Problem 1. Determine the prime numbers $a>b>c$ for which $a-b$, $b-c$, and $a-c$ are different prime numbers.
Solution. If $a-b, b-c, a-c$ are different prime numbers, then $a, b, c$ cannot all be odd. It follows that $c=2$ and $a-b=2$.

2 p

We have $a-b=2, b-2=x, a-2=y$, where $x$ and $y$ are prime numbers, and thus the numbers $a=b+2, x=b-2$ and $y=b$ are prime numbers. $2 p$

The prime numbers $b-2, b, b+2$ are consecutive odd numbers, so one of them is a multiple of 3, from which $b-2=3$, for which $b=5$ and $b+2=7$ are prime. $2 p$

Therefore, $a=7, b=5, c=2$ are the sought numbers. $1 p$