Special Problem 2:   Suppose that b is an integer greater than 2.   Carefully prove that the set

                                                    { r   |      0  <   r   <  b   and   (r, b) = 1 }

                                       has an even number of elements.
 

Special Problem 3:   Carefully prove that the equation   x² - 35y² = 11    has no solutions
                                       where x and y are integers.
 
 

Special Problem 4:    Suppose that  a is an integer. Suppose also that,  for every positive integer m ,
                                       the congruence  x² =  a  (mod m)   is solvable.   Prove that  a  must be a perfect
                                       square.
 
 

Special Problem 5:   (a)  Let p be a prime number.   Consider the congruence  x² = 2 (mod p).
                                               By experimentation,  find a simple rule involving p for determining
                                               whether this congruence can be solved.  The rule should be simple enough
                                               so that you can tell very quickly whether or not the congruence is solvable,
                                               even if the prime p is extremely large.

                                       (b)  Find a positive integer m so that the congruence  x² = 2 (mod m)   has
                                               exactly four incongruent solutions module m.