The Ramanujan Library - Automated Discovery on the Hypergraph of Integer Relations
Authors: Itay Beit Halachmi
Ido Kaminer
Venue: ICLR 2025 Poster (2025)
Keywords: Continued Fractions
Mathematical Constants
Integer Relations
Experimental Mathematics
Riemann Zeta Function
Irrational Number
PSLQ
AI In Mathematics
Automated Conjecture Generation
Primary area: other topics in machine learning (i.e., none of the above)
TL;DR: Automated generation of nonlinear formulas connecting mathematical constants, forming a publicly-available resource of constants and connections.

Abstract: Fundamental mathematical constants appear in nearly every field of science, from physics to biology. 
Formulas that connect different constants often bring great insight by hinting at connections between previously disparate fields.
Discoveries of such relations, however, have remained scarce events, relying on sporadic strokes of creativity by human mathematicians.
Recent developments of algorithms for automated conjecture generation have accelerated the discovery of formulas for specific constants.
Yet, the discovery of connections between constants has not been addressed.
In this paper, we present the first library dedicated to mathematical constants and their interrelations. This library can serve as a central repository of knowledge for scientists from different areas, and as a collaborative platform for development of new algorithms.
The library is based on a new representation that we propose for organizing the formulas of mathematical constants: 
a hypergraph, with each node representing a constant and each edge representing a formula.
Using this representation, we propose and demonstrate a systematic approach for automatically enriching this library using PSLQ, an integer relation algorithm based on QR decomposition and lattice construction. During its development and testing, our strategy led to the discovery of 75 previously unknown connections between constants, including a new formula for the `first continued fraction' constant $C_1$, novel formulas for natural logarithms, and new formulas connecting $\pi$ and $e$.
The latter formulas generalize a century-old relation between $\pi$ and $e$ by Ramanujan, which until now was considered a singular formula and is now found to be part of a broader mathematical structure. 
The code supporting this library is a public, open-source API that can serve researchers in experimental mathematics and other fields of science.

Published as a conference paper at ICLR 2025
THE RAMANUJAN LIBRARY AUTOMATED DISCOV- ERY ON THE HYPERGRAPH OF INTEGER RELATIONS
Itay Reit-Halachmi and Ido Kaminer
The Ramanujan Machine Team, Faculty of Electrical and Computer Engineering Technion Israel Institute of Technology
Haifa 3200003, Israel
itaybel www.eeaaaaust.cchiiooo... www.heeechhnn.eeee ac.il, ido. kaminaminneeeeee memeineree gma com
ABSTRACT
Fundamental mathematical constants appear in nearly every field of science, from physics to biology. Formulas that connect different constants often bring great in- sight by hinting at connections between previously disparate fields. Discoveries of such relations, however, have remained scarce events, relying on sporadic strokes of creativity by human mathematicians. Recent developments of algorithms for automated conjecture generation have accelerated the discovery of formulas for specific constants. Yet, the discovery of connections between constants has not been addressed. In this paper, we present the first library dedicated to mathemati- cal constants and their interrelations. This library can serve as a central repository of knowledge for scientists from different areas, and as a collaborative platform for development of new algorithms. The library is based on new representa- tion that we propose for organizing the formulas of mathematical constants: a hypergraph, with each node representing constant and each edge representing a formula. Using this representation, we propose and demonstrate systematic approach for automatically enriching this library using PSLQ, an integer relation algorithm based on QR decomposition and lattice construction. During its devel- opment and testing, our strategy led to the discovery of 75 previously unknown connections between constants, including a new formula for the 'first continued fraction' constant C1 novel formulas for natural logarithms, and new formulas connecting and The latter formulas generalize century-old relation between æ and by Ramanujan, which until now was considered a singular formula and is now found to part of a broader mathematical structure. The code supporting this library is a public, open-source API that can serve researchers in experimental mathematics and other fields of science.
INTRODUCTION
With the rise in computing power and the rise of artificial intelligence, efforts have been made to harness these tools to further scientific discovery. Some of the earliest examples of such automated discovery efforts include the Automated Mathematician and later Eurisko (Lenat and Brown, (984),, systems that automatically discovered concepts in mathematics and later in other scientific domains. Another notable early discovery system Graffiti (Fajtlowicz, 1988), which automatically generated conjectures in graph theory. Since then, the usage of computer algorithms as scientific tools (Wang 1960: Daviss and Lenat, 1982: Wolfram et all 2002),, and in particular as tools for aiding with math- ematical proofs (Appel and Haken, [1976; Zeilbergei, 1990; Wilf and Zeilberger, 1992: Buchherger all, 200G), has skyrocketed. Noteworthy recent examples include papers by Google DeepMind (Davies et all 2021; Fawzi ail 0022; Romera-Parede el all (0zz), which have utilized Al to find new algorithms for matrix multiplication, and generate conjectures in topology, representation theory, and combinatorics.
noteworthy example of computers assisting mathematicians is the Ramanujan Machine project (Raayomi all 2021: Razon et all, 2023: Elimelech et all 2022)), which specializes in automated conjecture generation in number theory. This project, named in honor of Srinivasa Ramanujan's 's sub- stantial mathematical contributions (Berndi, 2012) and his unconventional work style, demonstrated

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automated discovery of formulas involving fundamental mathematical constants. The discovered formulas and the properties they exhibited (Ben David et all 20724) enabled the construction a previously unknown mathematical structure (David, 2023: Elieelcc et all 2024) that forms sys- tematic framework for the generation of new formulas and for proofs of irrationality of constants, in places where systematic proofs were known (e.g.. Apéry (1197)); David (2022))
Fundamental mathematical constants like n. e, or ((3) are central in several branches of mathematics and in other fields of science (Einch, R(OK). The appearance of the same mathematical constant in different fields that may seem unrelated can lead to surprising connections. One of the earliest such connections is the Basel problem. posed in 1650 and solved by Euler (Emler, 1748; Ayoub, 19749, establishing that one of the values of the Riemann zeta function is ((2)
π //.6 /6, and also providing family of formulas expressing all positive even values ((2n) using 2t, Due the connection of the Riemann zeta function to the distribution of prime numbers, this formula established the profound and nontrivial importance of æ in regards to prime numbers. This example shows how discovery of new connections on fundamental constants has great potential in connecting different fields and leading to new discoveries. Another such example is a connection involving т and found by Ramanujan, presented in figure MI Though it is not known if and algebraically independent Morandi (1996), this is one of many formulas that show how else they can be connected.
Te
2
15
(2n 1)!!
14
2
1
+
1+
π
TEE
1
1/2
2
1/4
2
1+
1/6
1
3
1
1+
1/2n
1
Figure 1: Automatically discovered formulas, showing a section of the full hypergraph figure The first formula was found by Ramanujan (Berndt el all, 1999), and the second formula is novel.
In this paper, we propose a new approach to the discovery of relations between mathematical con- stants, and collect its results in pubsicly-aceessible database. This database is the first collect and organize the known mathematical constants and their relations, and will provide a unique resource for experimental mathematicians and number theorists worldwide.
We expand our database into library that also collects the known relations between the constants and organizes them into hypergraph, which we show be an effective representation of the struc- ture among formulas of mathematical constants and among the constants themselves. Going beyond the collection of known relations, we develop algorithms that the database search for new rela- tions between its constants, successfully generating 75 new formulas. These algorithms are used for further (automated) enrichment of the library. We present two enrichment strategies: (1) automated search of relations within the database using a polynomial integer relation algorithm (see section 2.11) and (2) the identify algorithm (see section H) executed on each new constant introduced the library to find its relations to existing constants and formulas in the database. Both strategies rely on the integer relation algorithm PSLQ (Ferguson and Bailey, 1922, Ferguson et all 1999). We developed new methodology for quantifying the PSLQ results and for determining which ones constitute new likely formulas, so they can be further analyzed to greater precision. Our algorithm the first to dis- cover nonlinear (polynomial) relations, rather than focusing linear relations. For comparison, show that identify succeeded in cases for which the state-of-the-are -heeh-rrt commercial solution (in Wolfram Alpha) unsuccessful.
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2 INTRODUCING THE HYPERGRAPH OF MATHEMATICAL CONSTANTS AND THEIR RELATIONS
Recall that an (ordinary) undirected graph on a vertex set is defined by an edge set E, cach connecting two vertices. An undirected hypergraph, then, generalizes the concept of edge so each one can connect more than two vertices. More explicitly, each edge is now a set of arbitrary size, instead of set of size 2. In our (undirected) hypergraph, the vertices will all be mathematical constants, either given explicitly (e.g., т, e, up to a user-defined precision) or given by formula converging to the constant. We choose represent formulas using canonical form that captures general continued fractions and infinite sums, namely the C-transforms of complex sequence fn:
f1
Cl = 1
f2
1
1+
f3
This definition can always be treated a formal expression, and when it converges it is instead defined as the limiting value. In practice, we generate each fn using a rational function. C-transforms (and continued fractions in general) are powerful as they contain, for example, all infinite sums (that can be converted using Euler's approach (Euler. 1748)). whereas not every continued fraction can be converted back to an infinite sum. The C-transform captures any continued fraction in canonical form that removes redundancy. The special case of fa being a rational function already generalizes all polynomial continued fractions (Khinchin, (964]) which are general mathematical constructs that capture all trigonometric functions, Bessel functions, generalized hypergeometric functions and much more. The C-transform notation helps facilitating the systematic automated research in our work (see appendix [ for further discussion).
The undirected edges (equivalently hyperedges) in our hypergraph are integer polynomial relations, forming representation that generalizes large number of formula structures. To define an integer polynomial relation, we first introduce integer relations: An integer relation on a vector of real numbers æ1, .2.... is a vector of integers 01, d22 .... not all of which are 0, such that 1, + 02
0. As an example, the formula 0, with the constants being Φ v5, and the integer coefficients being 2, -1,-1 -1. This example illustrates how integer relations are a general structure that captures mathematical equalities.
v5+1 is equivalent to the integer relation 2Φ /5
We expand on this definition by introducing polynomial relations, which have @1 02, as the integer coefficients of (nonzero) multi-variable polynomial p. looking for solution satisfying p(z1,I2, ...) = O. For each identified polynomial, we define its degree as the greatest sum of all exponents each monomial, and its order the largest exponent that appears in it. These quantities let us measure how "complex" given polynomial is, which we use for eliminating redundancies in our algorithm. As another example, consider the relation between ı and e, captured by the Ramanu- jan formula in figure II These types of relations constitute the edges of the hypergraph we construct automatically in this work.
To account for possible numerical inaccuracies arising from our computational methods, we propose a definition that allows for (typically small) error s == |0131 + dg42 + +...| 2 This definition also applies polynomial relations, in which case |p(21, ...).. Then, the precision of the relation is defined as [loogæ=]. With the ability to quantify each constant's s numerical error as E E2, ... we instead replace in the precision with max ),, allowing the discovered relation to account for the inaccuracy of its constants.
Hypergraph edges stand for either linear relations with integer coefficients, like that between v5 and 2
v5+1 or nonlinear (polynomial) relations, like that between æ and ((2) x"
6 Interestingly.
all edges representing línear relations satisfy a kind of transitivity in any subgraph they form: given set of constants A, and another on set of constants B, such that A and B share at least one constant æ, it is possible to construct a relation on (AUB) \ {z}. However, edges like the ones show between π and impose more intricate structure that does not conform to regular transitivity but requires its generalization. A simple example of such transitivity is the (non-linear) polynomial relations between and its higher powers, and its rational exponents.
an integer relation
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2.1 ALGORITHMS FOR AUTOMATED ENRICHMENT OF THE HYPERGRAPH
Our algorithms populate the hypergraph of integer relations by searching for sets of constants 01, 22, and polynomials p such that p(#1, 22, ...) up to some tolerance. Section 3. sum- marizes the results, with a full listing in appendix B. The underlying mechanism is as follows:
Beginning with totally disconnected hypergraph (i.e. no relations are known yet), choose the search space of constants, partitioned into user-defined subsets, one of which may be fundamental mathematical constants, and another may be constants provided as limits of formulas (generally described by C-transforms), Then, each run of the algorithm takes some or all of the subsets in the partition, filters them further, and then takes their product space. In addition, the algorithm requires a choice of maximum degree and order for the polynomials. For example, the hypergraph in figure and order
d resulted from a search that yielded an edge of degree on and e, which can be represented using simpler degree 5 edge on the constant me instead.
for the polynomials
Second, once the search space decided, each set of constants is run through an integer relation algorithm to obtain a polynomial relation, using multiple precision arithmetic (see section B for details). For this we have chosen the PSLQ algorithm, whose supporting theorems ensure that can recover a relation with the working precision for almost all real vectors (Ferguson et al, 1999), except for some special cases we note in appendix B. We discuss in the next section how we run PSLQ and what heuristic we use to filter integer relations that are likely to be false positives. This is the most resource-intensin part of the algorithm, with each run of PSLQ having a time complexity of (m*)) where is the maximal number of monomials in polynomial with degree d and order o (see appendix I for a more detailed analysis). However, each run of PSLQ is independent from every other run, making the algorithm embarrassingly parallel, allowing us scale the search space with the number of processors available.
Lastly, once significant relations have been collected in the previous step to populate the hypergraph, these integer relations are then used in future runs of the algorithm to save time: Given a set of constants X, the hypergraph already has an edge e S X whose degree and order are no more than the current maximal degree and order, then we do not run X through PSLQ again. Essentially, as the algorithm learns more edges of the hypergraph of integer relations, future runs become more efficient. This also means that the algorithm can accept a hypergraph of integer relations at any point (even if partially filled), using the given integer relations to save time.
PSLQ hyperedge
Fundamental
identify
In2
Constants
|5zy In 2 5x 2yl < 50000
Integer Relation Machine
yes, return
Check Return Investment
Library of Numerical Values and Integer Relations
Automated search
Continued
integer precision
Fractions
no
lengths
drop
Figure 2: Automated search of integer relations. The process begins by collecting fundamental constants and continued fractions from the literature and organizing them into a database. Then, the algorithm checks subsets of constants for polynomial relations using PSLQ. We identify that a relation is significant by high Return on Investment (Rol), as described in section 3. Such events are generally extremely rare, and each one is returned to the database and saved, enriching the hypergraph and adding to our knowledge about mathematical constants and their relations. Each such discovered relation also saves time on future runs of PSLQ. In addition, our novel identify utility (see section Al allows for manually adding constants and continued fractions to the database regardless of the automated search.
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3. IDENTIFYING INTEGER RELATIONS BETWEEN MATHEMATICAL CONSTANTS USING PSLQ
PSLQ is a numerically stable integer relation algorithm (Ferguson and Balle, 1992). This algo- rithm uses partial sum of squares scheme to manipulate lattice, in similar manner to the PSOS algorithm (Bailey and Ferguson, 1998), using L.Q matrix decomposition (transposed QR decompo- sition). Follow-up works on PSLQ applied it in wide range of settings (Bailey and Broadhursi 200T)). PSI.Q accepts a vector of real numbers Tj T2..... and returns integers d1, Q2,,... such that a + agg2 + ................................ 0 within user-set tolerance, maximum number of steps, and maximum absolute value for all d; (or nothing if no such relation exists), Ordinarily, any of the three user-set parameters can terminate the algorithm's 's calculations, but in this section, we demonstrate how and why we run PSLQ only until tolerance is exhausted, and present novel heuristic for deciding if its result is significant, which we have termed Return on Investment (Rol).
From a purely theoretical perspective, integer relations are either true or false, with the latter provid- ing no insight towards the former. That is, if a certain integer relation accurate to 100 digits of accuracy, but is then found to be incorrect at the 10 digit, it does not imply that any "similar" inte- ger relation will be fully accurate. To identify promising relations despite the finite precision, we rely on a perspective motivated by data compression and statistics: Given d binary digits, one can express about 2 integers (including negatives). Thus, if our working precision is d binary digits, and an integer relation algorithm yielded numbers with d binary digits, the result is most likely a false posi- tive. With a tighter analysis, if one splits this reasoning into n integers, each with d1, d2. d, binary digits, then the total amount of options for all integers is about 21+d; 2 21+d2.
22+dd+ As such, for precision of d binary digits, any integer relation algorithm yielding integers with d1, d2,... binary digits such that n + d1 + d2 + ... d is most likely a false positive.
Based on the above analysis, we introduce and apply the concept of Return on Investment (Rol) to analyse the validity of integer relations: Given an integer relation with a precision of d, on integers each with d1, d2. ... binary digits, the Rol is defined as ddad--a+.... With this definition in mind, it is reasonable to state that the higher the Rol. the more likely an integer relation is to be true. This assertion matches theory, as in that case, if an integer relation is theoretically true, then the Rol grows to co as the working precision increases.
d
Experimenting with the PSLQ algorithm to determine an empirical lower bound for Rol, beyond which integer relations can be considered sigmificant. shows that an Rol of 2 seems to comfortably separate random results from false positives, with a generous margin of error (see figure B). As such,
this has been decided as the lower cutoff for integer relations generated by PSI.Q to be considered significant as part of the algorithm in section P. In order to further show how general Rol can be. we performed a similar experiment using a different integer relation algorithm called LLL (Lensira aal (182), and described its findings in appendix D
3.1 CONTINUED FRACTION CALCULATION AND CONVERGENCE
Motivated by previous works of the Ramanujan Machine (Raayoni et all 2021: Elimelech et all 202Z), generalized continued fractions (represented as C-transforms) are highly expressive frame- work of generating constants. One of the challenges of working with such mathematical formulas is that it is not always known in advance whether certain formula converges to limit. Never- theless, our numerical experiments suggest that the asymptotics of each sequence determines its convergence properties and speed, leading to the following conjecture and analysis. This analysis aids in populating our database of continued fractions by rejecting formulas that do not convergence and helping with numerical identification tasks (see section () We propose the following conjecture, which generalizes on previous works (c.g.. (Raayoni et all, 2021: Ben David of all (024)), providing the complete convergence conditions that we use when given an arbitrary C-transform:
Conjecture 1 Let fn be real sequence. The formula C[ converges in the following cases, at rate provided by the error en == 11 f1/(1 + f2/(...+ +f))) c[f,]].
ff =(nt)) for some k < 0, then e, = ((n))*)
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(a)
[b)
2/3
Rel: 5905.2 confidence
4/15
6/35
Rol 2310.5, high confidence
2n/ (4m²
+
10
1
2/15
33, moderate confidence
6v 9
4/35
2/21
2n/ (4m² 8n 3)
-2a, minimal confidence
+
Rol
0.9
mumbers 11
average Rol
0.8
10 -
0.7
140
160
2
9
10 21
12
23
14
15
17
18
binary digits
0.5
numbers
Figure 3: Experimental analysis the Return on Investment (Rol) property, showing its use for identifying integer relations. (a) For each n, we ran PSLQ 100 times with a pre-selected binary precision of 50 5n. We present the average Rol for each The standard deviation for each presented as fading errorbars, with the half length of each darkest error bar being equal to one standard deviation. The lighter dashed line the constant the constant Rol of 1.5, which the plot suggests are viable options for minimum Rol for filtering integer relations. (b) Sampled formulas are listed with their Rol on panel (a). Thanks to their high precision, their Rol is much greater than
of 1.25 and the darker dashed line
we ran PSLQ 100 times until tolerance (equal 65% of the working precision, see appendix for more details), each with n numbers between 0 and 1, each with uniformly random binary digits. Then, we present average Rol across all 100 runs for each d, n. For a fixed the average Rol is close to constant in d.
recommended Rol cutoff. (c) For each
If1 + 4fn = C + o(1) ((1)for for some constant + 1, then En =0
L+vE
If1 + 4fn Cn o(n*) for some k (0, then En = 0 (0-in/
If1 O(n²) with & 0 (1, then en O ) for some O.
Otherwise, C[Sn] does not converge.
Another challenge we face when analyzing continued fraction formulas is explicitly determining the limit of converging C-transform. In lieu of this, a C-transform converges then it is sufficient to stop at some finite evaluation depth and use the resulting convergents to calculate an approximation of the limit. Then, though it would beneficial know the exact error of the resulting approxi- mation, it is much easier and readily available calculate error proxy based on the difference to the previous depth's s approximation. Table M shows a few examples this procedure, which also showcase the convergence model in action.
THE RAMANUIAN LIBRARY: PROVIDING PUBLIC ACCESS TO DATABASE OF INTEGER RELATIONS AND MATHEMATICAL CONSTANTS
The code we have written and the library we have curated are ooun-source and aucesily- I including the novel database of mathematical constants and the hypergraph of integer relations.

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Table 1: The predicted vs measured convergence properties of selected continued fractions in their canonical forms: error analysis. Without prior knowledge, it not possible to know exactly how inaccurate a given C-transform's s approximations are. Thankfully, the error predicted by the con- vergence model usually provides a good estimate of the error proxy, which is usually a lower bound on the true error. Both the error proxy and the true error are measured in terms of how many decimal digits agree between the approximation and the true limit. Each C-transform here from the literature, which case its limit is known, or otherwise found numerically, and then its limit denoted with an asterisk and awaits proof. Note the lack of predicted error for C(n²). due to no known formula for such C-transforms (see conjecture M...
either taken
C-transform
Evaluation depth
Predicted error
Error proxy
Limit
True error
c|//n|
2
2640
2644
e
2647
C[1]
2
428
427
(1 + /5)/2
428
C[n]]
2
27
26
26
3 orto /
C(n²]
220
N/A
5
1/In2 (*)
Table 2: Sampled formulas from the hypergraph integer relations. Each formula here taken from figure a As an example, consider how the first relation can be found using PSLQ: Assuming
polynomial with degree and order 1, and given Catalan's constant G and C
2v.
PSLQ can
find integers R1, T12 n3, n4 such that n2CC
2n
+
n2C
2n
+ n2GG + n4 = 0. In this
9vs 3n2 +1
Shy 3
case, find n 2, n2 mg 0, n1
2nd 3m= and G' are known to high precision, this is a signal that an integer relation exists. The other relations here can be captured in a similar way, using more complex integer polynomials.
and given that both
Relation
Icon
2n In 3n
G
+
2G
In 2
In2
1-x
25 100m
xC =
+ C [
TT
e
2
TT
(2+))
296 188
Dn 48an 2559 495
1444 912
((2)
Our code also contains specialized algorithms for working with the hypergraph and for testing new candidate constants and candidate formulas. Access the database from our code is established using psycopg2 and sqlalchemy (Bayer, 2012). Our code contains for example the automated search algorithm described in section R.I. along with additional utilities like the C-transform calculator and the numerical identification suite we call identify, which are described in this section.
The computation of continued fractions fairly straightforward thanks to the recursive relation mentioned in appendix AL and so the C-transform calculator was developed to allow to compute C-transforms to arbitrary depths. The calculator utilizes three libraries for its computation: sympy (Meurer et all 2017)), gmpy2, and mpmath (mpmath development team, 2023). Using the above libraries, computation up to a depth typical personal computer, for most continued fractions. Using the error proxy and convergence model mentioned in section B. we also give a preemptive analysis of any given C-transform of rational function, which is also useful to warn of C-transforms that expected to not converge
million can be achieved a few minutes at most on a
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or to have slow rate of convergence. This preemptive analysis also allows the user to specify their desired error proxy instead of having to guess an explicit depth in advance. Finally, when operating on C-transforms of rational functions specifically, the calculator automatically shifts the given function so as to avoid its singularities and zeroes, resulting in an equivalent C-transform up to some number of steps forwards or backwards (see figure 2 for demonstration of this equivalence).
The integer relation machine depicted figure @ has two uses, one of which has been described in section 2.11 and the other is identify: a novel numerical identification tool. _ The input to identify an array of decimal expansions representing one or more constants and/or their implicit C-transforms. The output the result of putting all values into the integer relation machine attempt to extract meaningful relations. If enabled, identify can also automatically fetch specific mathematical con- stants or C-transforms from our database and match them against its input, informing the user of what the database is familiar with. if applicable. If at any point, identify is given C-transform that the database is unfamiliar with, identify will then automatically upload to the database. Similarly, if a new relation is found in the process, it will also be uploaded to the database.
SELECTED DISCOVERED RELATIONS BETWEEN MATHEMATICAL CONSTANTS
The novel algorithm described in section 2 has yielded results which we summarize here. These results were found after running the algorithm on smaller scale (an 8-core AWS machine) for several months. The total compute time of our results is, thus, about 16 compute months. However, since the algorithm embarrassingly parallel, runtimes of the algorithm in practice go down with the number of CPU cores used. This shows the potential of our algorithm for running much larger compute resources. Figure E shows 118 of the relations found, of which 43 were known in the literature and 75 are novel to the best of our knowledge. Figure @ shows some example relations. Appendix IB catalogues all relations in detail. These results were found after starting with completely disconnected hypergraph, containing the constants of interest, and no initial relations. Here we present notable discovered relations involving म, e, (n2 and the Lemniscate constants.
We recall the following formula, attributed to Ramanujan etendrecal, 1999) (using the notation defined in equation II, section D):
πE 1 + 2C 2n)/(An(n + 1))l - 2
Cin]
The search job unearthed a family of conjectures that generalizes the second C-transform in this relation, given in table B
Next, we examined the family of constants Cin²/(k²(1 - 4n²))] for integers k 2 1, first noted in Ben David at (2024) (table 4 column 1). When they were first discovered, these C-transforms were known to converge but their limits were unknown. Our search algorithm connected three of
these C-transforms to 1n2 in pairs:
1
1
In2 2C[n2 (4(1 4n2))]
7C (49(1 4n?)]]
2
2
5C'n2/ 4n2))]
7C[n2 []n2//(49(] 4n²))]
1
2
C[n2/(4() - (4((((1212)
5C|n2/(251) 4n²))
Thanks to the high precision to which these equations have been verified, our Return Investment (Rol) measure (section ()) quantifies how unlikely they are to be false conjectures. These conjectures also listed in appendix E. Later investigation has revealed:
Conjecture For all k 2
2 Jn
2/k lack+1) in
"See README in the Github link, or the tutorial hosted at: https///colah.rssaacc.....o wwtpg//coo///ov//////PPX.X TAnnIAAAAAAAAAAAnnAA T TOTNODDDDDOSSSSWWNKKatll
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Table Discovered Ramanujan-like formulas for VTC. For brevity, we denote Ra erf vI The rows in this table represent the 8 formulas we discovered for Vire, generalizing Ramanujan's original formula that was so far unique. We proved the first 4 rows using transformations on Ramanu- jan's original formula. The latter 4 rows are still unproven (yet equivalent, such that proving any one of them will prove all four). The last row of each set of four is an infinite family of formulas: for any integer & 2 there exist integers a, 3, 7, that satisfy an equality between the two sides.
C-transform
Limit
C[(1 2n)/(4n(n++1))
(R2 + +1/2R2
C[ 2r)
(2R2 + +1/4
C[(1 (n)
[B2 + +1/
C
(aR₂ + +3)/(7R2 +
(nn
1/(2R2/ -1) R2
C[ + 1)/((n(n 1))]
( + ()
(2R2 + 1)/ + +1) (aR2 (> +5)
C[ + + (2/
As a final example of notable results, we compared the performance of identify with commercial nu- merical identification tools such as Wolfram Alpha B. Our identify method was successful in cases for which Wolfram Alpha was unsuccessful in identifying the constant. This was the case for formulas such as C[-(2n 3)2 /(18n(n + 1)]] (as an example of larger family of C-transforms), Executing 5A+08
identify found that the limit is 5.1-92 (to 50 digits of accuracy initially, and further thousands upon reconfirming), where A, B are respectively the first and second Lemniscate constants.
LIMITATIONS AND OUTLOOK
Our work presented a successful approach automated formula discovery in number theory. The numerical nature of our algorithms means that results are not theorems, but rather conjectures await- ing proofs. In many cases, the discovered relations between formulas led to patterns that enabled generalizing formulas into families. Some of these families are connected such that proving one for- mula proves the entire family. We provided proofs for selected cases (shown in appendix E), hinting that the other discovered formulas can also be proven. In similar vein, our findings reveal the com- plete rules of convergence of general continued fractions, yet these discovered rules lack a formal proof for many of the special cases. Future work by number theorists can follow on our groundwork to complete the proof and build the complete theory convergence of continued fractions.
The Return Investment (Rol) mechanism that we presented to quantify the success of the PSLQ algorithm can be directly extended to other integer relation algorithms such as LLL (nsstra et all 1982) (see also appendix DD.. This quantitative approach algorithms in a wider range of fields, and will benefit from a thorough theoretical investigation.
contribute to the application of such
Our approach and implementation implemented in a way that allows for distributed computing. We are now adapting the algorithm to utilize additional computing power using the online distributing network BOINC (Anderson, 2003). Such large-scale computing efforts will be very promising in generating new, more elaborate for- mulas and interrelations. The increase in the number of interrelations will require corresponding methods to climinate false positives, which could be accomplished automatically by retesting them over time with higher precision constants. Sufficient precision will eventually reveal each potential false positive. Moreover, the current automated search algorithm relies on pre-selected, static set of constants, which directly determines which relations can be found. Methods of automatically adding "interesting" constants to the database can improve the potential for finding novel relations.
benefit from many improvements. The search algorithm was
Looking forward, the publicly-available Ramanujan library that we developed can now serve number theorists for investigating mathematical constants with more powerful tools that are easily accessible. The hypergraph of integer relations may be viewed like knowledge graph, and as a public resource, it can help open such areas of investigations to a broader community with different levels of expertise.
00A0000100000881113313177 131 01113366666111111999999933333331 119933222316611

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Published as a conference paper at ICLR 2025
By presenting fascinating connections between constants, hope that this resource will invoke curiosity in younger audiences and encouraging mathematical experimentation as a path for learning mathematics. In its next generations, the Ramanujan library be improved by augmenting the hypergraph with written explanations on constants and links to relevant papers, forming public hub of knowledge on constants and their relations.
ACKNOWLEDGEMENTS
This research is supported by the generosity of Eric and Wendy Schmidt by recommendation of the Schmidt Futures Polymaths program.
This research received software engineering support from the Georgia Institute of Technology's Scientific Software Engineering supported by Schmidt Sciences, as part of Virtual Institute for Scientific Software (VISS) Program
In2
In2
C
5C
16n
25
100n
((5)
(1
The hypergraph of integer relations: each vertex is a constant, and each edge formula. The hypergraph summarizes our automated discovery of relations between mathematical constants, presenting selected formulas from database. This hypergraph does not show all dis- covered relations for clarity's sake. Empty circles denote constants (written inside). Colored circles denote continued fractions. Black denotes continued fractions whose limit is known in the broader literature. Green denotes continued fractions whose limit is found in (Raayoni et all, 2021). Blue de- notes continued fractions whose limit was unknown before our work, to the best of our knowledge. We denote order-1 connections (Mobius-liie like) with solid lines, and higher-order connections (constants may appear squared, cubed, etc.) with dashed lines. Edges that connect more than two vertices are marked with small empty squares, denoting formulas that involve more than two constants of formulas (example in the inset at the top right). Each algebraic constant has an edge of size 1, also marked with small empty square (placed at the top left). Such degenerate edges correspond to the minimal polynomial of the constant.
Figure 4:
10

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REFERENCES
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David Bailey and David Broadhurst. Parallel integer relation detection: Techniques and applications. Mathematics of Computation, 70:1719: 179-11736, 2001.
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Nadav Ben David, Guy Nimri, Uri Mendlovic, Yahel Manor, Carlos De la Cruz Mengual, and Ido Kaminer. On the Connection Between Irrationality Mcasures and Polynomial Continued Fractions. Arnold Mathematical Journal, 2024.
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Alex Davies, Petar Veličković, Lars Buesing, Sam Blackwell, Daniel Zheng, Nenad Tomašev, Richard Tanburn, Peter Battaglia, Charles Blundell, András Juhász, al. Advancing mathematics by guiding human intuition with AI. Nature, 600:70-74, 2021.
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Rotem Elimelech, Ofir David, Carlos De la Cruz Mengual, Rotem Kalisch, Wolfgang Berndt, Michael Shalyt, Mark Silberstein, Yaron Hadad, and Ido Kaminer. Algorithm-assisted discov- ery of intrinsic order among mathematical constants. Proceedings of the National Academy of Sciences, 121:c2321440121, 2024.
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Helaman Ferguson, David Bailey, and Steve Arno Analysis PSLQ, an integer relation finding algorithm. Mathematics of Computation, 68:351-369, 1999.

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A ON GENERALIZED CONTINUED FRACTIONS
generalized continued fraction, given two sequences on bas is the following formal expression:
b1
GCF bn) dg
a1 + b2
b3
02
In practice, one defines two sequences pu: In called the convergents, such that
b1
Pa
0g
b2
Gn
Q1
+
b3
Q2
+
bn
On
and then both sequences satisfy recurrence relation, summarized by the following matrix product:
ag
0
6,
Pn-1
Pn
9n-1
nn.
The special case of bn = 1 called (regular) continued fraction. C-transforms are the special case of On = 1. Using basic transformations, any generalized continued fraction can be converted into this latter form.
2n
2n
GCF
1. An²
GCF + 1,2n]
1n=
sqrt(e)
2n
2/15
2/15
6y/²
(2n 1))2n
2n +
4/35
(2n + 3) + 5)
2R +
(2n + 5)(2 7]
Figure 5: Equivalences between continued fractions. Expanding on figure MIl our representation of the continued fractions enables each integer relation to capture an infinite family of formulas. This redundancy is mostly eliminated by using the C-transform. Additional equivalences still exist, as discussed in section and shown by the second integer relation.
Our choice of C-transforms arises from the following equivalence transformation property of con- tinued fractions: Given any GCF(a,, bn] and nonzero sequence cn, the following equality holds:
b
cib
ag +
b2
ag
+
C1 c10222
al
Ci
+
ba
C20gb3
+
C202 +
Or. more compactly, c,GCF[aa, bnl GCFIc,an b,,b (see figure B for more concrete
amples). This equality holds even neither side converges, in which case the convergents still coin- cide for all As such, if a, * 0, the choice of c, on
yields 80
GCF (an, (ibb,/(an-1a 10))
13

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B REBOUNDING DETECTION
Since we have elected to nullify two of the three stopping conditions of PSLQ, we must consider whether or not the remaining stopping condition can always be achieved. Namely, can every run of PSLQ terminate when only stopping once tolerance has been achieved. Even with a fairly standard choice of 75% tolerance, the answer is no, at least for the implementation we use. However, our ex- periments show that all such cases appear exhibit the same general behavior: PSLQ first gradually increases the precision of the integer relation as usual, and at some point before reaching tolerance, it reverses course and starts reducing the precision (see figure G for example run).
Our implementation of PSLQ includes a failsafe that detects this behavior: As PSLQ iterates in the main algorithm, remember phest the best precision ever obtained in the current run. If at any point Peurrent the current precision satisfies 2pcurrent < Pbest. abort the algorithm and return no integer relation. However, this mechanism is only enabled after initial grace period of 100 steps. This grace period was chosen since it was observed that during the phase where the precision increases, does not always increase monotonically, and can fluctuate.
precision
80
60
identified abort
pression
40
20
PSLO gets stuck
20
60
100
step number
Figure 6: Example run of PSLQ showcasing the rebounding phenomenon. The axis represents the precision of the best integer relation that PSLQ finds each step number, which must reach the dashed line to terminate. The red dot where the failsafe would have triggered, except that is being prevented by the grace period to enable computing this example.
C TIME COMPLEXITY OF THE PSLQ ALGORITHM
Given ø vector of m real numbers, and assuming the minimal euclidean norm of all integer rela- tions on it is M E IR, running PSLQ to retrieve an integer relation on & has a time complexity of O(m² m log M2) (see (Ferguson all (199) corollary 2). To estimate the time complexity of our runs of PSLQ, we substitute M, with constant based on the working precision. When applying PSLQ large scale, choosing an upper bound for the precisions of the constants involved prevents this M2 term from dominating the runtime, and so we can asymptotically treat this as constant, leaving us with time complexity of O(m²).
Since we run PSLQ to detect specifically polynomial relations, we should evaluate m in terms of the amount of constants n, the maximal degree d, and the order the relation, as these dictate how many monomials we expect the polynomial to have, and in turn determine m. If we ignore the limitation on the order, counting all possible monomials of polynomial on n variables with degree d is identical to counting all multisets of size d on + I items, each of which represents either one of the variables, or a dummy variable that completes the monomial to degree d. This has closed form solution based the stars and bars method:
12 + --=
+
(n d)! d!n!
m
When including the limitation on the order, one obtains smaller value than this through a more delicate analysis, but to the best of our knowledge, no closed form solution exists. Regardless, this
means that each run of PSLQ will have a time complexity of O ((*te))
1)
14

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Published as a conference paper at ICLR 2025
D THE GENERALIZEABILITY OF RETURN ON INVESTMENT (ROI)
The Rol measure that we introduced section is defined on numerically-comjeeuuned integer re- lations. Our work relies on the PSLQ algorithm, which one of many integer relation algorithms, and so it is natural to ask if Rol can be applied to other integer relation algorithms and reach the same conclusions. Here we motivate a positive answer to this question by demonstrating the same experiment shown in section B (specifically figure ED, but performed on the LLL algorithm instead (Lenstra all (182).
Figure Z. which is structured in an identical manner to figure B. demonstrates nearly identical experiment, except the PSLQ algorithm is replaced by the LLL. algorithm. This experiment used the oll python package for implementation of LLL. The result shows how the algorithms are similar yet different in terms of Rol: as n increases, both algorithms exhibit greater average Rol, up to some apparent upper limit less than 1.5. However, PSLQ showed greater standard deviation than LLL, which is also visible as greater noise on each figure's 's (c) panel.
(a)
(b)
10a
2/3
: 5905.2, high confidence
V8
4/15
6/35
2310.5, high confidence
2n1/ (4m²
1)
1
10
1.6
1
14
2/15
6.
4/35
2/21
1.4
~3a, moderate confidence
8n
3)
1+'
Rol
1.2
(c)
1.0
1.0
averag perage nol
0.8
0.8
10
0.7
0.6
140
0.5
23
binary digits
numbers
Figure 7: Experimental analysis of the Return on Investment (Rol) property, demonstrated on the LLL algorithm. (a) For each
ran LLL 100 times, with preselected binary precision of 50 + 5n, and present the average Rol for each n. The standard deviation for each n presented as fading errorbars, with the half length each darkest error bar being equal to one standard deviation. The dashed line is the constant Rol of 1.3, which the plot suggests is a viable option for minimum Rol for filtering integer relations from LLL. (b) Sampled formulas listed with their Rol panel (a). Thanks to their high precision, their Rol is much larger than our recommended Rol cutoff. (c) For each n, we ran LLL 100 times, each with n numbers between and 1, each with uniformly random binary digits. Then, we present the average Rol across all 100 runs for each d, n. For fixed nı, the average Rol is close to constant in d.
E PROVEN FORMULAS
After having found the formulas shown in the hypergraph (figure D) proved some of them, and we list these proven formulas here. We also plan joint works with mathematicians that focus on the proofs of the formulas and their potential impact in the relevant fields of mathematics, to be submitted to mathematics journals. This effort complements our main contribution in this work, which is concerned with the automated generation of the formulas.
In total, we proved 47 formulas of the form an + b (a + b2/(... + b,/(a,, + ...))), of which 22 showcase in this table. of the formulas can be generalized to three parametric families,
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Published as a conference paper at ICLR 2025
presented in lines 1-9. Lines 10-25 describe additional proven formulas. The proofs rely (I) inverting Euler's s continued fraction formula with parameters h1, h2, f to (2) create an infinite sum, which we (3) translate into hypergeomettri-funntiie form, and (4) complete the proof using known hypergeometric identities.
Table 4: Selected list of formulas that we proved.
On
bn
h1
h2
f
Known limit of resulting infinite sum
n² wn
+
1
(1 + (2 (,, 2,- 1)
5
n² + 4n
n
+ 4
1
131 192 log
4
n² 3n
+ 3
1
3
1og
(wn 1)²
Ww 1
1
1
(w 1)(2
w+1 2w+1 , 11-1-111
2
4n² + 4n +
22
2n + 1
1
I
1
n² +2n +1
-n 1+11
1
log (2)
l-log2
n² (w + 1)n
-1:
(w
+ -1)
3
n² + 4n +3
+ 3
Ing
2
n² 33 +2
-n
9-
12 log
5
n² +2
n+2 2
+
-2
17
24 log
5
n + 4n +3
72
n n+3
n +
-3
5
1560
n +
-
n n
11 +
12 16 log
4
n² + 3n +
11
2
7++2 2
2
20-994 99 104
4
4n² + 4n
-2n
2n + 1
244
log
3
n
-
n
3-3 1l3og3 2
3
n + 2n +
72
n n+1 1
n +
3
3( 20 30
log 2)
3
n² 4n +
- 2
+
-6
21
+105 log
1
n² + 4n +
n
-n 2
1
-2
+-3log2
4
4n -
-2n +
2n 1
1
8
+1
+
2
4n 4n
- 2n +
2n -
1
+1
5
4n + 2n
-2n
2n +
1
-20 16V3
5
4n + 2n
-2n
2n
n
n+3
3
2
3
4n 2n
-2n +1
2n
1
2+ v2
2n
+
2n
+
n² n
1
2n
+
22 n1
1
((2)
F DETAILED INTEGER RELATIONS FROM THE HYPERGRAPH
This section provides the full data underlying the hypergraph (figure H. Recall that according to appendix A. each of the integer relations presented here can be connected to infinitely many gen- eralized continued fractions, and the C-transform is chosen as a representative for them all. Thus, every vertex in the hypergraph could represent infinite formulas that we consider trivially equivalent. This reasoning motivated the use of the C-transform as the canonical representative that defines the vertex. The edges in the hypergraph provide additional (non-trivial) equivalences. Some of the for- mulas and relations shown here can be found in the literature, and others are novel, to the best of our knowledge (as denoted in left-most column). Every integer relation has been rearranged for clarity, though binary precision and Rol (rounded down) still computed in terms of the original integer relation.
Novelty
Integer relation
Precision
Rol
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Published as a conference paper at ICLR 2025
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17

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Published as conference paper at ICLR 2025
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---

Published as a conference paper at ICLR 2025
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---

Published as a conference paper at ICLR 2025
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Reviews:
[{"review_id": "qsGAOLg888", "reviewer": "Reviewer_cctr", "summary": "The paper introduce a new search algorithm for integer polynomial relations between mathematical constants. The authors organize their algorithm into a library and use a hypergraph as data structure to store the relations. The library can interact with a relation finding machine to search for new relations.", "questions": "Some other minor comments and questions: \n- Line 42, I believe it is quite a stretch to claim that any computer assisted proof (which comprises of the majority of citations in this sentence are examples of \"usage of AI as a scientific tool\" \n- Line 101-104, it took me awhile to parse the sentences here. This can be solved by explicitly stating the definition of C-transform and how it differs from continued fraction since at first glance and without definition, it's not easy to distinguish. Alternatively, one can also just give an example of the difference. An example of why this is confusing is: it seems that C-transform, as stated on arbitrary function f_n, captures all continued fraction; yet some continue fractions cannot be converted to infinite sums when C-transforms can? \n- What is the question mark in table 2?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 2, "strengths": "Strengths:\n- The paper claims to have discovered new integer relations between constants that have not been discovered before. This is an exciting and relevant direction for publication in ICLR since machine assisted mathematics has many unrevealed potentials.\n- The methodology is sound and correct, since it is a combination of previously published and peer-reviewed systems. \n- The ROI scheme, while simple, may prove to be generalizable to other domains/applications.\n- The resulting code seems to be available publicly. \n- The paper is mostly written in a clear and comprehensive way, with appropriate figures and tables for demonstration.", "weaknesses": "Some weaknesses include:\n- There is almost no machine learning (emphasis on the 'learning' part) in the paper, as it seems to be an engineering/database product that interfaces with existing code base. This is, of course, not a weakness on the content of the paper, but on its fit for the venue of publication.\n- The use of hypergraphs as data structure, while theoretically clean, is perhaps only practically useful if the code can make use of graph symmetries (for instance in a graph learning framework). Due to the lack of practical ways to distinguish sets vs sequence in hardware, I very much doubt storing relations as hypergraphs is much more practically advantageous than any naive data structure. Of course, some data structure is needed to store the relations, but the authors also claimed that this is an \"effective representation\" (line 72) and overall feature it as one of their main contributions. \n- From my understanding, perhaps inherent to the problem the authors are trying to solve, until a proof is given, it is almost never guaranteed that any new relations found are actually correct. This makes the problem rather ill-defined since there are no performance metric attributed to the proposed methodology, making it hard, almost impossible, to judge its effectiveness, outside of engineering perspective. \n\nOverall, this is still an interesting paper, even as an engineering paper so I still advocate for publication just for the code and methodology to be published for more to see.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "UyDBWklx6I", "reviewer": "Reviewer_Bz6b", "summary": "This proposes an improved method to discover equations", "questions": "It may help to answer questions in my detailed comments, even if that is not guaranteed to clarify everything.", "limitations": "", "rating": 3, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "Automated equation discovery is interesting and providing a library with basic tools is certainly useful.\nThe text uses good english.", "weaknesses": "The text only provides a high level explanation of a few ingredients of the proposed method.  While it may be understandable for a few specialists, the text isnt accessible to the general logic or reasoning expert.\nThe text doesnt introduce the basics of the considered formalism, does not offer specifications nor examples of inputs or outputs of the system.\n\n\n\n#### Supplementary material\n\nThe code contains a Readme file but it does not comprehensively explain how to install,or use the code.  Despite looking around to the several files i didnt succeed to use it nor understand it.\n\n#### details\n\n* when defining hypergraph, make clear whether you consider directed / ordered or undirected edges.\n* Sec 2 mentions giving cnstants such as $\\pi$ or $e$ explicitly but does not say how to give an irrational number explicitly.\n* please specify how one can define a C teansform (as it contains an infinite number of parameters)\n* while libe 108 talks about polynomial relations, line 111 only gives the form of a linear relation\n* line 115 requires thar the absolute value of the linear expression equals exactly $\\epsilon$.  I suppose you mean $\\le$ instead.\n* the precision is defined as $\\log(\\epsilon)$, i guess this can lead to problems if $\\epsilon$ happens to be exactly 0 ?\n* line 119: \"we define its degree as the sum of all exponents in each monomial\": does this imply you consider homogeneous polynomials, as else the different monomials may have a different sum of exponents?\n* line 125: is an edge a hyperedge?\n* line 127: please define transitivity in this context\n* line 153: please define \"type\"\n* line 155: the expression \"product space on certain subtypes\" is very unclear.\n* line 159: it is unclear what is a combined constant, or what language or equations the user can use to define them.\n\nSection 5 may have provided useful insight in capabilities of the proposed system,if the used notations would have been clear.\n\nMaking abstraction of that problem i observe the displayed equations are rather complicated and hence the search space of possible conjectures must be huge.  Given one searches only approximate equalities under bounded precision it seems likely one will discover many incorrect conjectures.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "pKJXe14BUx", "reviewer": "Reviewer_AmTb", "summary": "The paper presents a new library for automatically discovering functional relations between mathematical constants. The paper outlines the structure of the library as well as the numerical approaches to discovering (polynomial) functional dependencies between fundamental mathematical constants. The authors discuss the convergence properties of the selected approach as well as some limitations and future directions.", "questions": "Would it be feasible and useful to create a knowledge graph that contains information about the constants and the relations between them? Initially, it will contain human/LLM written explanations, e.g. a short text with references to research papers that introduce the constant or relation and how it is used. When new relations are discovered, the graph will be automatically updated with a short description of the discovery process including computational statistics. The edges of the graph can be labeled with the different relation types (e.g. linear, polynomial, etc.) and/or the certainty of the discovered connection. Knowledge graphs are very popular and this would allow the usage of third-party software for a better visualization and understanding of the discovery process.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "I am not an expert in the area but I enjoyed reading the paper. There are quite a few things I like about it:\n\n- Fundamental mathematical constants are fascinating, they are often the cornerstones in many scientific disciplines. Discovering new complex relations between them can inspire interdisciplinary discoveries.\n- The presented approach for the representation of organizing the relations in a hypergraph and searching for new relations using numerical methods is innovative.\n- The implementation is publicly available and likely to benefit the broader scientific community.\n- The most time-consuming algorithm is embarrassingly parallel, so it is likely scalable.", "weaknesses": "I think the presentation can be improved. The paper is interesting but at times it felt more like reading an article from Quanta magazine. Here are a few suggestions:\n- Include a more thorough description of the setup. How was the hypergraph initially created, how many constants and relations were used, how much did the graph expand when new relations were discovered, etc? Some tables and figures can help.\n- Include some explanation about operating the library. For example, how can one include new constants and start searching for relations? I think some high-level pseudocode will help.\n- Add some formal results about the validity of results and rate of convergence. I think the discussion in section 3 can be organized in a more formal way using some lemmas. This will help the reader concentrate on the results and maybe think about improvements. \n- Maybe provide examples with a few more constants, I see only $\\pi$, $e$ and $\\zeta$.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)