HW #4. Due  Wed, Feb 18, 1998.   Given on Feb 11.
Math 365. Number Theory.  Spring 1998.

Use Matlab for the following  2  problems.
Problem 1.  Find the minimal twin pair of primes  >100000.
Problem 2*.   List all  numbers  <10000   which can be represented 
   as a sum  of two cubes (of pos. integers) in more than  one  way
   (the order is ignored; thus the representations   8+27  and  
    27+8  are considered to be the same).
_____________________________________
Without computers:

Problem 3.  Find the largest  n   for which   3^n  |  1000! .  
Problem 4*.  Denote  c(n)=(2n)!/(n!)^2.  Find  minimal   n   such that
    25 | c(n),  or prove that such  n   does not exist.
Problem 5.  Find the sum of the inverses of positive
   divisors of   3000.
Problem 6.  Let  p>10   be a prime.  Prove that the list  1,2,3,4,5,6   contains
at least  3  quadratic residues.
-----------------------------------------------
New challenge problem.  Prove:

Problem.  Prove:  The sum of   [n/k] mu(k)   over  k=1,2,..., n   is equal to  1,  
    for every   n>0.      (mu(x)  stands for the Mobius function.)