Exercises for the Lecture on Computational Number Theory

In this exercise n is a large integer. a) Suppose that there is an integer F = ∏k i=1 p ei i with known primes pi and ei > 0 such that F |(n − 1) and F > √ N . Let R = (n − 1)/F and assume that for each i there is an integer ai such that the following conditions hold: a RF i ≡ 1 mod n and (a R(F/pi) i − 1, n) = 1. Show that Z/(n ·Z)× contains a primitive F−th root of unity and conclude that n is prime. b) Let N = Z/(n · Z) and s > √ n be an integer with t = ords(n) the order of n mod s. Let Φs(X) ∈ Z[X] be the s−th cyclotomic polynomial, i.e. the minimal polynomial of a primitive s−th root of units, e.g. e2πı/s. Assume that there is a polynomial Ψ ∈ N [X] of degree t such that (i) Ψ(X) divides Φs(X) over N . (ii) Let ζ = X + (Ψ(X)) ∈ N [X]/(Ψ(X)). Then Ψ(ζni) = 0 for i = 1, 2, . . . , t. Then r|n iff r ≡ ni mod s for some i ∈ {1, 2, . . . , t}.