# Thread: What is the significance of a prime number - 1?

1. ## What is the significance of a prime number - 1?

What is the significance of a prime number - 1? Such as 43-1 or 29 - 1?

2. Originally Posted by dragstang86
What is the significance of a prime number - 1? Such as 43-1 or 29 - 1?
What is the context? Lots of things are true, for example $p-1$ is always even for $p \neq 2$, so for $p\neq 3$, $p-1$ is not prime. This is somewhat uninteresting though.

However, what is interesting is that $a^{p-1} \equiv 1 \text{ mod } p$. This is called Fermat's Little Theorem.

3. This reminds me a humorous theorem that all natural numbers are interesting. Indeed, assume the contrary, that the set of boring numbers is nonempty. Then there is the least boring number. But hey, this is interesting!

4. Originally Posted by emakarov
This reminds me a humorous theorem that all natural numbers are interesting. Indeed, assume the contrary, that the set of boring numbers is nonempty. Then there is the least boring number. But hey, this is interesting!
Apply the well-ordering principle, then everything is interesting!