Consider modulo , for withOriginally Posted by DenMac21
. You will find
is never divisible by and so cannot be divisible by .
If,Originally Posted by DenMac21
Noticing that 5 is a prime, look at its discrimanant ( .
The above congruence has a solution if and only if its discrimant is a quadradic residue.
Apply Euler's Criterion to determine if something is a quadradic residue we get that,
Thus, is never divisible by 5, thus how can it be divisible by ?
Thus, we have shown it is impossible.
To remind you Euler's Criterion states, is a quadradic residue of if and only if,