# Popular-Lonely primes understood

Standard

While reading standup mathematician Matt Parker‘s book Things to Make and do in Fourth Dimension, I found answer (on pp. 146) to the question I raised 7 months ago.

When the grid happens to be a multiple of 6 wide, suddenly all primes snap into dead-straight lines. All primes (except 2 and 3) are one more or less than a multiple of 6. (© Matt Parker, 2014)

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 $p$ 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 $p^2-1=(p-1)(p+1)$ is a multiple of 24.

Note that, $p^2-1$ 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 $p^2-1$ is divisible by 8.

Observe that exactly one of three consecutive numbers, $p-1,p,p+1$ must be divisible by 3. Since $p$ is an odd prime different from 3, one of $p-1$ or $p+1$ must be divisible by 3. Hence we conclude that $p^2-1$ 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“.