On the Euler Function of the Catalan Numbers

We study the solutions of the equation φ(Cm)/φ(Cn) = r, where r is a fixed rational number, Ck is the kth Catalan number and φ is the Euler function. We note that the number r = 4 is special for this problem and for it we construct solutions (m,n) to the above equation which are related to primes p such that 2p− 1 or 4p− 3 is also prime. 1. An observation concerning φ(Cn+1)/φ(Cn) For a positive integer n, let (1) Cn = 1 n+ 1 ( 2n n ) be the n-th Catalan number. For a positive integer m we put φ(m) for the Euler function of m. A Carmichael’s conjecture [5], which is still open, states that for every n it is possible to find an m 6= n such that φ(m) = φ(n). Since this problem seems to be currently out of reach, one would look at the behavior of the Euler’s phi function φ(•), or at quotients φ(•)/φ(•), when the arguments belong to some smaller classes of integers, like the binomial coefficients, binary recurrent sequences, or even Catalan numbers. In fact, there is a growing literature on arithmetic functions with binomial coefficients [12], [15] and [17], or on arithmetic functions with members of binary recurrent sequences [3, 11, 13, 14, 16, 18, 19, 20, 21], etc. At first, we wanted to test whether we could find distinct m and n such that φ(Cm) = φ(Cn) but did not find solutions other than the trivial solution φ(C1) = φ(C2) = 1. So, we checked numerically for the values of the ratios φ(Cm)/φ(Cn) for m 6= n. While computing such ratios for small values of m and n, we first noted, then we proved, the following result. Theorem 1.1. The equality (2) φ(Cn+1) = 4φ(Cn) holds in each of the following two instances: Date: February 26, 2012.