We designed an Excel programme in order to find the examples that the smaller triples differ by one so that we could, hopefully, find the pattern of them and find out a general solution of them.
Let the length of the adjacent side (the smallest side) be n
Then, the length of the opposite side would be n+1
The first column is the value of n.
The second column is the value of n^2.
The third column is the value of (n+1).
The fourth column is the value of (n+1)^2.
The fifth column is the summation of second and fourth.
The sixth takes the root of the fifth column.
The seventh one checks whether the numbers in the sixth column is an integer.
Considering the 65534 numbers, which is the maximum number of rows that Excel provides, we have found that there are 7 sets of solutions.
We, after finding these, now try to find the pattern of the numbers, but we hope that our findings can serve as the first step for others who would like to investigate this mathematics problem, just like us.