HW #8. Due  Fri, Apr 17, 1998.   
Given on Mon, Apr 13, 1998
Math 365. Number Theory.  Spring 1998.

Denote by  phi(n) = card(U(n))  the Euler function.

1.  Prove that   phi(n)/n > 1/6   for all  
     positive integers
           n<200000000=2*10^8.
2.  Prove that   phi(n)/n < 1/100   has 
     infinitely  many solutions.
3.  Let  p=9999991.  Find all solution  
     mod p  for:        (a)  x^3=1;
        (b)  x^4=1;     (c)  x^5=1.
4.  Let   p(n)   stands for the n-th prime 
     number.  Thus   p(4)=7.  Prove that  
     the  inequality
              p(n+1)<(p(n)+p(n+2))/2
      (a)  holds  for an infinite set of  n;
      (b)  fails  for an infinite set of  n.
5.  Find an increasing arithmetical sequence 
      of length  L=8  of prime numbers.
6* (optional). The same  with  L=10.
              
New m-files:    
     ordermod.m    a2bmodc.m    a2bmodc2.m

New Chal.  Problem.  
   Prove that the following  (special) 
   Fib. sequence  
           g(n) = {1,3,4,7,11...}
     has the property    g(p)=1  mod p
     for every prime  p.