
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
2
and Catalan’s constant, π
3
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-
ematics
At the core of our algorithm lies a novel observation about the greatest common divisors (GCDs) g
n
of the
numerators p
n
and denominators q
n
of the convergents of polynomial continued fractions. The convergents
are the rational numbers p
n
/q
n
found by truncating an infinite continued fraction. More precisely, if a
n
and
b
n
are the partial denominators and numerators of a continued fraction respectively (as in the rightmost
term of Eq. 2), then
p
n
q
n
= a
0
+
b
1
a
1
+
b
2
a
2
+
.
.
.
+
b
n
a
n
. (3)
These p
n
and q
n
are defined by the recursion u
n
= a
n
u
n−1
+ b
n
u
n−2
with initial conditions p
−1
= 1, p
0
= a
0
and q
−1
= 0, q
0
= 1. The growth rate of the resulting convergent numerator p
n
and denominator q
n
is a
power of a factorial, i.e. p
n
, q
n
∼ (n!)
d
for some positive integer d. See [6,25,40,41] for additional background
on continued fractions.
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