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Statement

Example (p. 295). Every integer satisfies one of the five congruences $x\equiv0\pmod2$, $x\equiv0\pmod3$, $x\equiv1\pmod4$, $x\equiv5\pmod8$, $x\equiv7\pmod{12}$.

Problem 14 (p. 295). Erdős asks whether, for every positive integer $n_0$, there are an integer $k$ and a system of $k$ congruences $x\equiv a_i\pmod{n_i}$ with $n_0<n_1<\cdots<n_k$ such that every integer satisfies at least one of them. He records that Davenport and he found such a system with $n_1=3$, and that Swift (oral communication) found systems with $n_1=4$ and with $n_1=6$, but that no general methods are known. He then asks whether there is a system in which all the $n_i$ are odd. He refers to [15] and [17] of the paper for the literature.

The paper poses both questions and resolves neither.

Source. P. Erdős, Some unsolved problems, Michigan Math. J. 4 (1957), 291--300; §A, Problem 14, p. 295. The edition read is identified on the source card.

Read depth. Claims checked: the item was read clause by clause on the page images of the journal print. The systems with $n_1=3,4,6$ are reported, not given.

Dependencies

None.

Bears on

• Problem 2: the first question asks for covering systems with distinct moduli all exceeding any given $n_0$, which is the problem's corrected Statement. The paper records smallest moduli $3$, $4$ and $6$ and does not resolve it.
• Problem 7: the second question, a system of this kind with all moduli odd, is the problem. The paper does not resolve it.