Problem 9.4. A prime number $p$ is such that the number $p+25$ is the seventh power of a prime number. What can $p$ be? List all possible options.
Answer: 103

Solution. We will use the fact that the only even prime number is 2.

- Let $p=2$, then $p+25=27$, which is not a seventh power. Contradiction.
- Let $p>2$, then $p$ is odd, and $p+25$ is even. Since $p+25$ is even and is a seventh power of a prime number, this prime number must be 2. Therefore, $p+25=2^{7}=128$, from which we get $p=103$.