Given any prime expressed as a+b, is there always some a,b such that 2^a*3^b is
one away from a prime?

I doubt it but have yet to find a counterexample.

Below are primes < 3187 which produced five or fewer solutions. [prime, number
of times that 2^a*3^b is one away from a prime]

[2, 2] [3, 3] [5, 5] [11, 5] [317, 5] [347, 3] [677, 4] [739, 4] [809,
5] [857, 5] [1033, 5] [1229, 5] [1291, 5] [1319, 5] [1451, 2] [1471, 5]
[1663, 3] [1721, 5] [2069, 4] [2477, 5] [2659, 4] [2677, 3]

Mark