Question
Please help!
Let and let p be an odd prime that does not divide a. This exercise shows that finding a solution of is equivalent to finding a solution of simpler quadratic congruence of the form . Prove that the congruence is equivalent to . Prove that the congruence is equivalent to the congruence . (Note that the last congruence is of the form ). Let y = 2ax + b and d = b2 - 4ac. Let x0 be a solution of the congruence . Prove that y0 = 2ax0 + b is a solution of . With y = 2ax + b and d = b2 - 4ac, let y0 be a solution of the congruence . Prove that any solution x0 of the congruence is a solution of . With d = b2 -4ac, prove that has no solution if d is a quadratic non-residue modulo p, one incongruent solution modulo p if p | d, and two incongruent solutions modulo p if d is a quadratic residue modulo p.
Explanation / Answer
Please help! Let and let p be an odd prime that does not divide a. This exercise