Given that and are primes, use the fact that to calculate .
I am a bit confused about the question to start with, is about fractions or quadratic residues?
How can I approach this question?
Those will be quadratic residues if only because otherwise the parentheses would be redundant.
Furthermore, it's not an interesting problem if they were only regular fractions.
The fact that the difference is requested, suggest that from each possible quadratic residue of the first, you can draw a conclusion about the quadratic residue of the second.
So if there is an x such that it should follow that:
- either you can also find an y with:
- or you can proof that the congruence with y does not hold for any y.
And if there is no such x, the inverse should hold.
Your result would then be either -2, 0, or 2.
To be honest, this is as far as I got.
I missed something - there is actually a Lemma in my lecture notes for this.
For and primes with , .
It is straightforward to solve this problem using this Lemma.
The proof of the Lemma is said to be difficult.