The problem only has one solution, B = 32, A = 3626.

I have not been schooled in Diophantine equations, as you probably have. My approach is this.

I substituted A = 1813B/16 into the expression (A*A-4)/B and set the result equal to N. Solving for B gives

B = 32(4N + √(1813^2 + 16N^2))/1813^2 = 32m

I solved for N and found

N = (3286969*m^2 - 1)/(8*m)

The only value of m that can satisfy this is m=1.

I have not been schooled in Diophantine equations, as you probably have. My approach is this.

I substituted A = 1813B/16 into the expression (A*A-4)/B and set the result equal to N. Solving for B gives

B = 32(4N + √(1813^2 + 16N^2))/1813^2 = 32m

I solved for N and found

N = (3286969*m^2 - 1)/(8*m)

The only value of m that can satisfy this is m=1.