If $p$ is prime, with $p> 3$, then $p^{2} \equiv1\pmod{24}$

Question

I tried to deny the thesis and show that $p$ is not prime. Like this, if $24\nmid p^{2}-1$ then $p^{2}-1=24q+r$, $0<r<24$, but it did not look so cool.

Answer 1

First, it's well known that $p^2 \equiv 1 \pmod 8$. (Try letting $p = 8k + m$ for odd $m$ and compute the binomial expansion of $p^2$.)

Then, since $3 \nmid p$, $p \equiv \pm1 \pmod 3$, which implies $p^2 \equiv 1 \pmod 3$.

Finally, finish this question with Chinese Remainder Theorem: as $\gcd(3,8) = 1$, $p^2 = 1 \pmod{24}$ is equivalent to $p^2 \equiv 1 \pmod 8$ and $p^2 \equiv 1 \pmod 3$.

Answer 2

If $p$ is prime and $p>3$ then $p \equiv1\pmod{4}$ or $p \equiv3\pmod{4}$ (otherwise $p$ is an even number, not true because $2$ is the only even prime number).

Because of that, either $n-1$ or $n+1$ is divisible by $4$ (but not both), the other number is also even (divisible by $2$), which implies $(n-1)(n+1)$ or $n^2-1$ is divisible by $8$.

Also if $p$ is prime and $p>3$ then $p$ is not divisible by $3$, so either either $n-1$ or $n+1$ is divisible by $3$ (but not both)$\Rightarrow n^2-1$ is divisible by 3.

Because $GCD(3;8)=1$, $n^2-1$ is divisible by $3 \times 8 =24$.

Answer 3

If $p$ is prime we know that it must have odd residue.

But it can't be $\{3, 9, 15, 21\}$, as $24k + r$ with one of those is divisible by $3$.

This leaves $\{1, 5, 7, 11, 13, 17, 19, 23\}$, all of which when squared have remainder $1 \bmod 24$.

Answer 4

The squares modulo $24$ are $0, 1, 4, 9, 12, 16$. There are no other possible values (see Sloane's OEIS).

If $n^2 \equiv 9 \pmod{24}$, that suggests $n$ is a multiple of $3$, and since you stipulated $p > 3$, this rules out $n = \pm 3$, the only primes divisible by $3$. And $0, 4, 12, 16$ are all even, but since you stipulated $p > 3$, $n = \pm 2$ is also ruled out.

That leaves us with $n^2 \equiv 1 \pmod{24}$. So if $p^2 \not \equiv 0, 4, 9, 12$ or $16$, what can it possibly be?

Look at the squares of a few small primes: $5^2 = 24 + 1$, $7^2 = 2 \times 24 + 1$, $11^2 = 5 \times 24 + 1$, etc.