Let n | A2 + kB2 where A,B are natural numbers so that gcd(A,kB)=1
 and k belongs to {1,2,-2} .Prove that there
 exist two integer
 numbers C and D such that n = C2 + kD2.
          Applying this event to deduce that all prime number form 4k+1 is 
a Pitagore number ie...there exist C,D are integer numbers and    p = C2 + kD2.
   

    Prove that for any given positive integer number n and k is
 natural then none of the 2n following numbers is integer  
                    1       1       1             1                         
           A  =   ----- + ----- + ----- + ... + -----       
                   k+1     k+2     k+3           k+n

 

   Let n be poitive integer.Prove that

    13 + 23 + ... + n3 divides by 3(15 + 25 + ... + n5 )

    Prove that for all m ,a are positive integer and a>1 we must have 
                      am - 1     
                 gcd(------- , a - 1 ) = gcd(m,a - 1)
                      a - 1

     Find the smallest positive rotional r so that it can epressed
 in the form 
                   x     y
              r = --- + --- 
                   a     b
 where a,b are given positive integers and x,y are integer.
    Hint:Note that gcd(a,b) is the smallest positive integer so that in 
has form ax+by where x ,y are integers!

   It is well-known that if we write the number:
                    1     1            1        M
                A= --- + --- + ... + ----- =   ---
                    1     2           p-1       N
   where p is an odd prime number and gcd(M,N)=1 then p divides by M.
   Now we consider the more genaral position: Prove or disprove that if 
                    1     1            1        U     
                B= --- + --- + ... + ----- =   ---
                    1k     2k         (p-1)k     V
  where k is positive integer,p is prime p>3 and gcd(U,V)=1 then p|U

   Prove that there exist positive integer N such that the decimal 
representation of the number 2000N starts with the sequence of the 
digits 200120012001.

  (i) Prove that sqrt(2)+sqrt(3)+sqrt(5)+sqrt(7) is not 
rational
  (ii)Prove that sqrt(n-1)+sqrt(n+1) is not rational for all n is positive integer. 

  Find all possible pairs (x,y) belong to Z+ such that  yx2 = xy+2

   Solve the following diophantine equation:
                           x^2=y^z-3
   for z#1(mod 4)  

   Determine all pairs (x,y) of positive integers such that 
         x2y + x + y is divisible by xy2 + y +7

   Determine all pairs(a,b) of real numbers such that a[bn] =b[an] for all
 positive integers m, n. 
   Note that [x] denotes the greatest integer less than or equal to x

   Find two disjoint sets A and B whose union is the non-negative integers such 
that every positive integer can be expressed as the
sum of two distinct elements 
of A in the same number of ways as it can be expressed as the sum of two 
distinct elements of 

   Let a1 ,...,an+1 be integers such that 1 <= a1 < ... < an+1 <= 2n.Prove that
 there is i,j,k such that ai = aj + ak.

     For every positive integer n, there is a prime number p such that n <= p <= 2n

         Let a,b be positive integers such that
               i) gcd(a,b)=1
              ii) if p is a prime number such that p<=sqrt(a^2+b^2), then p|ab
     Prove that a^2+b^2<49
    (You can use  Bertrand Conjecture to prove it) 

    Let S be a set of integers (not necessarily positive) such that 
          i)There exist a, b with gcd(a,b)= gcd(a-2,b-2)=1
         ii)If x and y are elements of S(possibly equal), then x^2-y also belongs
 to S. 
     Prove that S is the set of all integers.

   Prove that (n!) is not a perfect square with n>2 a natural number.

    Let a,b,c and m,n,p be natural numbers with gcd(a,b)=gcd(b,c)=gcd(c,a)=1 
and gcd(m,n)=gcd(n,p)=gcd(p,m)=1 . Prove that there exist infinitely many natural 
numbers x,y,z so that a/(x^m)+b/(y^n)=c/(z^p)  .

      Let m,n be natural numbers such that 
                       (m+3)n + 1 
                    A= --------         is an integer
                          3m
     Prove that it is odd.