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Campbell 2026 binary digits erdos borwein constant
John M. Campbell, On the binary digits of the Erdős-Borwein constant. arXiv preprint (2026). arXiv:2605.24160.
The copy read for this card is arXiv:2605.24160v1 (22 May 2026). Its Theorem 1 (p. 12) answers an open problem of Crandall (2012), also noted by Shallit, in the affirmative: the block 11 appears infinitely often in the binary expansion of the Erdos-Borwein constant E, the sum of 1/(2^n - 1) that Erdos showed to be irrational. The argument builds a system of congruences by the Chinese remainder theorem, in the manner of Erdos's irrationality proof, and counts primes in arithmetic progressions with a lemma of Alford, Granville and Pomerance (Lemma 1, taken in Vandehey's form); the author records that it was developed through extensive interaction with GPT-5.5 Pro. The paper does not mention the series of problem 249; for that problem it is adjacent work, on the binary digits of E = sum d(n)/2^n (the paper's (1)), the divisor-function analogue of the sum of phi(n)/2^n, and evidence of an active research thread.
Source: https://arxiv.org/abs/2605.24160. The arXiv record (https://arxiv.org/abs/2605.24160, read 2026-10-02) names the Creative Commons Attribution 4.0 license.
Bears on. #249