Prime Numbers and the Convergents of a Continued Fraction
Continued fractions offer a concrete representation for arbitrary real numbers. The continued fraction expansion of a real number is an alternative to the representation of such a number as a possibly infinite decimal. Continued fractions are important in many branches in mathematics. Since ancient times they played an important role in the approximation to real numbers by rational numbers, using convergents. In 1939 P. Erdos and K. Mahler showed that there are irrational numbers for which each of the denominators of the convergents of their continued fraction expansion is a prime number. Using the techniques presented in their paper we showed that "for almost all" real numbers the greatest prime factor of the numerators of the n-th convergent of the corresponding continued fraction increases rapidly with n. In this talk we present this and other results of our investigation on the numbers for which each of the numerators of the convergents of their continued fraction expansion are prime numbers.