Triangular Triples
Entire pursuit inspired by this James Tanton tweet.
Summary of my efforts documented in this pdf.
It's well-known that there are infinitely-many "Pythagorean Triples" that satisfy A²+B²=C² for positive integers A, B, and C, and there exist formulas to generate all of them.
If we care to, we may define the "squaring function" S(N)=N² and express the Pythagorean Theorem as S(A)+S(B)=S(C).
Now consider the "triangular number function" T(N) = 1+2+3+…+N = N(N+1)/2.
The referenced tweet invites the challenge: How many "Triangular Triples" satisfy T(A)+T(B)=T(C) for positive integers A, B, and C, and does there exist a formula or algorithm to generate all of them?
I'm confident that I've found such an algorithm that finds all such Triangular Triples, and I've implemented it into the GeoGebra construction below. However, it falls short of a tidy closed-form formula that I'd consider to be ideal. Rather, at one point in my algorithm, a pair of linear Diophantine equations must be solved. It doesn't take much computing power to churn out the solutions to the Diophantine equations, but it still strikes me as a tad unsophisticated.
Also worth noting: If we consider {A, B, C} to be the same triple as {B, A, C}, then such duplicates do appear in this algorithm. For example, m=3 & n=2 & u=0 yields solution {4, 9, 10}, and m=4 & n=1 & u=0 yields solution {9, 4, 10}.
If using a keyboard, finer control of the sliders below may be achieved by first clicking on the slider and then using ←→ or ↓↑ arrow keys to decrement/increment the values.
While I'm content to consider my work on this challenge to be complete, if I ever get so inspired to tackle it again and if I can overcome this hurdle, it will certainly be the highlight of my life.