2^4=16+105=121, 2^6+105=169, 2^8+105=361.

Analysis of the behaviour of  2^n+105 mod9 gives the cycle {8,1,5,4,2,7} showing that the power of 2 must be even, and hence a square itself. Hence only the first 5 even powers of 2 need to be checked, as the gap between 2^12=64^2 and the consecutive square already exceeds 105.