Incredible prime 2!(part 2)

Number Theory Level 1

Let $S$ be the set of all positive integers $n$ , such that $n^2 - 1$ is a prime.

Find the smallest element of $S$ .

Also try Incredible prime 2! This question is a part of this set

1 solution

Nihar Mahajan
Feb 8, 2015

We all know -

$\huge n^2 - 1 = (n + 1)(n - 1)$

This can be only prime if one of its factors is $\huge 1$ . So ,

$\huge Case 1:$

$\huge n +1 = 1 \Rightarrow n = 0 \Rightarrow -1 ,is , prime$ , which is not possible.

$\huge Case 2:$

$\huge n - 1 = 1 \Rightarrow n =2 \Rightarrow 3, is , prime$ , which is true.

So , $\huge n = 2$ is the only unique solution.

Also , $\huge 3$ is the only example of a prime being one less than a perfect square.

Nice solution, but no need to use \huge for everything.

Siddhartha Srivastava - 6 years, 4 months ago

the answer should be $-2$ , (there's no mention of positive solution)

tasmeem reza - 6 years, 4 months ago

Yeah! Sorry about that. I edited the question.

Nihar Mahajan - 6 years, 4 months ago