Reciprocal Sums and the Sylvester Recurrence

Will Cook

Short paper

Reciprocal Sums and the Sylvester Recurrence

Erdős #243 By Will Cook September 2026 Authorship, AI use and citation

PDFOpen the PDF, 21 pages TeXLaTeX source Source on GitHub Problem page

Summary. For strictly increasing positive integers with $a_{n}^{2} / a_{n + 1} = 1 + 3 / n + o (n^{- 3})$ , the reciprocal sum is irrational. The argument extracts a cubic polynomial from an integer tail numerator, restricts its coefficients using a square in a cubic field, and excludes the remaining cases modulo seven. Under rationality and the quadratic growth limit, separate criteria using bounded numerator increases and weighted first crossings force the eventual Sylvester recurrence. The unrestricted Erdős–Graham question remains open.

This paper covers cubic-rate irrationality, nonintegral regular-rate exclusion and sufficient tail criteria for the Sylvester recurrence; the unrestricted Erdős–Graham question remains open.

It does not establish the validity of claims tagged Lean, which belongs to the cited kernel-checked source, or a solution to Erdős #243, which remains open.

Start here, by the editorial route

In this paper

Tail numerators

For the Sylvester sequence $2, 3, 7, 43, \dots$ , whose reciprocal sum is $1$ , the successive tails are $1, 1 / 2, 1 / 6, 1 / 42, \dots$ . Multiplication by the products $1, 2, 6, 42, \dots$ leaves the constant integer numerator $1$ . For any rational reciprocal sum, including a denominator of the whole sum in these products again gives integer numerators. We keep these fractions unreduced: the exact multiplicative denominator update will transfer the growth rate to the numerator.

Assume now that $a_{n}$ are strictly increasing positive integers, $a_{n + 1} / a_{n}^{2} \to 1$ , and $\sum_{n \geq 1} 1 / a_{n} = p / q$ with positive integers $p, q$ . All indices in this construction start at $1$ . Put

x_{n} = \sum_{k \geq n} \frac{1}{a_{k}}, P_{n} = \prod_{1 \leq j < n} a_{j}, D_{n} = q P_{n}, C_{n} = D_{n} x_{n} .

Here $P_{1} = 1$ , $D_{1} = q$ and $C_{1} = p$ . Since every $a_{k}$ with $k < n$ divides $P_{n}$ , subtracting the finite prefix from $p / q$ gives

C_{n} = p P_{n} - q \sum_{1 \leq k < n} \frac{P_{n}}{a_{k}} \in N_{> 0} .

Positivity comes from $C_{n} = D_{n} x_{n}$ . Removing $1 / a_{n}$ from the tail gives

\begin{matrix} (1) & C_{n + 1} = a_{n} C_{n} - D_{n}, D_{n + 1} = a_{n} D_{n} . \end{matrix}

Koizumi constructs these coordinates in Lemma 4 [10]. On a pseudo-greedy tail his $c_{n}, d_{n}, e_{n}$ correspond to $C_{n}, D_{n}, D_{n} - (a_{n} - 1) C_{n}$ . If the tail is restarted with its value in lowest terms, both coordinates are divided by the same fixed integer. Reducing at every step would instead introduce varying cancellation factors into the numerator ratio. We therefore extract the polynomial before dividing by the eventual fixed gcd in Section 3.2.

For large $n$ , $a_{n + 1} \geq a_{n}^{2} / 2 \geq 2 a_{n}$ . The terms after $1 / a_{n + 1}$ sum to at most $2 / a_{n + 2} \leq 4 / a_{n + 1}^{2}$ , so

\begin{matrix} (2) & x_{n} = \frac{1}{a_{n}} + \frac{1}{a_{n + 1}} + O (a_{n + 1}^{- 2}), \frac{C_{n + 1}}{C_{n}} = \frac{a_{n}^{2}}{a_{n + 1}} + O (1 / a_{n}) . \end{matrix}

For the second estimate, the first gives $a_{n} x_{n} = 1 + O (1 / a_{n})$ . Use this in

\frac{C_{n + 1}}{C_{n}} = \frac{a_{n}^{2}}{a_{n + 1}} \frac{a_{n + 1} x_{n + 1}}{a_{n} x_{n}} .

The last factor is $1 + O (1 / a_{n})$ , and $a_{n}^{2} / a_{n + 1} \to 1$ , giving the stated additive error. Quadratic growth makes $1 / a_{n} = o (n^{- k})$ for every fixed $k$ , so this error preserves the precision required in the cubic theorem. We keep the original index $n$ throughout that proof. Although an eventual recurrence survives removal of a finite prefix, the expression $1 + 3 / n + o (n^{- 3})$ changes its lower-order terms under an index shift.

About this paper

Authorship and AI use. Will Cook built and directed the research infrastructure and maintains the public release. He reviewed claims when he could. AI agents did most of the research and drafting. Cook did not independently verify every claim.

Cite and contact. Cite this paper by its title, author and date above, with its PDF; cite the earlier sources it uses for a mathematical result. For software, use the release citation and give the commit used. Contact Will with questions or corrections.

Prefer the manuscript?

PDFOpen the PDF, 21 pages TeXOpen the LaTeX source Problem page

Equations typeset from the exact TeX. Rendered from the LaTeX manuscript source with file fingerprint sha256:b393a9190d51137f. Its file fingerprint matched the published source record, so the PDF above, the LaTeX, and this page are one manuscript. Redeploying the site regenerates this page from whatever the public repository holds at that moment.