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Question - Math 324 Assignment 7 (due November 5)

1. (11.1 # 12 b)) Consider the quadratic congruence ax2 + bx + c â‰¡ 0 mod p, where p is an
odd prime and a, b, c are integers with (a, p) = 1. Let d = b2 âˆ’ 4ac, and show that
a) the congruence ax2 + bx + c â‰¡ 0 mod p is equivalent to y2 â‰¡ d mod p, with y = 2ax + b.
b) if d â‰¡ 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 âŽŸâŽŸ
âŽ
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âŽœâŽœ
âŽ
âŽ›âˆ’
p
2 = 1 if p â‰¡ 1, 3 mod 8, and = âˆ’ 1 if p â‰¡ âˆ’ 1, âˆ’ 3 mod 8.

3. (11.1 #14). Show that, if p â‰¥ 7 is prime, then ...Read More

Solution Preview - c and y = 2ax + b, we get an identity y2 âˆ’ d = 4a(ax2 + bx + c). Since (4a, p) = 1, it follows that x solves ax2 + bx + c â‰¡ 0 mod p â‡” y solves y2 â‰¡ 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 â€˜completing the squareâ€™. b)

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