Thread: Quadratic Residues

We know that there are always 2 consecutive quadratic residues of p if p >= 7 (and p is a prime).

1.) Show this is true for two diff. primes >= 7

2.) Prove why this true. (Hint: you will first want to show that at least one of 2, 5, and 10 is a quad. residue of p).

Originally Posted by math19091

We know that there are always 2 consecutive quadratic residues of p if p >= 7 (and p is a prime).

1.) Show this is true for two diff. primes >= 7

2.) Prove why this true. (Hint: you will first want to show that at least one of 2, 5, and 10 is a quad. residue of p).

An integeristing problem .

Theorem: Let p be an odd prime the number of quadradic residues (modulo classes) is the same as the number of quadradic non-residues.

Proof: I can write the prove, but I am sure you learned this theorem.

Consider our incongruents integers:
1,2,...,p-1

The idea is that since the number of quadradic residues is equal to non-residues they MUST be next to each other unless they alternate:

residue, non-residue, ...
(Sine 1 is residue).

But that cannot happen.

See if you can finish it.