Bremner-Macleod numbers

are the smallest positive integer solutions \((a,b,c)\) of the following :
$$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=N$$
where \(N\) is a positive integer. The equation and the sizes of its solutions were discussed by Andrew Bremner and Allan Macleod.Andrew Bremner and Allan Macleod, An unusual cubic representation problem
==Size==
For all odd \(N\) and some even \(N\), the equation has no positive integer solutions.
The Bremner-Macleod number for \(N=4\) is (a,b,c) =
For \(N=6\) it is
For \(N=10\) it is
For \(N\le200\), the amounts of digits in \(\max\{a,b,c\}\) are shown below:
N 	# digits
4 	81
6 	134
10 	190
12 	2707
14 	1876
16 	414
18 	10323
24 	33644
28 	81853
32 	14836
38 	1584369
42 	886344
46 	198771
48 	418086
58 	244860
60 	9188
66 	215532
76 	23662
82 	85465
92 	252817
102	625533
112	935970
116	112519
126	196670
130	8572242
132	3607937
136	26942
146	259164
156	12046628
158	15097279
162	1265063
178	398605460
182	2828781
184	20770896
186	5442988
196	11323026
198	726373
200	71225279
For \(N=896\), \(a\) has more than 2.187 trillion digits.
==Sources==
Category:Numbers
Category:Class 2
Category:Class 3
Category:Lower class 3
Category:Humor