Consecutive Numbers pt. 1

Number Theory Level 3

For every positive integer $n,$ does there always exist $n$ consecutive positive integers that are all composite?

Yes No

1 solution

Steven Yuan
Jul 20, 2017

The sequence

$(n + 1)! + 2, (n + 1)! + 3, \dots, (n + 1)! + (n + 1)$

consists of $n$ consecutive positive integers that are all composite, because $(n + 1)! = (n + 1) \times n \times (n - 1) \times \cdots \times 3 \times 2 \times 1$ shares a factor with each of $2, 3, 4, \dots, n + 1. \, \, \blacksquare$

Consecutive Numbers pt. 1

1 solution

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