| Here are the solutions for sides <1,000,000, including the degenerate
solution. The general solution is that B is an integer solution of the
quadratic diophantine equation
2 2
B = 3Y + 4,
where Y is also an integer. I don't have my Pell equation solving program
handy, so this will have to do. - Lynn Yarbrough
A B C Area
--------------------------------------------------------
1 2 3 0
3 4 5 6
13 14 15 84
51 52 53 1170
193 194 195 16296
723 724 725 226974
2701 2702 2703 3161340
10083 10084 10085 44031786
37633 37634 37635 613283664
140451 140452 140453 8541939510
|
| I found a reference to the problem in Dickson, History of the
the Theory of Numbers, volume 2, page 197.
If the sides are n-1, n, n+1, then n satisfies the recurrence
n[0]=2, n[1]=4, n[k+2]=4*n[k+1]-n[k].
An explicit formula for n is
n = (2 + sqrt(3) )^k + (2 - sqrt(3) )^k .
|