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# Missing answers for the equation (n(n + 1)/2)(m(m + 1)/2) = x(x + 1)/2?

Posted 10 years ago
 When I inputed the above equation, it did not give me all the integer solutions, or showed a way to get them. I wished to answer the question: "Which integer triangular numbers are the product of two other integer triangular numbers?" Wolfram Alpha returns only four integer solutions. However, the integer solutions x and n of n(n + 1)/2 = 2(x(x + 1)/2) + 1 generate infinite amounts of integer solutions to "(triangular number) multiplied by (other triangular number) = (another triangular number)". i.e. 10 * 21 = 210. All three numbers in this equation were triangular. Why doesn't Wolfram Alpha output formulas to generate all the other integer solutions to (n(n + 1)/2)(m(m + 1)/2) = x(x + 1)/2?
 I am a hobbyist in number theory. Suggestion from the math majors are welcomed. n(n + 1)/2 = 2(x(x + 1)/2) + 1 is a Pell-like equation and WL/Mathematica can recognize the pattern and solve it with Reduce: Reduce[n (n + 1)/2 == 2 (x (x + 1)/2) + 1, {x, n}, Integers] (*same implementation in W|A*) Your question seems much more complicated and challenging for finding a general solution. I am not sure if there is a generic quadratic Diophantine equation solver for this type of problem with 3 unknowns. I tried something here: For (n(n + 1)/2)(m(m + 1)/2) = x(x + 1)/2 <-> T(m)*(n*(n+1)/2) = (x+1)*x/2 if T(i) == i*(i+1)/2 I can do some algebra and make the rhs a perfect square. Recall: (2 x + 1)^2 = 4 x^2 + 4 x +1 I hence transfer the triangle number identity into the following: p^2 - T(m)*q^2 == -T(m)+1 with p == 2*x+1 and q = 2*n+1, which is of the following form: p^2 - D q^2 ==c with |c| = T(m)-1 > Sqrt(T(m)) Unfortunately this is really something like "Hilbert 10th" problem for generic solution. Math world also gives the following quote: If |c|>sqrt(D), the procedure is significantly more complicated (Beiler 1966, p. 265; Dickson 2005, pp. 387-388) and is discussed by Gérardin (1910) and Chrystal (1961). Yet with given numbers, you can find the solution: d = (m*(m + 1))/2 /. m -> 7; Reduce[p^2 - d q^2 == -d + 1 && p > 0 && q > 0, {p, q}, Integers] A sample notebook is also attached. Attachments: