Task solved by cyrix

# Task 2 - 261222 

Determine all triples $(p, q, r)$ of prime numbers that satisfy the following conditions (1), (2):

(1) In the sequence of all prime numbers, $p, q, r$ are consecutive prime numbers in this order.

(2) The number $s=p^{2}+q^{2}+r^{2}$ is a prime number.
I. If $(p, q, r)$ is a triple that satisfies conditions (1) and (2), then $(p, q, r)$ is not the triple $(2,3,5)$; because this does not satisfy condition (2) due to $2^{2}+3^{2}+5^{2}=38$.

Furthermore, it follows that:

$(p, q, r)$ is not a triple with $p>3$ (and thus also $q>3, r>3$); because every prime number greater than 3, when divided by 3, leaves a remainder of either 1 or 2, and thus its square leaves a remainder of 1 in both cases.

For every triple $(p, q, r)$ of prime numbers $p, q, r>3$, $p^{2}+q^{2}+r^{2}$ is therefore divisible by 3 (and greater than 3), and thus not a prime number.

By (1), the only remaining possibility is that $p, q, r$ is the triple $(3,5,7)$.

II. This triple satisfies condition (1) as a triple of three consecutive prime numbers, and since $3^{2}+5^{2}+7^{2}=83$, it also satisfies condition (2).

With I. and II., it is shown that exactly the triple $(3,5,7)$ satisfies conditions (1) and (2).

## Adopted from $[5]$

#### Abstract