Task B-2.5. The ages of a father and his two children (they are not twins) are powers of the same prime number. A year ago, the ages of each of them were prime numbers. How old is the father, and how old is each of his two children?
## Solution.

Let $p^{a}, p^{b}, p^{c}$ be the ages we are looking for. Then the numbers $p^{a}-1, p^{b}-1, p^{c}-1$ are prime.

If $p=2$, the powers of 2 are $2,4,8,16,32,64,128, \ldots$

Among them, we are looking for those which, when we subtract 1, yield prime numbers. These are the numbers 4, 8, and 32, respectively. It is unlikely that someone can be older than 120 years, so the only possibility is that the father is 32 years old, and the children are 4 and 8 years old.

If the base $p \geq 3$, the numbers $p^{a}, p^{b}, p^{c}$ are all odd, and they are all even when "one year younger," so they cannot be prime numbers. This means that the only solution is $32,4,8$.

(1 point)