Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 267

claims/: The 6 claim pages of Problem 267, one per claimant's result; the problem's standing derives from them.

Statement. Let $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$ be the Fibonacci sequence. Let $n_1<n_2<\cdots $ be an infinite sequence with $n_{k+1}/n_k \geq c>1$. Must

\sum_k\frac{1}{F_{n_k}}

be irrational?

Status. Claimed. The site labels the problem OPEN (page last edited 18 January 2026). The case $c\ge2$ is settled by Badea's 1993 corollary, the accepted partial claim on the Badea claim page, and the site's commentary records the case $1<c<2$ as open. One pending full claim, a Lean 4 proof of 2026, asserts the answer yes for every $c>1$ ; the frontmatter standing follows from it.

Source. erdosproblems.com/267, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #267, https://www.erdosproblems.com/267.

References.

[An89] André-Jeannin, Richard, Irrationalité de la somme des inverses de certaines suites récurrentes. C. R. Acad. Sci. Paris Sér. I Math. (1989), 539-541.
[Ba87] Badea, C., The irrationality of certain infinite series. Glasgow Math. J. (1987), 221-228.
[Ba93] Badea, C., A theorem on irrationality of infinite series and applications. Acta Arith. (1993), 313-323.
[BiHo76] Hoggatt, Jr., V. E. and Bicknell, Marjorie, A reciprocal series of Fibonacci numbers with subscripts $2^nk$ . Fibonacci Quart. (1976), 453-455.
[Go74] Good, I. J., A reciprocal series of Fibonacci numbers. Fibonacci Quart. (1974), 346.

Formalization. Statement in formal-conjectures (pinned at the repository's commit of 2026-09-18), which tags the statement research solved with answer yes and cites in its formal_proof attribute the Lean 4 proof recorded on the claim page; its variant specialization_pow_two, the instance $n_k=2^k$ , cites a formal proof by AlphaProof, recorded on its claim page. The corpus records no build or audit of either.

Current assessment

The site records Problem 267 as OPEN (page last edited 18 January 2026). The Progress note below records the two literature results and the index range they leave open, $1<c<2$ ; the corpus records a check of their statements against the papers, no check of their proofs, and no literature search beyond the sources named below. The pending claim described next would close that range.

Proof claims on the site. The proof-claims tab carries one full claim, submitted 2026-07-15 by Colin Snyder with an AI system credited for the proof: a Lean 4 development stating that the sum is irrational for every index sequence with $n_{k+1}/n_k\ge c$ for some real $c>1$ , which would close the range $1<c<2$ . The claim page records the formal statement and its standing: the site labels the problem OPEN, the proof-claim entry has no comments, the only referee report is an automated one on the hosting site, and the formal-conjectures catalog tags the statement solved and cites the proof. The corpus records no build or audit of the Lean files. The problem's discussion thread carries four remarks, all on results within $c\ge2$ or within Badea's condition: Kevin Barreto (2026-01-01) on Badea's proofs of the $2^k+1$ and Lucas $2^k$ instances and of the case $c\ge2$ , Alfaiz (2026-02-23) on Nguyen's transcendence result, and Alfaiz and Terence Tao (2026-04-30) on the Chattopadhyay instance $n_k=n^k$ , which Tao notes is much easier because each denominator divides the next.

Progress

Badea's 1993 Corollary 3.2, recorded on [[problems/irrationality/E0267/claims/1993_01_01_badea|the accepted partial claim page]] and on the card badea_1993_theorem_irrationality_infinite_series_applications, proves the sum irrational whenever $n(k+1)\ge2n(k)-1$ for all large $k$ , and Badea notes that this answers the problem for every $c\ge2$ . Nguyen's Theorem 1.2 adds that the sum is transcendental whenever the index ratio is at least a fixed $c>2$ , even if Fibonacci and Lucas reciprocals are mixed; his introduction notes that irrationality in this range follows already from a simpler denominator-and-tail estimate. The question remains open for index ratios bounded below by a constant in $(1,2)$ . Good's evaluation for indices $2^k$ gives a quadratic irrational, consistent with the original question.

Known Results

Every $c\ge2$ is settled by Badea 1993, the accepted partial claim on the Badea page. Its condition also contains the instance results the site and its thread record, each an accepted partial claim of its own: Good 1974 on the Good page and Hoggatt and Bicknell 1976 on the Hoggatt–Bicknell page ( $n_k=2^k$ , with the value $(7-\sqrt5)/2$ , and $n_k=2^jk$ for every fixed $k$ ), and Badea 1987 on the Badea 1987 page ( $n_k=2^k+1$ , which meets Badea's 1993 condition with equality although its ratios are below $2$ ). The formal-conjectures variant specialization_pow_two states the instance $n_k=2^k$ and cites a formal proof by AlphaProof, the pending partial claim on the AlphaProof page. Chattopadhyay's $n_k=n^k$ for integers $n\ge2$ , cited in the thread on 2026-04-30, also lies inside Badea's condition. Nguyen 2022 strengthens irrationality to transcendence for $c>2$ . André-Jeannin's 1989 irrationality of $\sum1/F_n$ itself concerns the full sequence, whose index ratios tend to $1$ , outside the problem's hypothesis, so it settles no instance and has no claim page. No result among the sources named on this page settles the whole range $1<c<2$ , which the site's commentary records as open; the pending Snyder claim asserts the answer yes there.

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 (21)

Good's evaluation of the sum over the indices 2^k Hoggatt and Bicknell's evaluation over the indices 2^n k Badea's irrationality for the indices 2^n + 1 Badea's irrationality for index ratios at least two AlphaProof's Lean proof for the indices 2^k A Lean proof of irrationality for every ratio gap c greater than one problems/irrationality/E0267/claims 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 Corollary 4 (p. 227): the sum of 1/F_{2^n+1} is irrational Corollary 5 (p. 227): the sum of 1/L_{2^n} is irrational 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/badea_1993_theorem_irrationality_infinite_series_applications irrationality/good_1974_reciprocal_series_fibonacci_numbers Theorem (p. 346, unnumbered): the sum of 1/F_(2^k) over k >= 0 is (7 - sqrt 5)/2 irrationality/hoggattjr_1976_reciprocal_series_fibonacci_numbers_subscripts Theorem (p. 455, unnumbered): the sum of 1/F over the indices 2^n k in closed form Nguyen (2022): Sparse Fibonacci reciprocal series Theorem 1.2: sparse Fibonacci and Lucas reciprocal sums are transcendental Theorem 1.3: transcendence of sparse sums over a real quadratic unit

Graph