There is one and only one pair of natural numbers m and n (15 and 21) such that: m and n are triangular numbers; the sum of their squares is a triangular number (666); and the sum of the triangular numbers m(m+1)/2 and n(n+1)/2 is a triangular number k(k+1)/2 (k=26). I have tested pairs of numbers from m+n=2 to m+n=100 000 000 using a computer program. (Notice that 15 20 25 is a Pythagorean triple, giving the sides of a triangle that's similar to the sacred Egyptian triangle 3 4 5.) I know, 666 is the number of the beast and all that stuff, but this is a serious mathematical problem.
I tried but it is difficult to find a proof for this conjecture. It is interesting to note that this conjecture or something similar is not valid for any other pair of triangular numbers.
It's a great observation! Should m and n be consecutive triangular numbers?
Not necessarily, but notice that 15+21 is another triangular number 36 and 666 is the 36th triangular number.
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