HW #7. Due  Wed, Feb 25, 1998.   Given on Fri, Apr 3, 1998
Math 365. Number Theory.  Spring 1998.

1.  Find the number of solutions to   
               x^2+y^2+z^3=0   
     mod  p   (a given prime).
2.  Find the number of solutions to   
             x+y^2+z^3+t^4=5   
     mod  1000.
3.  Let   M   be the set of positive integers 
     which are not divisible by any integer  n, 
     1< n <10.      Find  d(M).
4.  Let  M  be the set of odd  integers  n   for 
     which  the Jacobi symbol   (3/n)   is  >=0 
     (larger or equal  0.)      Find  d(M).
5.  Let  M  be a subset of positive integers 
     such that  the sum of  1/m,  m  runs 
     over  M,  is  finite.  Prove that   d(M)=0.
6. Is the converse to  5.  true?  

New Chal. Problem. Denote by  w(n)  the period
     of Fib. sequence mod n.  Prove that for every
     prime  p,  not  5,  
     either   w(p)|(p-1),  or  w(p)|(p+1).
     Try to characterize  pšs  corresponding to
      each case.