High School Mathematics Extensions/Further Modular Arithmetic/Problem Set

1. Suppose in mod m arithmetic we know x &ne; y and
 * $$y^2 \equiv x^2 \pmod{m} \!$$

find at least 2 divisors of m.

2. Derive the formula for the Carmichael function, &lambda;(m) = smallest number such that a&lambda;(m) &equiv; 1 (mod m).

3. Let p be prime such that p = 2s + 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 &equiv; g, then g is a generator mod p. That is gq &ne; 1 for all q < p - 1.