Hi, I got an answer for you, which (I think) is the only one:43 1 67
61 37 13
7 73 31
I don't know of any big secret for how to solve the puzzle, just trial and error. To start with, there are only 6 combinations of two primes p1 and p2 such that p1 + p2 + 1 = 111, so those are the only numbers you can use in the first row and the middle column (p1 and p2 can be: (3,107), (7,103), (13,97), (31,79), (37,73), or (43,67)). Also, since you can't use 1 anywhere else in the puzzle, the two primes you pick for the top row can't be the same two you use in the middle column if you want the diagonals to add up to 111 as well as the rows.
If you start with 3 and 107 in the top row and two other primes from the list above in the middle column, you can tell after a few trials that adding numbers to 107 quickly gets you over 111, so you don't have many options for the other boxes. Eventually you figure out that you need to use the two pairs of primes which are closest together, which are (43,67), and (37,73). Sorry if that explanation is confusing, hope it makes at least some sense