Prove this diophantine equation $2^a-3^b=5~，a,b\in N^{+} $ has no postive integers solution

Question

show that the diophantine equation $$2^a-3^b=5~~~~,a>5,b>3，a,b\in N^{+} $$ has no postive integers solution

maybe is old problem,But I try somedays,can't solve it by now

Answer 1

Please see this paper of Herschfeld (pages 232 and 233) for a proof. The machinery used is elementary. I think the result was first proved by Pillai.