The chinese remainder theorem

Reference no: EM13521443

Exercise 1. Using the Chinese remainder theorem, how many (incongruent) solutions does x2 ≡ -1 (mod m) have when:
(a) m = 30? (b) m = 65?
Exercise 2. It turns out that every nonnegative integer can be written as a sum of four perfect squares. In contrast, show that if n ≡ 7 (mod 8), then n cannot be written as a sum of three perfect squares.
Exercise 3. For a ∈ Z and p a prime, consider the equation
x2 = [a]p (1)
in Z/pZ.
(a) As a and p are allowed to vary, what are the possibilities for the number of solutions of (1)? For each possibility, illustrate with an example of a and p such that (1) has precisely the given number of solutions.
(b) Give an example of a prime p such that if only a varies, not every possible number of solutions listed in part (a) can occur for (1).
(c) Show that if we restrict to a which are not multiples of p, then for any fixed choice of p, not every possibility listed in (a) will occur as the number of solutions of (1).

Reference no: EM13521443

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