The theory of continued fractions has applications in cryptographic problems and in solution methods for Diophantine equations. We will first examine the basic properties of continued fractions such as convergents and approximations to real numbers. Then we will discuss a computationally efficient attack on the RSA cryptosystem (Wiener\u27s attack) based on continued fractions. Finally, using continued fractions and solutions of Pell\u27s equation, we will show that the Diophantine equation
x^4-kx^2y^2+y^4 = 2^j (k,j are natural numbers)
has no nontrivial solutions for j=9,10,11 given that k \u3e 2 and k is not a perfect square
