This is a near-prime square

Number Theory Level 3

Let $p_i$ be the $i$ -th prime number such that $p_i + 1$ is a square number. Give your answer as the least upper bound to the set $\{ p_i \}_i$ .

The answer is 3.

1 solution

Henry U
Dec 22, 2018

By the condition, $p_i + 1 = n^2, n \in \mathbb{N}$ . This is equivalent to $p_1 = n^2 - 1 = (n-1) (n+1)$ . But a prime can't be the product of two positive integers, unless one of them is 1. Since $n-1$ is the smaller of the two, it has to be 1, and hence $n = 2$ gives the only prime of this type, $p_1 = 2^2-1 = \boxed{3}$ .

This is a near-prime square

The answer is 3.

1 solution

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