Question

If p is a prime. Prove 12 is a factor of p^2-1, p>=5.

Answer 1

fermat's little theorem with p = 2

Answer 2

2 isn't greater than or equal to 5

Answer 3

oh you are right this requires some thought. but i am fairly certain it is a direct application of fermat

Answer 4

Probably

Answer 5

but you are right, not trivial one. requires thinking

Answer 6

oh no might not, might be easier.
\[p^2-1=(p-1)(p+1)\] and p is odd so both factors are even and so the product is divisible by 4

Answer 7

now we have to show it is divisible by 3 and then we are done right?

Answer 8

and of course p is not divisible by 3 (it is prime) so
\[p=3k+1\] or
\[p=3k+2\] and now you can show that no matter which,
\[p^2-1\] is a multiple of 3

Answer 9

for example, if
\[p=3k+1\] then
\[(p+1)(p-1)=3k(p+1)\] hence divisible by 3

Answer 10

last one is similar. let me know if this is not clear

Answer 11

Nice!

Answer 12

merci

Answer 13

Well played. :)