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Question - Math 324 Assignment 7 (due November 5) 1. (11.1 # 12 b)) Consider the quadratic congruence ax2 + bx + c ≡ 0 mod p, where p is an odd prime and a, b, c are integers with (a, p) = 1. Let d = b2 − 4ac, and show that a) the congruence ax2 + bx + c ≡ 0 mod p is equivalent to y2 ≡ d mod p, with y = 2ax + b. b) if d ≡ 0 mod p there is exactly one solution x mod p; if d is a quadratic residue there are exactly two solutions mod p; and if d is a quadratic non-residue there is no solution. 2. Show that ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛− p 2 = 1 if p ≡ 1, 3 mod 8, and = − 1 if p ≡ − 1, − 3 mod 8. 3. (11.1 #14). Show that, if p ≥ 7 is prime, then there are always two consecutive quadratic residues mod p. Hint: First s ...Read More

Solution Preview - c and y = 2ax + b, we get an identity y2 − d = 4a(ax2 + bx + c). Since (4a, p) = 1, it follows that x solves ax2 + bx + c ≡ 0 mod p ⇔ y solves y2 ≡ d mod p. Also y = 2ax + b, with (2a, p) =1, implies that the number of solutions x mod p is the same as the number of solutions y mod p. Remark: This is just ‘completing the square’. b)

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