## Task 1 - 230931

Determine all triples $(x, y, z)$ of natural numbers with the following properties:

(1) $x, y$ and $z$ are prime numbers.

(2) Each digit in the decimal representations of $x, y$ and $z$ is a prime number.

(3) $x < y$.

(4) $x + y = z$.
Due to (1) and (4), $z \geq 2+2>2$, hence $z$ is odd. Therefore, exactly one of the two summands $x$ or $y$ is even, and thus, due to (1), it must be 2, the smallest prime number. Therefore, $x=2$ according to (3). This makes $y$ and $z$ twin primes.

Multi-digit twin primes have the ending digits (9 and 1), (1 and 3), or (7 and 9), because the ending digit 5 would lead to divisibility by 5 and thus, due to the multi-digit nature, to a contradiction with (1). Therefore, $y<z$ is a single-digit number, and the solution triples are $(2,3,5)$ and $(2,5,7)$.