  We study the set D of positive integers d for which the equation
$\phi(a)-\phi(b)=d$ has infinitely many solution pairs (a,b), where $\phi$ is
Euler's totient function. We show that the minumum of D is at most 154, exhibit
a specific A so that every multiple of A is in D, and show that any progression
a mod d with 4|a and 4|d, contains infinitely many elements of D. We also show
that the Generalized Elliott-Halberstam Conjecture, as defined in [6], implies
that D equals the set of all positive, even integers.
