On the Connection Between Irrationality Measures and Polynomial Continued Fractions

Linear recursions with integer coefficients, such as the one generating the Fibonacci sequence, have been intensely studied over millennia and yet still hide new mathematics. Such a recursion was used by Ap\'ery in his proof of the irrationality of $\zeta(3)$, later named the Ap\'ery constant. Ap\'ery's proof used a specific linear recursion containing integer polynomials forming a continued fraction; called polynomial continued fractions (PCFs). Similar polynomial recursions prove the irrationality of other mathematical constants such as $\pi$ and $e$. More generally, the sequences generated by PCFs form Diophantine approximations (DAs), which are ubiquitous in areas of math such as number theory. It is not known which polynomial recursions create useful DAs and whether they prove irrationality. Here, we present general conclusions and conjectures about DAs created from PCFs. Specifically, we generalize Ap\'ery's work, going beyond his particular choice of PCF, finding the conditions under which a PCF proves irrationality or provides an efficient DA. To provide concrete examples, we apply our findings to PCFs found by the Ramanujan Machine algorithms to represent fundamental constants such as $\pi$, $e$, $\zeta(3)$, and the Catalan constant G. For each such PCF, we demonstrate the extraction of its convergence rate and efficiency, as well as the bound it provides for the irrationality measure of the fundamental constant. We further propose new DA conjectures based on PCFs. Our findings motivate future research on sequences created by any linear recursions with integer coefficients, to aid the development of systematic algorithms for finding DAs of fundamental constants. Consequently, our study may contribute to ongoing efforts to answer open questions, such as the proof of the irrationality of the Catalan constant or of values of the Riemann zeta function (e.g., $\zeta(5)$).


Apéry's constant and his polynomial continued fraction (PCF)
In his paper [1,2], Apéry ingeniously presented a specific linear recursion with integer polynomial coefficients that proves the irrationality of (3). This polynomial recursion generated two sequences , (given different initial values) such that / →∞ → 6/ (3), i.e., it constituted a Diophantine approximation of (3). Apéry then showed that this specific sequence proved the irrationality of the number to which it converged.
He also demonstrated [1] that the linear recursion was equivalent to the following polynomial continued fraction (PCF). Apéry's paper inspired other researchers to apply related strategies to other problems in Diophantine approximations, to study irrationality measures of other constants, and to find applications in other fields [3][4][5][6][7][8][9][10].
Apéry's result hints at a more general question: Which PCFs prove the irrationality of the number to which they converge? In other words: Which pairs of integer polynomials (such as − 6 and 34 3 + 51 2 + 27 + 5 in Apéry's case) can be used to prove irrationality? This question is directly related to the intrinsic properties of PCFs, specifically, their rate of convergence and the properties of the Diophantine approximation sequences they create.

Polynomial continued fractions (PCF)
In their most general form, PCFs denote a generalized continued fraction in which At each finite step , the PCF is a rational number / , where and are the numerator and denominator of the th convergent, respectively. Both and can be shown to satisfy the same recursion of depth 2: with different initial conditions, −1 = 1, 0 = 0 −1 = 0, 0 = 1.
The limit of / is the value of the PCF. There exist Mobius transformations with integer coefficients that transform between the limits of / for different initial conditionsfor any two pairs of rational, linearly-independent initial values [11].
PCFs appear in a wide range of fields of mathematics and are related to many special functions, including all trigonometric functions, exponentials, Bessel functions, generalized hypergeometric functions, and the Riemann zeta function, and many other important functions such as erf and log [12][13][14]. Moreover, any infinite sum can be converted into a continued fraction (known as Euler's continued fraction). The other direction is not correctnot every continued fraction can be converted into an infinite sum. The space of PCFs also contains all linear recursions of depth 2 with rational polynomial coefficients (and some of their generalizations). In his study, Apéry developed a linear recursion with rational polynomials, and since it was of depth 2, he was able to convert it to a PCF, using the standard definition above.

The goals of this paper
Looking at the bigger picture, it is interesting to generalize Apéry's PCF. Consider an arbitrary linear integer recursion (of any order) used to create the numerators and denominators in a sequence of rational numbers. In other words, provided two sets of initial conditions, for the numerator and denominator, the linear recursion creates a Diophantine approximation sequence. Each such sequence may provide an efficient representation of the limit of the sequence. Intuitively, the efficiency is described by the rate (as a function of ) at which the sequence converges relatively to the sizes of the denominators. What can be said about the resulting sequence? What condition should the linear recursion fulfill for the generated sequence to prove that its limit is irrational?
More generally, what bounds on irrationality measures does each linear recursion create?
In this paper, we describe the construction of a systematic method to find, for each PCF, the efficiency of its limit approximation, i.e., the lower bound it provides for the irrationality measure (we address a lower bound that simultaneously provides an upper bound [2]). We develop a criterion on the PCF for proving the irrationality of its limit.
Specifically, Theorem 2 states a formula for the irrationality bound for each PCF, states that, for the growth rate of the GCD to be sufficient for an irrationality proof, the polynomial must be a product of two rational polynomials of equal degrees.
An important advantage of this approach is that it does not require the determination of the PCF limit or any knowledge of it. PCFs that yield efficient Diophantine approximations are in general also better for computing more quickly the numbers to which they converge. Consequently, the results of our study could be used to develop faster means for high precision calculations of fundamental constants, such as attempts to compute more digits and study the normality of such constants [15][16][17][18][19][20][21][22].
Any mathematical expression that can be converted to PCFs, such as infinite sums used for the computation of fundamental constants [15][16][17]20,21], could be analyzed with the approach that we present in this paper. The conjectures that arise from our study hint at a general theory that goes beyond PCFs to any polynomial recursion, and maybe eventually beyond it to any linear recursion with rational coefficients.
Some of the conclusions of our study presented below go beyond PCFs and beyond the motivation of irrationality proofs. In general, when given any linear recursive formula with integer coefficients, not necessarily one representing a PCF, it is interesting to study the GCD of two (or maybe more) sequences arising from the same recursion with different initial conditions. We find the solution for special cases of linear recursions, showing the rate of growth of the GCD. We hope that our study will contribute to efforts toward finding the general rules for GCDs of arbitrary linear recursions.

Motivation and potential applications
Many of the PCF formulas that led us to the conjectures and proofs in this paper were originally found in the Ramanujan Machine project [10], which employs computer algorithms to find conjectured formulas for fundamental constants. Various algorithms are being developed as part of that project, and so far they all focus on formulas in the form of PCFs. Since the algorithms check candidate formulas by their numerical match to target constants, the results are always in the form of conjectures rather than proven theorems. The first algorithms succeeded in finding conjectured PCF formulas for , , values of the Riemann zeta function , and the Catalan constant [10]. These latter formulas led to a new world record for the irrationality bound of the Catalan constant.
The theorems and conjectures below can also help improve future algorithms that search for such conjectures.
We point to three interesting challenges that motivate this work, each having prospects in Diophantine approximations, as well as in experimental mathematics, i.e., computation-driven mathematical research (e.g., [10,23,24]): (1) Given the polynomials , of the PCF, determine whether the PCF provides a bound on the irrationality measure, and if so, then find the bound analytically from , .
(2) Estimate the efficiency of each PCF for computing fundamental constants to high precision.
(3) Develop faster algorithms to compute PCFs; more generally, compute any polynomial recursion more efficiently.

The measure of irrationality of a number
The irrationality measure of a number is the largest for which there exists a sequence of rational numbers / ≠ s.t.
This maximal is called the irrationality measure of the number [2,15], or the Liouville-Roth exponents. For irrational numbers, this maximum can be obtained by the regular continued fraction of ; however, its closed formula is often unknown (e.g., in the case of ). The Diophantine approximation is thought of as more efficient when is larger. Rational numbers have an irrationality measure 0, meaning that they cannot be approximated efficiently by other rational numbers. This property is part of the irrationality criterion: if there exists a sequence / for which this inequality holds for some > 0, then is irrational. Further, if there exists a sequence / for which this inequality holds for some > 1, then is transcendental by the Siegel-Roth theorem.
Finally, if the inequality holds for arbitrarily large values of , then is a Liouville number (infinite irrationality measure) [15]. Intuitively, for a sequence that proves irrationality, the growth of the denominator should be sufficiently slow in relation to the convergence rate of the PCF. For our purposes below, the sequences / are generated by PCFs.
In the rest of the paper, we use the symbol = { / } to denote the largest that satisfies the inequality for a specific sequence of rationals { / } and almost all values (also called an effective irrationality measure) [10,15]. For each sequence, there always exists a sequence for which = 0 (for a rational ) or ≥ 1 (for an irrational ).
However, the largest known can be smaller or larger than 0. To find even one explicit sequence that reaches the maximal value is challenging. This challenge continues to motivate searches for new sequences { / } for constants, from which one can extract larger lower bounds for . Each constant for which the rationality or irrationality is still unknown, has all its known sequences with ≤ 0 (as in the case of the Catalan constant [19][20][21][22]). Then, finding one PCF for which > 0 will directly prove irrationality. When is known to be positive, as in , it is still interesting to find better PCFs with larger values, because it improves the bounds on the constant's irrationality measure (e.g., 's upper bound [25] by Zeilberger and Zudilin). Therefore, it is of interest to find sequences for which is as large as possible, even when the value is negative.
This paper shows the conditions on and for which PCF[ , ] provides nontrivial (larger than −1) and presents certain conjectures for the dependence of on the choice of , . To find these conditions and conjectures, we predict the PCF convergence rate | − | and the rate of growth of the reduced denominator gcd [ , ] .
We present a criterion for the growth rate of the greatest common divisor GCD[n] ≝ gcd [ , ], which is necessary and sufficient for a nontrivial : ln( ) ≝ limsup 1 ln ( GCD ! deg ) > −∞. We use this criterion to calculate and provide conjectures for its dependence on , .

Summary of the main results
Unless stated otherwise, we focus on "balanced-degree PCFs", where This PCF type is arguably the most common in the literature related to mathematical constants (see Appendix A, Ref. [10], and further references therein). We show that the growth rate (as a function of ) of the GCD is key to the analysis of PCFs of this type.
We find special interest in cases of PCF [ , ] for which the GCD grows so fast that it reduces most of the denominator : that is, while the denominator grows as some power of !, the reduced denominator /GCD[ , ] is of exponential order. We call this phenomenon factorial reduction (FR).
Below, we prove that for a PCF to provide a nontrivial value, FR is necessary (Theorem 1). We also derive formulas for these s (Theorem 2), which could help provide irrationality proofs. The other results of our work are conjectures which attempt to provide a complete characterization of PCFs with nontrivial . All the conjectures are backed with extensive, computer-based, numerical tests and await a formal proof. , for = 17 + 12√2, which exactly matches our general expression (see Theorem 2). Below, we generalize this process and conclusions to all PCFs, classify their different GCDs, and present a criterion for the PCF that allows us to prove irrationality.

Theorems about factorial reduction (FR)
We tested many PCFs for FR and identified a surprising phenomenon: despite the rarity of FR in an experimentally random PCF, we have so far found FR in every PCF that converges to a fundamental constant (we tested PCFs that converge to , , (3), (5), and the Catalan constant ). Specifically, we tested all the PCFs found so far in the Ramanujan Machine project [10] and many other PCF formulas. This relation between FR and PCFs of fundamental constants is surprising because the algorithmic search in [10] did not favor PCFs that have FR. This intriguing fact hints at an underlying structure of PCFs that is required for formulas that converge to certain mathematical constants.
for some subsequence of indexes. FR means that > 0.
The notation ≐ represents that lim →∞ 1 log = 0. That is, and agree in their exponentials but may still differ in slower than exponential portion (e.g., they may differ in polynomial pre-factors before their exponentials).
To set the ground for Theorem 2, we denote to be the larger solution (in absolute value) of the equation where and are the leading coefficients of and , respectively.

Theorem 2 (A formula for δ). For a PCF[ , ] of the second type with FR, the effective irrationality measure is
The case of ≤ 2 /4 remains for future study.
This formula connects the GCD to the value of . Note that larger values of imply larger values of (beneficial for proving irrationality). The maximum possible value is , which is positive for ≥ 3 and thus proves irrationality for all these PCFs' limits. Table 1 These examples emphasize the strength of our approach: the determination of an irrationality measure without a need to find a closed formula for a PCF sequence or to find the PCF limit.

Conditions for the existence of factorial reduction (FR)
Theorems 1 and 2 leave us with two important questions: (1) which PCFs have FR and if so, then (2) what are their exponential orders . Following many computer tests, the following conjecture is an effort to answer the first question. The second will be discussed later.
The complete structure for deg = 2 and deg = 1 Having performed many numerical tests, we propose the next general conjecture for the families for which the PCF has FR for a given splittable of degree 2.  .
We tested many of these PCFs numerically and indeed they all have FR.

Summary of our main conjectures regarding polynomial continued fractions (PCFs)
The above examples summarize the four aspects of our conjectures so far:  i.e., a rational root. It is interesting to try generalizing the above conjectures to discover the most general rules of this mathematical structure. Additional hints of the mathematical complexity of the yet unknown general structure are related to the existence of generalized Pythagorean triples (see Section 2.7 below).

Closed-form formula of the GCD and the effective irrationality measure
The goal of considering the next sections is to predict exponentially tight formulas for the GCDs, i.e., up to a slower than exponential factor. For each PCF[ , ], we aim to find both the exponential order and the closed-form expression for the GCD that yields this . Representative examples are provided in Table 1  "Zebra" pattern (see below). see Appendix C Table 1: PCF examples with different GCD formulas. The presented GCD formulas differ from the exact GCD by sub-exponential factors.
The term "exponentially coprime to the GCD" generalizes the idea of a coprime and means that the highest powers of dividing the GCD for the terms in the sequence increase sub-exponentially. This statement implies that a certain prime does not affect the reduction.
Generalizing from these examples and many more (Appendix E), a conjectured structure of the exact GCD forms (up to sub-exponential factors) is presented next. Note that there exist multiple equivalent ways to present some of the forms, for example using LCM [2 ] (as shown in Table 1).

Conjecture 2 (The exact forms of the GCD). We can represent every GCD as a multiple of two parts, a factorial and an exponential expression:
➢ The factorial part in general appears in the form ( + )! ( )^d eg( ), where , ∈ ℕ and (⋅)! ( ) is multifactorial of order .
(The special case of ! corresponds to = 1, = 0, and deg( ) = 1.) ➢ The exponential part takes one of the following forms or their multiples: • Power of a prime ( ) Table 1 shows a case of LCM [2 ]. This is seen only in denominators.
• Zebra: There is an additional pattern for which we lack an explicit formula. We find this pattern in the denominators. We can identify this pattern in many PCFs but do not entirely understand it. The investigation of the Zebra pattern is left to future work.
For computational simplicity, most of our numerical analysis is focused on PCFs of deg = 2 and deg = 1. Based on this analysis and additional simulations, we conjecture that the above description captures any GCD sequence of a PCF, also of the higher order , . Furthermore, we expect analogous mathematical structures to exist in the GCDs of any linear recursion with polynomial coefficients, the investigation of which remains for future work. Note that, for PCFs without FR, numerical analyses show that Θ( ) primes in [2, ] are exponentially coprime to GCD.

Fast calculation of PCFs using simplified recursion formulas and FR
In this section, we discuss an application of the ability to predict the exact formulas of FR and other forms of reduction. Provided we have a closed-form formula for the GCD, we can apply the reduction straight to the recursion, so that the computation is performed with smaller integer values. Such simplified recursions enable faster estimation of the PCF limit. The computation advantage of such recursion is substantial with FR: it requires only manipulating sequences that grow exponentially with the PCF depth (instead of super-exponentially).
Example: A simple recursion for the reduced numerator and denominator For = 2 2 + , = , we find (numerically) that the GCD is ! 2 (up to a subexponential factor), and therefore, there exist integer sequences ′ and ′ such that In other words, GCD[ , ] ≐

Hints for a deeper mathematical structure
This section provides additional examples of special mathematical properties that we found numerically and hint at a much wider theory that still awaits discovery. A generalized proof for this case, even without knowing the PCFs limits, is available in Appendix D. The theorem in the next section shows how almost any constructs a PCF that proves the irrationality of its limit, although, apart from ln 2, the identity of these irrational limits is still unknown to us.

Infinite s that prove irrationality for a given
The next section shows infinite families of PCFs that prove irrationality of certain numbers. Specifically, we conjecture that for any , there exists an infinite set of s such that each constructs a PCF that proves the irrationality of its limit. In particular, for large enough s, the limit will be irrational since > 0.
Proof: (Straightforward) If → ∞, then the leading coefficient of ( ) uphold → ∞ and the characteristic equation 2 = + has a solution that certifies → ∞. proves the irrationality of its limit. As for higher degrees, we conjecture the existence of similar structures.

Additional properties of the greatest common divisors
We By this definition and the recursion formula for and , one can show that GCD2 | GCD2 +1 .
Since GCD2 | GCD , part of the reduction may be explained by the GCD2. It remains to be seen what part of the FR and its exponential part is contained in GCD2. Having inspected many PCFs numerically, with or without FR, we conjecture the following. The meaning of this exponentially tight equality is that all the theorems and conjectures presented here may apply also for GCD2. Specifically, if FR exists, then both the factorial and the exponential part of the GCD will exist in GCD2. The important consequence is that we can use either GCD or GCD2 for purposes of irrationality proofs, such as Theorem 2.
This conjecture enables us to treat the GCD as a growing product of some integer series and, at a given depth , calculate and reduce only one integer term: Moreover, this definition is advantageous because GCD2|GCD, and it thus sorts out sub-exponential factors that have no effect on proving irrationality. This observation facilitates the numerical analysis and helps identify the exact formula for the GCD.
As a side note, Conjecture 3 helps show that the GCD of PCFs that have FR always has a factorial term such as ( !) , rather than a term such as ⋅ (which also grows like ( !) up to an exponential factor by Stirling's approximation). In fact, all the PCFs with FR that we encountered could be written as ( !) • ( ) ( ) with ( ) and ( ) being integer sequences that grow exponentially. Some cases are more complex, such as when GCD ≐ (3 + 1)‼!, but these do not contradict the above statement. It would be interesting to try to prove this phenomenon.

Outlook and motivation
By their further development, the conjectures presented can provide useful tools for irrationality proofs, as well as for fast calculations of polynomial integer recursions of mathematical constants.
Specifically, the results related to FR can be applied to shrink the search space of the Ramanujan Machine algorithms [10]. By focusing on PCFs with FR, the algorithms would have a better chance to find new conjectures that are simultaneously of a relatively fast computation time and have nontrivial s that we can extract. That is, removal of the cases that have no FR avoids all the hard-to-compute PCFs that also provide trivial s.
Looking forward, we believe that by generalizing the mathematical structure of PCFs with FR, it would be possible to find universal structures in PCFs made from arbitrarydegree polynomials. As a more ambitious step, it is interesting to consider deeper linear recursions (beyond depth 2), which can also be harnessed to find new conjectures. One can search for analogous algebraic structures and ideas as presented above.
In the following, we present several ideas and open questions that arise from our mathematical experiments and from our conjectures. These open questions may be simple or complex, and we hope that they can engender more ideas for future research in different communities.

Implications of FR for a faster computation of PCFs
Once a closed formula for the GCD has been found, numerical calculations of PCFs will become easier and faster since the FR decreases considerably the numbers participating in the arithmetic operations. In particular, PCFs with FR benefit greatly from this reduction since and decrease from a super-exponentially (factorial) growth to exponential growth. In other words, finding the exact formula for the reduction enables one to construct a simpler recursion formula that directly gives the reduced numerators and denominators.

Families of PCFs
Following Conjecture 1.2, it is natural to try to generalize the families of , for higher degrees. What affects the number of families and subfamilies? Conjecture 1.4 claims that only two families exist for the discussed degrees, and one of them is branched into several subfamilies. In this case, the number of subfamilies depends merely on the number of divisors of (the leading coefficient of ). We do not yet have a solid and more general conjecture that relates to all degrees. Another question regarding families of or is whether a relation exists between the limits of any sibling PCFs. For example, if this relation hints that the limits are equivalent, for proofs of irrationality, it will suffice to find just one limit and use Theorem 3 (infinite s that proves irrationality).

Finding and proving the exact form of the GCD
We did not find the exact form of the GCD, but nevertheless tried to list the different types of expressions that comprise it. The motivation to find the closed form of the GCD is the possibility of writing a reduced recursion that yields the reduced numerators and denominators, which can simplify any numerical calculation of the PCF. Moreover, a closed-form formula would also directly predict the effective irrationality measure given by the PCF.
We note that Apéry proved his case by finding an explicit expression for the PCF at each depth. As an example of taking a more general approach, in Appendix D we address a family of GCDs and bypass the need for an explicit expression. As examples that can promote future research, we present in Appendix E a set of unproven examples that yield precisely the same simplified recurrence relations.

Predicting the exponential order :
To search for conjectures in the form of PCFs that prove the irrationality of constants, it suffices to predict only the exponential order . Using this value, Theorem 2 calculates the effective irrationality measure . It remains to find a direct relation from , to .

Appendix A: Classification of polynomial continued fractions (PCFs)
All PCFs can be split into three types by the ratio of the degrees of the polynomials , (  Table 2: Summary of the three types of PCFs, partitioned by the ratio of the degrees of , . We show the conditions for each type that leads to a nontrivial effective irrationality measure . Note the crucial role of FR in the second type (middle column), which is at the core of this manuscript. Proofs of part of the regimes of convergence can be found in [10]. *There are cases for which we do not know the conditions for convergence.
In most of this paper, we focused on the second PCF type, that of balanced-degree PCFs. PCFs of this type are those that prove the irrationality of (3), (2), and ln(2) in Apéry's work [1,2]

Appendix B: Proof for Theorems 1 and 2
For each PCF, we estimate the convergence ratio in relation to the denominator growth rate.
Denote the partial numerators and denominators of the PCF by and ; these answer the following recursion formula: Here, we encounter an assumption that is justified below. If and since +1 ≐ , we use ∏ 2 . If the assumption is wrong (Case 1 below), then by numerical tests we conjecture to be polynomial.
Denoting = deg and = deg , one can estimate the polynomials using their leading coefficients, , , and some positive constant : Therefore, Now, we need to split the fractions into three cases by the ratio . Each case differs by the growth rate of , which is affected by the significant element(s) in the next equation: Note that the split occurs because, in either case, ≐ • ! ⋅ for some , and can be justified by estimating the multiplicative error as presented below for the second case only.
First case: > 2, and therefore, the significant element is : Second case: = 2, and therefore, both and are significant and the PCF is "balanced". Now, assume inductively that the above inequality holds till some ≥ 0 , and for + 1 By combining these two, we obtain as required for +1 .
Third case: < 2, and therefore, the significant element is .
Step 3: Combining these results with the irrationality criterion.
First, without reducing , and second, when we reduce by GCD = gcd (p n , q n ), we obtain better results. The finite calculation error remains the same since the reduced fraction represents the same number. However, the real denominator becomes smaller, and therefore, only the right hand side changes, and the inequality transforms to Notice that, as expected, the bigger the GCD, the bigger is . We use the equality sign for the limit (inferior) as tend to infinity, as this value yields a lower bound on the irrationality measure.
First case: and therefore, the assumption is not justified, and the convergence rate is subexponential. For this reason, to provide nontrivial , the GCD must be exponentially equal to , so that both sides of the inequality will decrease sub-exponentially, and a more delicate analysis is required. In conclusion, the condition GCD ≐ ≐ /2 • ! /2 is necessary but not sufficient for yielding nontrivial .
Second case: and the assumption holds, except for | | = 2 . With reduction: To change the limit above, the GCD must be of factorial order. If it is, and the factorial power is (the exponential factors have no effect), then which is better than the without reduction. Here, if = , then is arbitrarily large and the limit is a Liouville number.

Appendix C: Inflation and deflation of continued fractions
In his paper, Apéry showed a linear recursion of depth 2 with rational function coefficients (ratio of two polynomial) and a related PCF. The direct translation of Apéry's recursion into a continued fraction has and as rational functions and not integer polynomials. However, they can be converted to a PCF form. To see the conversion, we multiply and by a non-zero sequence, thus converting them to integer polynomials without changing the limit. We call this process "inflation". This process is also needed when some rational coefficient is used in Conjecture 1.4.
Conversely, any PCF that has been multiplied by a non-zero sequence can be simplified by removing that sequence. We call this process deflation. Deflating makes the PCFs' and smaller (possibly of a lower degree), and most importantly, helps simplify the GCDs, despite not changing the induced . This process can explain some powers of prime in Table 1. Applying Identity 1, we can inflate this continued fraction using the denominators of and , i.e., the sequence = ( + 1) 3 , and obtain the PCF 6 (3) = 5 − that is, Apéry's PCF, which is presented in our introduction.
The effect of these processes on the GCD remains to be seen. For this reason, the following theorem is presented.