Prime Numbers and the Convergents of a Continued Fraction
Cahlen Charles Humphreys · 2013
Continued fractions offer a concrete representation of arbitrary real numbers, where in the past such numbers were represented in decimal format. Continued fractions are found useful in many different areas of mathematics and science. Since ancient times they have 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, and through Theorem 3.5 and Corollary 3.6, we show that for almost all real numbers the greatest prime factor of the numerator of the nth convergent of the corresponding continued fraction increases rapidly with n.