23. Find all such prime numbers, so that each of them can be represented as the sum and difference of two prime numbers.
23. Since the desired number $p$ must be the sum of two prime numbers, $p>2$. All such prime numbers are odd. To get odd numbers when adding and subtracting two numbers, one of the components must be odd and the other even. Since the only even prime number is 2, then $p=p_{1}+2$ and $p=p_{2}-2$. Therefore, the prime numbers $p, p_{1}$, and $p_{2}$ are consecutive odd numbers, and the only such numbers are $3, 5, 7; p=5; 5=3+2$ and $5=7-2$.