How Do You Calculate Degeneracy in a 2D Particle in a Box?

[HAMJOOP]

Question
Particle in a box (2D)
Determine the energy levels (degeneracy) of the lowest three


I found that E = A (4a^2 + b^2)
where A is a constant
a and b are positive integers (principle quantum number)


My steps
I assume 4a^2 + b^2 = k
where k is also a positive integer

The minimum value of a and b are 1, so k ≥ 5

I would like to find the combination of a and b for a given k.
But I don't know how to solve integral solution.

Is there any systematic methods apart from trial and error ?

 

[DrClaude]

Not trial and error, but a systematic exploration of all possibilities is quite easy here. Since the problem asks for the three lowest levels, $b \le3$ and you need only calculate a few values.