Erdos 1981 sur l irrationalite d une certaine

../

conjecture_p765: Erdős expects that a-n over n tending to infinity is already sufficient for the sum of a-n over two to the a-n to be irrational, and records that he found no sequence with unbounded gaps and a rational sum.

remark_p768: In the closing paragraph Erdős records that he tried several times, without success, to settle the arithmetic nature of the sum of g-n over two to the g-n, where g-n runs through the squarefree numbers.

theorem_p765: For every increasing sequence of integers whose consecutive differences tend to infinity, the sum over n of a-n divided by two to the a-n is irrational.

---

P. Erdős: Sur l'irrationalité d'une certaine série (in French), C. R. Acad. Sci. Paris, Sér. I Math. 292 (1981) no. 17, 765--768; MR 82g:10052; Zentralblatt 466.10028.

The note's single Théorème, unnumbered (p. 765), states that if a_1 < a_2 < ... are integers with a_{n+1} - a_n → +∞, then the sum over n ≥ 1 of a_n/2^{a_n} is irrational. In the next paragraph Erdős adds, without proof, that the theorem remains true with 2 replaced by an integer t > 1; he notes that he conjectured the result more than twenty years earlier and that previously only the case a_n > cn log n was known. The proof (pp. 765-767) is by contradiction: assuming the sum equals u/(v 2^r) with v odd, take k large, let a_n = 2^k - s be the largest term below 2^k, multiply through by v 2^{a_n}, and the quantity in equation (2) must be a positive integer, hence at least 1. If t_1 + s > k for infinitely many k (case (3)), a fractional-part estimate gives the lower bound (4), 1/(2v^2), on a tail that equations (5)-(6) show tends to 0; otherwise t_1 + s ≤ k, the sum splits as in (7)-(9), and the fractional-part bound (13) gives the contradiction.

On p. 765 Erdős conjectures that lim a_n/n = +∞ is very likely a sufficient condition for irrationality; in the same paragraph he says he could not find a sequence with lim sup(a_{n+1} - a_n) = +∞ and a rational sum, though he thinks one exists and could be easy to find. On p. 767, after the proof, he says the gap condition can be weakened a little but not yet substantially. The note closes (pp. 767-768) with two questions and a remark: for a rational α < 1 written as the sum of a_n/2^{a_n} by the greedy algorithm, whether some α has unbounded a_{n+1} - a_n; whether for every c > 0 there is a sequence of rationals (or reals) u_n > (1 + c)u_{n+1} such that a rational sum of u_{n_i} forces n_{i+1} - n_i to be bounded; and the remark that Erdős tried several times, without success, to prove a statement about the sum of g_n/2^{g_n} over the squarefree numbers g_n, printed as "est rationnel" [sic], where the context suggests "irrationnel".

Source: https://users.renyi.hu/~p_erdos/1981-36.pdf. The file prints the journal header "C. R. Acad. Sc. Paris, t. 292 (11 mai 1981)" and "Série I — 765" and no copyright or license line on any of its four pages; the hosting archive's site footer speaks for the site, not the paper ("(C) 2005-2007 All rights reserved. All material on this site is for scientifics purposes only.", https://users.renyi.hu/~p_erdos/, read 2026-10-02); the 1981 volume is not on the publisher's current platform and no publisher page was read, and no Crossref license is recorded; the term is unstated.

The copy read for this card is the four-page reprint at the address above.

Results.

• Théorème (p. 765): the sum of a_n/2^{a_n} is irrational for integers a_1 < a_2 < ... with a_{n+1} - a_n → +∞.
• Conjecture (p. 765): lim a_n/n = +∞ should suffice for irrationality.
• Remark (p. 768): the unsettled sum of g_n/2^{g_n} over the squarefree numbers.

Read status. Claims checked: the statements above were read clause by clause on the printed pages; the proof was read for structure only.

Bears on. #260 (the problem's question is the paper's p. 765 conjecture; the Théorème proves irrationality for the sequences whose gaps tend to infinity, which satisfy a_n/n → ∞, and says nothing about other such sequences), #259 (the p. 768 remark records that Erdős could not settle the problem's series, the sum of g_n/2^{g_n} over the squarefree numbers; the paper proves nothing about it)

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