Badea 1987 irrationality certain infinite series

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corollary_1: Badea's Corollary 1: a sequence of positive integers with a_{n+1} > a_n^2 - a_n + 1 for all large n has irrational reciprocal sum, and the sequence 2, 3, 7, 43, ... shows the strict inequality cannot be relaxed.

corollary_4: Badea's Corollary 4: the sum over n of the reciprocals of the Fibonacci numbers F_{2^n+1} is irrational, answering a question of Erdős and Graham.

corollary_5: Badea's Corollary 5: the sum over n of the reciprocals of the Lucas numbers L_{2^n} is irrational, answering a question of Erdős and Graham.

proposition: Badea's Proposition: every convergent infinite series of positive rationals has infinitely many pairwise disjoint subseries whose sums are irrational.

theorem: Badea's main Theorem: for sequences of positive integers a_n and b_n with a_{n+1} > (b_{n+1}/b_n) a_n^2 - (b_{n+1}/b_n) a_n + 1 for every large n, the sum of b_n/a_n is irrational.

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Badea, C., The irrationality of certain infinite series. Glasgow Math. J. 29 (1987), 221--228.

The main Theorem states that if (a_n) and (b_n) are sequences of positive integers with a_{n+1} > (b_{n+1}/b_n) a_n^2 - (b_{n+1}/b_n) a_n + 1 for all large n, then sum b_n/a_n is irrational; the proof rewrites the sum as a limit of rationals A_n/P_n with P_n = a_1...a_n and applies Brun's convexity criterion for irrationality of limits of increasing rational sequences. Corollaries 1-3 are criteria parallel to the earlier ones of Erdos-Straus, Erdos and Sandor, each gaining something and losing something against them, not recovering them (Corollary 1: a_{n+1} > a_n^2 - a_n + 1 for all large n forces sum 1/a_n irrational, and the recursion c_{n+1} = c_n^2 - c_n + 1 shows it is best possible in a certain sense, since > cannot be replaced by >=), and a Proposition shows that any convergent series of positive rationals contains infinitely many pairwise disjoint subseries whose sums are irrational. Section 5 answers the two Erdos-Graham problems: Corollary 4 shows sum_{n>=1} 1/F_{2^n + 1} is irrational and Corollary 5 shows sum_{n>=1} 1/L_{2^n} is irrational, both by verifying the criterion through Fibonacci identities such as F_{2k+1} = F_k^2 + F_{k+1}^2 and F_p^2 - F_{p+1}F_{p-1} = (-1)^{p+1}. Corollary 4 is the paper's bearing on problem 267, which asks whether sum_k 1/F_{n_k} must be irrational whenever n_{k+1}/n_k >= c > 1: it settles the single instance n_k = 2^k + 1. Corollary 5 concerns Lucas numbers and is not an instance of problem 267.

Source: https://doi.org/10.1017/S0017089500006868. The copy read prints only its "Published online by Cambridge University Press" footer; the journal's article page (https://www.cambridge.org/core/product/identifier/S0017089500006868/type/journal_article, read 2026-10-02) states "Copyright © Glasgow Mathematical Journal Trust 1987" and does not mark the article Open Access, every other right reserved.

Bears on. #267 (Corollary 4 proves $\sum_{n\ge1}1/F_{2^n+1}$ irrational, the single instance $n_k=2^k+1$ of the problem; Corollary 5 is over Lucas numbers and is not an instance), #243 (Corollary 1 gives that a sequence of positive integers with rational reciprocal sum has $a_{n+1}\le a_n^2-a_n+1$ for infinitely many $n$; the paper does not give the problem's conclusion), #263 (context only: the problem page records a thread remark calling the main Theorem a stronger classical criterion; the paper addresses neither of the problem's questions)

Results.

• Theorem (main result, p. 222): if $(a_n)$ and $(b_n)$ are sequences of positive integers with $a_{n+1}>(b_{n+1}/b_n)a_n^2-(b_{n+1}/b_n)a_n+1$ for every large $n$, then $\sum b_n/a_n$ is irrational.
• Corollary 1 (p. 224): if $(a_n)$ is a sequence of positive integers with $a_{n+1}>a_n^2-a_n+1$ for all large $n$, then $\sum 1/a_n$ is irrational; the sequence $c_1=2$, $c_{n+1}=c_n^2-c_n+1$ has $\sum 1/c_n=1$, so the inequality cannot be weakened to $\ge$.
• Proposition (Section 4, p. 225): "Every convergent infinite series of positive rationals has infinitely many disjoint subseries with irrational sums."
• Corollary 4 (p. 227): $\sum_{n\ge1}1/F_{2^n+1}$ is irrational.
• Corollary 5 (p. 227): $\sum_{n\ge1}1/L_{2^n}$ is irrational, where $L_n$ is the $n$th Lucas number.

Corollaries 2 and 3 (pp. 224--225) are further consequences of the Theorem, parallel to the theorems of Erdős and of Sándor that the paper cites; they have no result page here.

Read status. Claims checked: the statements of the Theorem, Corollaries 1, 4 and 5 and the Proposition were read clause by clause on the printed pages; the proofs were read for structure only.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.