Now 5 is not a factor of 2p-1, so 5 must divide evenly into p.
Since p is prime, we conclude that the only case where the expression can be a prime power
with p odd is p = 5.
But for that case, the expression is equal 32 + 243 = 275, which is not a perfect power.
Hence 2p + 3p is never a perfect power for any prime p.