Claims

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2015_06_27_gurev_korsky: Problem 4 of ELMO 2015, proposed by Gurev with Korsky's official solution: for every integer a > 1 some 2^(2^n) + a is composite, which answers question (iii.a) no for every shift at least 2; pending.

2026_04_08_bloom: The site curator's commentary construction, primes a_k with q_k dividing a_k + k, growing as fast as desired, with 0 the only integer shift making every term prime: (i) answered no, and (ii) for nonnegative shifts.

2026_04_15_barschkis: Barschkis's forum note of 15 April 2026, with a Lean file, answering the two general questions and the always-prime question no; the site's commentary carries the construction and credits the order argument.

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