HW #3. Due  Wed, Feb 11, 1998.   Given on Feb 4.
Math 365. Number Theory.  Spring 1998.

Use Matlab for the following  4  problems.
Problem 1.  Find the largest prime  <100000.
Problem 2.  Find the  1000-th prime number.
Problem 3. List all the primes <100000  whose last
   four digits (in the decimal system)  are  9999.
Problem 4.  What is the largest gap between two consequtive
   primes  < 100000.
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Let  f( )  stands for the Euler function,
f(n)=cardinality(U(n)). 
Problem 5. Find all  n>1  for which  
   (a) f(n)=4;  (b) f(n)=12.
Problem 6. Find the number of positive integers 
   n<1000000   which have an odd number of
   positive integer divisors.
Problem 7. Find the number of solutions to
x^2=1  (mod  N)   where   N   stands for the product 
of first  10  primes.
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New challenge problem.

Problem 4. Let  x=sqrt(2),  y=sqrt(2)+2.
Prove that   A = {[nx] | n>0}   and  B = {[ny] | n>0} .
partition the set of  pos. integers,  i.e.  A  and  B  donšt
intersect and their union is  Z+.