Assignment #2 Due: Wednesday June 1 You will need to use some sort of computational aide for some of these problems. I've provided Sage cells with sample inputs below. Values of \(\phi\) Prove whether there are infinitely many values of \(\phi(n)\) that end in zero. Prove that for any positive integer \(t\), there exists a positive integer \(n\) such that the base 10 representation of \(\phi(n)\) ends with \(t\) zeros. Find one such integer where \(t=25\). Primitive Roots How many primitive roots are there mod 12251? Determine whether 10 and 11 are primitive roots mod 12251 using the criterion described in Algorithm 2.5.16 of Stein. Include a summary of the calculations you needed to do. Prove that an odd primitive root modulo an odd prime is still a primitive root for twice that modulus. ( This is a case I overlooked in describing the moduli that have primitive roots in class on May 23) (Stein 2.26) For every positive integer \(b\), show that there exists a positive integer \(n\) such that the polynomial \(x^2-1\) over \(Z/nZ\) has at least \(b\) roots Sage Cell with input that computes modular powers Sage Cell with inputs that compute \(\phi(n)\) and factors \(n-1\). You might not need to do both of these at once.