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Danube Mathematical Competition, 2014

Number theory Difficulty 4.6 AIME Prove it Romania

Given any integer $n \ge 2$ , show that there exists a set of $n$ pairwise coprime composite integers in arithmetic progression.

Solution

Fix a prime $p > n$ and an integer $N \ge p + (n-1)n!$ and consider the arithmetic progression of length $n$ consisting of $N! + p + kn!$ , $k = 0, 1, \dots, n-1$ .

Suppose, if possible, that $q$ is a prime factor of two of these numbers. Then $q$ divides their difference which is of the form $kn!$ , for some positive integer $k < n$ . It follows that $q$ does not exceed $n$ , so $n!$ and $N!$ are both divisible by $q$ , and consequently so is $p$ — a contradiction.

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