Unit fractions with shifted prime denominators

We prove that any positive rational number is the sum of distinct unit fractions with denominators in -1 : p prime\ . The same conclusion holds for the set -h : p prime\ for any h∈ Z \0\ , provided a necessary congruence condition is satisfied. We also prove that this is true for any subset of the primes of relative positive density, provided a necessary congruence condition is satisfied.