Does there exist a positive irrational number &#x03B1;<!--

Roland Waters

Roland Waters

Answered question

2022-06-20

Does there exist a positive irrational number α, such that for any positive integer n the number n α is not a prime?

Answer & Explanation

Abigail Palmer

Abigail Palmer

Beginner2022-06-21Added 30 answers

A Beatty sequence is what we have here. Throughout the sequence, there will in fact always be a prime. The bound for the OP's sequence (provided α > 1) is
p  L 35  16 ϵ α 2 ( 1  ϵ ) p m + l 1 + ϵ 
where L = log  ( 2 α )p n denotes the numerator of the n th consistent with the ongoing, regular fraction increase of α, and m is the unique integer such that p m  L 16 α 2 < p m + 1 ϵ can be selected at will, although l relies on ϵ.

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