Reciprocal Sums and the Sylvester Recurrence: Further Results and Proofs

Classical criteria and pseudo-greedy expansions.

The rational-tail method predates these coordinates. Erdős and Straus [15] use eventual integer remainders under a small-numerator hypothesis. In polynomial Cantor series, Hančl and Tijdeman split the numerator into finitely many shifted products. Rationality is then characterised by the vanishing of the resulting polynomial sum [16]. Their theorem provides methodological context, not a result for arbitrary arrays of signed integer coefficients: the rearrangement of the infinite sum needs boundary control, supplied there by the polynomial hypothesis. Neither result supplies the first-crossing argument for a lower error bound proved below.

Several classical results give sufficient conditions directly on the sequence $(a_n)$. Their hypotheses use different quantities, so the comparisons must be made separately. Koizumi’s product criterion is implied by eventual nonnegativity of the integer error $E_n$, under the identification of coordinates given below. It therefore already covers the descent argument in Section 8 (see [12]). The original Erdős–Straus criterion instead uses a least common multiple. Erdős and Straus proved that if $lim{a}_n/a_{n-1}^2=1$, the reciprocal sum is rational, and $(a_n)$ has no Sylvester tail, then

where $[a_1,\dots ,a_n]$ is the least common multiple [1]. This is the indexing in the original theorem. With $A_r=\mathrm{lcm}(a_1,\dots ,a_{r-1})$ and $r=n+1$, its expression is exactly $(A_r/a_r)(a_r^2/a_{r+1}-1)$, the LCM quantity in the short note. The index change is not an additional difference between those two criteria. Erdős’s 1988 survey, followed by the catalogue summary, instead prints $a_n^2/a_{n+1}-1$ in the second factor while retaining the same prefix-LCM quotient; that displayed summary is off by one and is not followed here [6][13]. Koizumi records the convenient sufficient rate $a_n^2/a_{n+1}=1+o(1/n)$ under which the Erdős–Straus criterion settles the problem [12]. Tijdeman and Yuan give a criterion of the same kind for Ahmes series with positive integer numerators, weighted by the least common multiple of the earlier denominators [4].

Duverney’s signed criterion gives a further comparison. For positive integers $a_n\to \mathrm{\infty }$ and signs ${ϵ}_n\in \{-1,1\}$, its printed hypothesis (3.6) is the one-sided condition

without absolute values. The stated conclusion is that $\sum _{n\ge 0}{ϵ}_n/a_n$ is rational if and only if

for all large $n$ [2]. For the comparison here we impose the stronger sufficient hypothesis $\sum _{n\ge 0}|a_{n+1}/a_n^2-1|<\mathrm{\infty }$ and take all signs positive. The distinction concerns one step of the printed proof. On pp. 299–300, the reduced auxiliary fractions $p_n^{\prime }/q_n^{\prime }$ satisfy $p_{n+1}^{\prime }\mid q_n^{\prime }$ and

The required boundedness follows if $\prod _{k\ge N}(p_k^{\prime }/q_k^{\prime })$ has a positive limit. Absolute convergence of the growth-defect series ensures this, using the summable error in Duverney’s estimate (3.3). Signed convergence alone does not justify that product step, even for positive rational factors. For each integer $m\ge 2$, take the pair $1+1/m$, $1-1/m$ successively $m$ times. The deviations cancel after every pair, and the intervening partial sums are $1/m\to 0$, so their series converges. But the product through the block $m=N$ is

the equality follows by cancelling powers of consecutive integers. This example does not impose $p_{n+1}^{\prime }\mid q_n^{\prime }$. It refutes only the general product inference, not Duverney’s arithmetic criterion. We therefore use only the absolute-convergence form; the interpretation with merely signed convergence is not needed in any proof here.

In the all-positive case ${ϵ}_n=1$, absolute convergence also gives a direct comparison with the classical product criterion. With the one-based notation $P_n=\prod _{1\le j<n}{a}_j$ used above,

The ratios on the right have an absolutely convergent sum of deviations from $1$, because the same is true of their reciprocals and those reciprocals tend to $1$. Thus $P_n/a_n$ has a positive finite limit, and its increments tend to zero. This case already satisfies the classical nonpositive-upper-limit condition; the bounded-increment result allows a larger class of sequences.

A different quantitative question is treated by Duverney, Kurosawa and Shiokawa [17]. For rational $x_n>1$, their result assumes eventual $x_{n+1}\ge x_n^2$ and control of the accumulated denominators of $x_{n+1}/x_n^2$; it computes the irrationality exponent of a signed reciprocal series. We use it only as a restricted comparison. In particular, the eventual Sylvester recurrence and the cubic rate considered here approach quadratic growth from the other side.

Badea’s positive-term criterion is adjacent but different. For a convergent series $\sum _n{b}_n/a_n$ with $a_n,b_n$ positive integers, eventual strict inequality

forces irrationality, while rationality under the corresponding non-strict inequality forces eventual equality [3]. For $b_n=1$, its hypothesis is $a_{n+1}\ge a_n^2-a_n+1$, an inequality between consecutive denominators, not the one-step sign condition $E_n\ge 0$ used in integer descent. Under the standing rational-tail and growth hypotheses, however, either inequality imposed eventually forces the Sylvester recurrence and hence the other. Their eventual forms are equivalent in this setting; that fact does not supply either condition for a general signed error.

Koizumi’s pseudo-greedy expansion [12] chooses $a_n$ by rounding $x_n^{-1}+1$ to the nearest integer, with a half-integer rounded upwards. Here $x_n$ is the remaining sum before subtracting $1/a_n$. Thus $a_n=\lfloor x_n^{-1}+3/2\rfloor$, and the gap ${\epsilon }_n=x_n^{-1}+1-a_n$ is the signed rounding error. Write the rational initial sum as $r=p/q$ with positive integers $p,q$. His Lemma 4  [12] produces integers $c_n>0$, $d_n$ and $e_n$ with $x_n=c_n/d_n$ and ${\epsilon }_n=e_n/c_n$, satisfying

and the proof of that lemma also gives $c_{n+1}=a_n{c}_n-d_n$. These are the recurrences of Section 4, after matching the starting index and the initial normalization. For a sequence satisfying the hypotheses of Problem 1.1, the rounding rule is guaranteed only after a finite restart [12]. Restarting there with the tail in lowest terms gives $(c_n,d_n,e_n)$; the original $(C_n,D_n,E_n)$ on that tail is a fixed positive integer multiple of this triple, with the indices matched. Thus $E_n/C_n={\epsilon }_n$ is unchanged by the restart. The integer $E_n$ is not necessarily a numerator in lowest terms, and the rounding range is not asserted before the restart. The first identity above gives $E_n=D_n-(a_n-1)C_n$; the remaining identities are exactly the numerator and denominator recurrences, including Proposition 4.1. On an all-negative tail we later write $e_n=-E_n>0$ for the magnitude. That local use of lower-case $e_n$ has the opposite sign to Koizumi’s signed integer $e_n$.

His Theorem 3 [12] proves the equivalence of two assertions. Conjecture 1 [12] asks whether, for a positive rational $r$, the condition ${\epsilon }_n\to 0$ forces ${\epsilon }_n=0$ eventually. Question 1 [12] is the question of Erdős and Graham stated as Problem 1.1 above. Two implications used below already appear in these coordinates: his Lemma 3, that ${\epsilon }_n=0$ forces ${\epsilon }_{n+1}=0$, is the absorption of Theorem 5.8, and his Proposition 1(2), that ${\epsilon }_n\ge 0$ for all large $n$ forces ${\epsilon }_n=0$ for all large $n$, is the descent of Theorem 8.1. Both statements are given in [12]. Koizumi attributes Proposition 1(2) to Badea. Its contrapositive says that a counterexample must have negative errors infinitely often; it assumes the sign is eventually nonnegative, while Theorem 12.1 below assumes only that the negative part is eventually bounded. It permits negative errors between $-B$ and $0$ and imposes no independent upper bound on positive errors; the relative-error limit is still required.

The growth hypothesis in Problem 1.1 is calibrated by two facts about Sylvester’s sequence. With $a_1=2$, it satisfies $a_n\sim c_0^{2^n}$ with $c_0=1.2640847\dots$. Deleting initial terms and reindexing produces sequences with $a_n\sim C^{2^n}$ for arbitrarily large $C$ whose reciprocals still sum to a rational number [7]. The classical sufficient condition for irrationality, $\underset{n}{lim}{a}_n^{1/2^n}=\mathrm{\infty }$, is therefore sharp. Kovač and Tao identify that condition as folklore  [7]; they attribute the sharpness observation to Erdős (1975). A sequence with $a_n/a_{n-1}^2\to 1$ has $a_n^{1/2^n}$ convergent, so that criterion says nothing about the sequences of Problem 1.1, and rationality is genuinely possible there. Sylvester’s sequence is A000058 in the OEIS; A129871 is the variant $1,2,3,7,43,\dots$ with an initial $1$ prepended. For the recent literature on irrationality of Ahmes series we refer to Kovač and Tao [7], who resolve several problems of Erdős and Graham drawn from the same two sources cited above, and whose introduction gives a sample of the intermediate work, including Sándor (1984) and Badea (1987). They do not treat Problem #243; the rigidity conclusion asked for there is not among their results.

Crmarić and Kovač [18] address the neighbouring Problem #270, not Problem #243. Allowing integer $f(n)\to \mathrm{\infty }$ in $\sum _n(\prod _{j=1}^{f(n)}(n+j){)}^{-1}$ gives every positive real value; imposing nondecreasing $f$ gives a measure-zero value set. The latter assertion does not rule out individual rational values. Their extension of Kakeya’s subsum argument explains why decay alone need not force irrationality when the summands remain freely selectable. The present exact denominator recurrence imposes additional compatibility, so neither direction is an implication between their theorem and ours.

The Formal Conjectures collection contains a mathematically equivalent unproved declaration, up to its zero-based indexing  [14]. Its summand is $\mathbb{Q}$-valued, so Lean’s Summable hypothesis asserts the existence of a sum in $\mathbb{Q}$; the finite indexing shift changes that sum only by a rational prefix. The declaration therefore does encode the rationality premise, but its proof is sorry. It states the problem without proving it, and the development in this note is independent of it. Earlier formal work on the remainder method should also be distinguished from a solution of this problem: Koutsoukou-Argyraki and Li’s Archive of Formal Proofs entry [19] records an Isabelle/HOL formalisation of Erdős–Straus (1974), Theorem 2.1, Corollary 2.10 and Theorem 3.1. That is formal prior art for those classical criteria, not an existing formal proof of Erdős #243.

Each criterion still requires an additional bound or convergence estimate under the unrestricted hypotheses. In the stable-gcd case, for example, the record-increment theorem gives a positive lower bound for the log-log coefficient, rather than proving it infinite. All conclusions about a persistently nonzero error assume failure of eventual Sylvester behaviour. We collect them in Section 14, where mixed-sign tails remain unclassified. The two recurrence proofs below are independent of the cubic argument: one uses a stable gcd and CRT, the other an infinite product and integer descent.