---

Statement

Setting (p. 120). The paper considers finite systems of congruences $a_i\pmod{n_i}$ with $n_1<n_2<\cdots<n_k$, so with distinct moduli, such that every integer satisfies at least one of them. It lists two examples, one containing the modulus $2$ and one without it, which are recorded on the Theorem 3 page.

Conjecture (p. 120, quoted). "It seems likely that for every $c$ there exists such a system all the moduli of which are $>c$."

Consequence drawn (p. 120). The paper says the conjecture would imply, by the argument that proves Theorem 3, that for every $c'$ there is an arithmetic progression no term of which is of the form $2^k+u$ with $v(u)<c'$, where $v(u)$ is the number of distinct prime factors of $u$ (defined on p. 117).

Scope

This is a conjecture the paper poses; it proves nothing on it.

Read depth. Claims checked: the setting, the conjecture and the consequence were read clause by clause on p. 120 of the print.

Source. P. Erdős, On integers of the form $2^k+p$ and some related problems, Summa Brasil. Math. 2 (1950), fasc. 8, 113--123; the edition read is named on the source card.

Bears on

• Problem 2: the conjecture asserts the yes answer to the problem's corrected statement, covering systems with distinct moduli and arbitrarily large smallest modulus; the paper prints the distinctness as $n_1<n_2<\cdots<n_k$. The problem page records the later work on the question.