Let $\phi$ denote Euler’s phi function. For a fixed odd prime $q$ we investigate the first and second order terms of the asymptotic series expansion for the number of $n\leqslant x$ such that $q\nmid \phi (n)$ . Part of the analysis involves a careful study of the Euler-Kronecker constants for cyclotomic fields. In particular, we show that the Hardy-Littlewood conjecture about counts of prime $k$ -tuples and a conjecture of Ihara about the distribution of these Euler-Kronecker constants cannot be both true.

The famous problem of determining all perfect powers in the Fi- bonacci sequence and the Lucas sequence has recently been resolved by three of the present authors. We sketch the proof of this result, and we apply it to show that the only Fibonacci numbers Fn such that Fn ±1 is a perfect power are 0, 1, 2, 3, 5 and 8. The proof of the Fibonacci Perfect Powers Theorem involves very deep mathematics, combining the modular approach used in the proof of Fermat's Last Theorem with Baker's Theory. By contrast, using the knowledge of the all perfect powers in the Fibonacci and Lucas sequences, the determination of the perfect powers among the numbers Fn ± 1 is quite elementary. We consider the Fibonacci sequence (Fn)n 0 and the Lucas sequence (Ln)n 0 both of which are solutions to the linear recurrence un+2 = un+1 + un, with the initial conditions F0 = 0, F1 = 1 and, respectively, L0 = 2, L1 = 1.

We show that the only Fibonacci numbers that are concatenations of two repdigits are 13, 21, 34, 55, 89, 144, 233, 377.