorithm to a relatively small precision in order to reduce the
search space and eliminate mismatches. It increases the precision with a larger number of GCF iterations
on the remaining hits to validate them as new conjectured RFs, and is therefore called MITM-RF. The
second algorithm uses an optimization-based method, which we call Descent&Repel, converging to integer
lattice points that deﬁne new conjectured RFs.
By regular formulas we refer to any mathematical expression or equality that is inﬁnite in nature but can be encapsulated
using a ﬁnite expression.