Question

Let a and b be integers not divisible by an odd prime p. Show that either one or all three of the integers a, b, and ab are quadratic residues mod p.

Answer 1

To show that either one or all three of the integers a, b, and ab are quadratic residues mod p, we consider two cases.

Step-by-step explanation:

To show that either one or all three of the integers a, b, and ab are quadratic residues mod p, we need to consider two cases:

Case 1: If a and b are quadratic residues mod p, then ab is also a quadratic residue mod p. This is because the product of two quadratic residues is also a quadratic residue.

Case 2: If either a or b is a quadratic residue mod p, then both ab and the non-residue are quadratic residues mod p. This is because the product of a residue and a non-residue is a non-residue.

Hence, in both cases, either one or all three of the integers a, b, and ab are quadratic residues mod p.