HW #5. Due  Wed, Feb 25, 1998.   Given on Feb 18.
Math 365. Number Theory.  Spring 1998.

1. Find a prime number   p>13  (any one!) for which all the numbers   -1,1,2,3,4,5,6,7,8,9,10,11,12
    are quadratic residues.  Explain your choice.  

2. Prove that there are infinitely many primes satisfying the condition of the previous problem.
    (You can use the results stated in the class without proof.  Alternatively,  you can use the results found in any book.)

    For a integer  n>0,  denote by   d=FD(n)  the first digit of  n  (d  is a integer between  1  and  9).

3.  Let   S={ k>0  |  FD(2^k)=9}.            (a) Find  min(S).  
       (b)Prove that   S   is infinite.      (c) List all    k<200  in  S.


4.  Prove that for every integer   N>1
              o{0<k<N |  FD(k)=1}  >=  o{0<k<N |  FD(k)=9}.
     For which   N    does the equality hold?

5.  Prove that the Jacobi symbol  (2x/(x^2-1))   is  1   for every even  x>0.

Remark. See Summary for new function in Matlab  (in particular,  jac(p,q),  for computing
     the Jacobi symbol of p mod q,  a2bmodc(a,b,c),  divisors(n), euler(n), phi(n)).