## Task 23/88

Determine all prime twin pairs $\left(p_{1} ; p_{2}\right)$ for which

$$
P=p_{1}^{2}-p_{1} p_{2}+p_{2}^{2}
$$

is either a prime number or a power $x^{y}$ with $x ; y \in N, x ; y>1$!
If we count $(2 ; 3)$ among the twin prime pairs, then $P=2^{2}-2 \cdot 3+3^{2}=7$ is a solution. For the twin prime pair $(3 ; 5)$, we get $P=3^{2}-3 \cdot 5+5^{2}=19$, which is also a solution.

If $p>5$, then for every twin prime pair $\left(p_{1} ; p_{2}\right)$, it holds that $p_{1}=6 n-1, p_{2}=6 n+1$ with $n \in N$. Thus, we have

$$
P=(6 n-1)^{2}-(6 n-1)(6 n+1)+(6 n+1)^{2}=3(12 n+1)
$$

Therefore, $P>3$ is divisible by 3 and consequently not a prime number. Since the second factor $\left(12 n^{2}+1\right)$ is certainly not divisible by 3, $P$ contains the prime factor 3 only to the first power. Thus, $P$ cannot be a power $x y$ with $x ; y \in N, x ; y>1$; because every power $x^{y}$ with $x ; y \in N$ contains each prime factor to the $y$-th power. Apart from the already determined solutions, there are therefore no further solutions.