### Smooth Carmichael and Square Euler

Igor Shparlinski
*Macquarie, Australia; visiting Royal Holloway and Oxford*

We outline some recent results about arithmetic properties
of the values of the Carmichael $\lambda$ and Euler $\phi$ functions.
For example, we study how often $\lambda(n)$ is smooth
(i.e. free of large prime factors) and show that this doesn't
happen too often (but more frequently than for a random
integer).
We also study how often $\phi(n)$ is a perfect square and show
that surprisingly enough this is quite common
(much more frequent than for a random integer).
We also show relevance of these and several related questions to
cryptography.
Several useful (but rather simple) number theoretic techniques,
used in the proofs, will be outlined.
The talk is based on joint work (in progress) with
Bill Banks, John Friedlander and Carl Pomerance.