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Updated Oct 8

Problem 243

claims/: The 4 claim pages of Problem 243, one per claimant's result; the problem's standing derives from them.

Statement. Let $1\leq a_1<a_2<\cdots$ be a sequence of integers such that

\lim_{n\to \infty}\frac{a_n}{a_{n-1}^2}=1

and $\sum\frac{1}{a_n}\in \mathbb{Q}$ . Then, for all sufficiently large $n\geq 1$,

a_n = a_{n-1}^2-a_{n-1}+1.

Status. Open.

Source. erdosproblems.com/243, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #243, https://www.erdosproblems.com/243.

References.

[Du01] Duverney, Daniel, Irrationality of fast converging series of rational numbers. J. Math. Sci. Univ. Tokyo (2001), 275-316.
[ErSt64] Erdős, P. and Straus, E. G., On the irrationality of certain Ahmes series. J. Indian Math. Soc. (N.S.) (1964), 129-133.
[ErGr80] Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monogr. Enseign. Math. 28 (1980).
[Er88c] Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986), Cambridge Univ. Press (1988), 102--109.
[Ko26] Koizumi, J., Irrationality of the reciprocal sum of doubly exponential sequences. INTEGERS 26 (2026), #A28; arXiv:2504.05933.

Formalization. Statement in formal-conjectures.

Current assessment

The site's formulation (page last edited 21 January 2026) states that an increasing sequence of positive integers with $a_n/a_{n-1}^2\to1$ and rational reciprocal sum satisfies $a_n=a_{n-1}^2-a_{n-1}+1$ for all large $n$ , and labels the problem OPEN; the sequences obeying the recurrence from the start are Sylvester's sequence $2,3,7,43,\ldots$ and its tails. No claim answers the statement for every sequence. Four claim pages record results that decide it for classes of sequences, and no full claim is pending.

Accepted partial claims (refereed).

Erdős and Straus 1964, Theorem 3 of [ErSt64] (printed p. 132): with $N_k=\operatorname{lcm}(n_1,\ldots,n_k)$ , the conclusion holds whenever $\limsup(N_k/n_{k+1})(n_{k+1}^2/n_{k+2}-1)\le0$ , in particular whenever $N_k/n_{k+1}$ is bounded (Theorem 1, p. 129). The site's remark states the theorem in contrapositive form and writes the factor as $(a_n^2/a_{n+1}-1)$ , one index earlier than the paper's.
Duverney 2001, Corollary 3.2 of [Du01] (printed p. 287): the conclusion holds whenever $\sum_n(a_{n+1}/a_n^2-1)$ converges, with signs $\pm1$ on the terms allowed; the paper calls it a partial answer to Erdős's question, which it locates at p. 64 of [ErGr80] and p. 105 of [Er88c]. The site's remark records the corollary after a thread comment of 11 October 2025 pointed to it.
Koizumi 2025, Corollary 1 of [Ko26]: a sequence of positive integers with $2/3\le a_n^2/a_{n+1}\le4/3$ for every $n$ and reciprocal sum $1$ is Sylvester's sequence. The same paper's Theorem 3 shows the problem equivalent to its Conjecture 1 on pseudo-greedy expansions of rationals, which the author checked by computer for $p/q$ with $0<p\le q\le10^5$ ; a reduction and evidence, not a settled part. V. Kovač's thread comment of 11 September 2025 pointed to the paper, after T. Tao had posted the same observations; Tao then recorded that the paper already contains them.

Pending partial claim. Cook 2026, Corollary 1.1 of a note posted on the thread on 11 September 2026, written up by the AI system Astra at the author's direction: with $P_n=\prod_{j<n}a_j$ , the conclusion holds whenever $(P_n/a_n)(a_n^2/a_{n+1}-1)$ is bounded above. Not refereed, not on arXiv, no check recorded; its Lean file covers the integer-state theorem only and is not built here. The note cites I. O. Bado, Prime-support rigidity and primitive pseudo-greedy dynamics: partial progress on Erdős Problem #243, an author-posted preprint registered in September 2026 under DOI 10.13140/RG.2.2.36612.08325, for a theorem under a two-sided bounded error (its Theorem 5.1). That preprint is recorded from Cook's citation alone: it is available only through its ResearchGate record, its statements are not recorded here, and it has no claim page for that reason.

Further refereed special cases that the site does not credit are recorded on the card Tijdeman and Yuan, Corollary 4.1: Badea's criterion (Acta Arith. 63 (1993)), the conclusion whenever $a_{n+1}\ge a_n^2-a_n+1$ for all large $n$ , and an improvement of the Erdős--Straus Theorem 3. Koizumi's Corollary 4 recovers the Badea and Erdős--Straus cases within his framework.

Dated search scope (2026-10-07). The site's page, its discussion thread (comments from 2025 to 11 September 2026, after which posting was suspended), its proof-claims tab (none registered) and the formal-conjectures statement file record no full claim and no dispute of the results above; no wider literature search was made. The site's label OPEN matches the derived standing: every claim is partial.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (31)

Erdős and Straus's recurrence criterion under an lcm condition Duverney's recurrence criterion under a summable error Koizumi's characterization of Sylvester's sequence Cook's recurrence criterion under a bounded one-sided error problems/irrationality/E0243/claims Irrationality and Diophantine Approximation Irrationality and Diophantine Approximation irrationality/badea_1987_irrationality_certain_infinite_series Corollary 1 (p. 224): sum 1/a_n is irrational when a_{n+1} > a_n^2 - a_n + 1 for all large n Theorem (p. 222): sum b_n/a_n is irrational when a_{n+1} > (b_{n+1}/b_n)(a_n^2 - a_n) + 1 for all large n irrationality/duverney_2001_irrationality_fast_converging_series_rational_numbers Corollary 3.2 (p. 287): the Sylvester recurrence under a summable relative error Theorem 3.1 (pp. 285-286): a rational fast converging series forces a quadratic recurrence irrationality/erdos_1964_irrationality_certain_ahmes_series Example 1 (p. 132): the sum of 1/(a^(2^k) + b_k) is irrational when the sum of |b_k| a^(-2^k) converges Examples (p. 131): rational Ahmes series with bounded N_k/n_(k+1) and limsup n_k^2/n_(k+1) = a, without the Sylvester recurrence Theorem 1 (p. 129): under limsup n_k^2/n_(k+1) <= 1 and bounded N_k/n_(k+1), the sum of 1/n_k is rational only for an eventual Sylvester recurrence Theorem 2 (p. 131): with n_k^2/n_(k+1) and n_1...n_k/n_(k+1) bounded, a rational sum of 1/n_k forces finitely many rational limit points of n_k^2/n_(k+1)Theorem 3 (p. 132): Theorem 1 with the bound on N_k/n_(k+1) replaced by limsup (N_k/n_(k+1))(n_(k+1)^2/n_(k+2) - 1) <= 0 irrationality/tijdeman_2002_rationality_cantor_ahmes_series Corollary 4.1: growth conditions under which an Ahmes-type series is rational exactly when a_(n+1) equals the Sylvester recurrence Corollary 4.2 (pp. 9-10): an lcm and gcd lower bound on a_(n+1) becomes an equality when the sum of b_n over a_n is rational Proposition 4.1 (p. 10): infinitely many gcd failures in the equality case of Corollary 4.2 force limsup a_n^2/a_(n+1) above one Theorem 4.1 (p. 6): an lcm-weighted limsup condition makes the sum of b_n over a_n rational exactly when a_(n+1) follows the Sylvester-type recurrence unit_fractions/koizumi_2025_irrationality_reciprocal_sum_doubly_exponential_sequences Conjecture 6: a vanishing gap sequence of a rational's pseudo-greedy expansion is eventually zero Corollary 2: Sylvester's sequence is the only one with a_n^2/a_{n+1} in [2/3,4/3] and reciprocal sum 1 Corollary 20: the Erdős–Straus and Badea cases of the Sylvester-recurrence question Remark 22: an affirmative answer to Question 5 would make 2^{2^n} a Type 2 irrationality sequence Theorem 1: ratios a_n^2/a_{n+1} near beta pin each large term to the reciprocal sum Theorem 16: Erdős and Graham's Sylvester-recurrence question is equivalent to Conjecture 6

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