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Source. Theorem 2, p. 356, of Yong-Gao Chen, On integers of the form $k2^n+1$, Proceedings of the American Mathematical Society 129(2), 355--361 (electronically published 28 August 2000), https://doi.org/10.1090/s0002-9939-00-05916-5, the edition named on the source card. The proof is on pp. 358--359.

Read depth. Claims checked: the statement and the definitions it uses were read clause by clause on the printed pages; the proof (pp. 358--359) was read. Nothing here is independently reviewed.

Statement

A $(2,1)$-primitive $r$-covering system is defined on the Theorem 1 page (Definitions 1--3, p. 356): residue classes $a_i\ (\mathrm{mod}\ n_i)$, $1\le i\le t$, covering every integer at least $r$ times, with distinct primes $p_i$ such that $2$ has order exactly $n_i$ modulo $p_i$.

Theorem 2 (p. 356). "The following statements are equivalent to each other: (i) there exists a $(2,1)$-primitive $r$-covering system; (ii) there exist an odd integer $k$ and a finite set $\{p_1,\cdots,p_t\}$ of distinct primes such that $k2^n+1$ is divisible by at least $r$ of $p_1,\cdots,p_t$ for all positive integers $n$; (iii) there exist an odd integer $k$ and a finite set $\{p_1,\cdots,p_t\}$ of distinct primes such that $k-2^n$ is divisible by at least $r$ of $p_1,\cdots,p_t$ for all positive integers $n$."

Proof pointer

(i) implies (ii) and (iii) by the construction in the proof of Theorem 1 (p. 358). For (ii) implies (i) (pp. 358--359), equation (5) takes for each $p_i$ the order $n_i$ of $2$ modulo $p_i$ and the least positive $a_i$ with $p_i\mid k2^{a_i}+1$; if $p_i\mid k2^n+1$ then $p_i\nmid k$ and $2^{n-a_i}\equiv1\pmod{p_i}$, so $n\equiv a_i\pmod{n_i}$ by (7). Hence the classes $a_i\ (\mathrm{mod}\ n_i)$ cover every positive integer at least $r$ times, which the paper takes as an $r$-covering system. (iii) implies (i) in the same way.

Dependencies

Theorem 1 (its proof, for the forward direction).

Bears on

• Problem 1113: the paper does not mention the problem. With $r=1$, the argument for (ii) implies (i) turns any finite covering set of primes for a coefficient $k$ (over exponents $n\ge1$) into a covering of the exponents by the classes $a_i\ (\mathrm{mod}\ \operatorname{ord}_{p_i}2)$; conversely, if such classes cover the exponents, the primes $p_i$ cover the terms, as in the proof of Theorem 1. So, over exponents $n\ge1$, a coefficient has a finite covering set exactly when finitely many of these order-residue classes cover the exponents. The theorem does not decide whether every Sierpiński number has one, which is the open question.