Math 430 Problem Set 2

                                                            Math 430 Problem Set 2

 You may discuss these problems with each other, and with me, but with no one else.  There will be one additional problem on the exam.  Notes will not be permitted during the exam.

(1)  Let p be an odd prime, and n=2p.  Show that a^n-1 º a  (mod n) for any integer a.

(2)  Let p be an odd prime.  An element a of (Z/pZ)* is called a fifth power if there exists b Î (Z/pZ)* with a = b⁵. 

a)  If p º2 (mod 5) how many elements of (Z/pZ)*  are fifth powers?

b)  If p º1 (mod 5) how many elements of (Z/pZ)*  are fifth powers?

 (3)  Let S(n) = ∑m(d)s(d), where the sum is taken over all divisors d of n.  Find a formula for S(n) in terms of the prime factorization of n.

(4)  Let p be a prime, pº3 (mod 4).  Prove that the product of all even integers less than p is  º 1  or (-1) (mod p).