You know, this is some kind of discrete optimization problem, because we know neither a, b or c. If we neglect the given characteristics of each parameter, there would be infinite possibilities.
e. g.
a + b + c = 10 with a = 3
So we know b + c = 10-3 = 7
This equation is satisfied with:
b=1,c=6: 1+6=7
b=6,c=1: 6+1=7
b=2,c=5: 2+5=7
and just to exaggerate (not discrete)
b=0.9,c=6.1: 0.9+6.1=7
b=0.99,c=6.01: 0.99+6.01=7
etc.
So using some methods like intelligent water drops or particle swarm optimization we could expect good results.
Here we have a quite large range of possibilities. Your only option is to insert and increment the numbers systematically.
At least you could try to simplify the problem.
If we set a to 1 (first possible value) we'll get:
b^2 + c^2 = d^2 - 1 with d = 427
=> b^2 + c^2 = 427^2 - 1 = 182329 - 1 = 182328
Futhermore:
Max. value of b^2 and c^2: 575^2 = 330625
Min. value of b^2 and c^2: 24^2 = 576
Watch out: 330625 > 182328. So the real max. value of b^2 and c^2 would be 182328-576 = 181752
-> You have to subtract 576 (=24^2) because b or c has to be at least 24.
So max. value out of the range for b and c: sqrt(181752) = 426 (discrete!)
With 426^2 = 181476 < 181752, the min value of b and c has to be 25.
Even though we don't have to go through 24 to 575 anymore, there is no guaranteed solution.
Well, at least the possible values for you:
a=6; b=58; c=423
a=6; b=87; c=418
a=6; b=418; c=87
a=6; b=423; c=58
a=18; b=103; c=414
a=18; b=166; c=393
a=18; b=274; c=327
a=18; b=327; c=274
a=18; b=393; c=166
a=18; b=414; c=103
a=21; b=168; c=392
a=21; b=392; c=168
a=22; b=54; c=423
a=22; b=297; c=306
a=22; b=306; c=297
a=22; b=423; c=54
a=23; b=102; c=414
a=23; b=270; c=330
a=23; b=330; c=270
a=23; b=414; c=102
+259 
+104 
