Is it safe to say the following about the odd prime numbers other than 5.

Question

I am studying basic number theory and have a habit of writing down interesting facts whenever I conclude something from the text or a problem itself. I was wondering whether I can write it down too:

All odd prime numbers other than 5, either themselves are one less or more than a multiple of 10, or their square is 1 less than a multiple of 10.

What do you say?

Answer 1

Yes, you don't even need the numbers to be prime. This can be proven via basic modular arithmetic. All $1,3,7,9 \mod 10$ follow the above rules.

Edit:- So you don't need modular arithmetic, but it does make your life easier. First, note that if any odd number ends in $1$ or $9$ then the first property is satisfied. Second, if it ends in $3$ or $7$, write it as $10n+3$ or $10n+7$ and square it to conclude that second property is satisfied