He also proves the following surprising theorem:
The square of every prime number greater than 3 is one more than a multiple of 24.
Let be an odd prime not equal to 3. Now we subtract one from the square of this prime number. Therefore, we wish to prove that is a multiple of 24.
Note that, is a product of two even numbers. In particular, one of these two even numbers must be a multiple of 4, as they are consecutive even numbers and every other even number is divisible by 4. Hence we conclude that is divisible by 8.
Observe that exactly one of three consecutive numbers, must be divisible by 3. Since is an odd prime different from 3, one of or must be divisible by 3. Hence we conclude that is divisible by 3.
Combining both the conclusions made above, we complete proof of our statement (since 2 and 3 are coprime).
Edit[19 April 2017]: Today I discovered that this theorem is exercise 68 in “The USSR Olympiad Problem Book“.