High School Mathematics Extensions/Further Modular Arithmetic/Problem Set

1. Suppose in mod m arithmetic we know x ≠ y and

$y^2\equiv x^2(\mathrm{mod}m)$

find at least 2 divisors of m.

2. Derive the formula for the Carmichael function, λ(m) = smallest number such that a^λ(m) ≡ 1 (mod m).

3. Let p be prime such that p = 2^s + 1 for some positive integer s. Show that if g is not a square in mod p, i.e. there's no h such that h² ≡ g, then g is a generator mod p. That is g^q ≠ 1 for all q < p - 1.