Question

8. Define (n) to be the number of positive integers less than n and n. That is, (n) = {x e Z; 1 < x< n and gcd(x, n) = 1}|. Notice that U (n) |= ¢(n). For example U( 10) = {1, 3,7, 9} and therefore (10)= 4. It is well known that (n) is multiplicative. That is, if m, n are (mn) (m)¢(n). In general, (p") p" -p Also it's well known that there are relatively prime, then infinitely many prime numbers p for which p +2 is also a prime number. They are called as"twin primes." (a) Prove that "If d is a positive divisor of n, the number of elements of order d in cyclic group of order n is ¢(d). a (b) Prove that in a finite group, the number of elements of order d is a multiple of (c) Derive a formula for ¢(pq) where p and q are twin primes. Hint: Let q > p, then q p+ 2.] (d) Use the above formula you derived to find the twin primes p and q when 288

Answer 1

(a) A cyclic A cyclic group has at least one generator, but it is finite , then it will have exactly & (n) otal, where Girs & our group and n=1Gl. So equivalently we could define the finite cyclic group of order as the group with exactly &(n)generators, but it would need to be proven that this is equivalent to being generated by a single element. An element of oredere a generated a subgroup of Ordered which is has since las is unique. A subgroup of orderc d e isomorphic to za it [n] generates The Eld, [1] = 0 [n] this is equivalent to saying 1 = mnted, this ged(d, n) = 1. Thes there exists & (d) generators. M &(Pq) = 8(P) +2, where p is Prime 493 p+2.. since & (P) = P-1 , fore paime po & (P+2) = (+2) - 1 (P-1) +2 > $(P+2) = $(P) #2 (p9)= 4 (0)419)=0CP) $ (P+2) = 4(P)/ 0(p) +9) (d) & (pq) = 288 = $ CD) ( 0C P) +2) = 288 -> (4CP)) + a $(P) = 288 OCP) = 16 = 17-1 so p=17,9=17+2=19.