Let a triangular number T(n) = n*(n+1)/2 be factored into the product A*B with A less or equal to B. Let gcd(x,y) be the greatest common divisor of x and y For each of pair (A,B) define C,D,E,F as follows C = (gcd(A,n+1))^2, D = 2*(gcd(B,n))^2, E = 2*(gcd(A,n))^2, F = (gcd(B,n+1))^2. As an example. Let n = 36 then T(n) = 666 which can be factored into 6 distinct pairs A,B. The six sets (A,B,C,D,E,F) are as follows 1. (1,666,1,648, 1369,2) 2. (2,333,1,162,1369,8) 3. (3,222,1,72,1369,18) 4. (6,111,1,18,1369,72) 5. (9,74,1,8,1369,162) 6. (18,37,1,2,1369,648) My conjecture is that the products (A+Ct)*(B+Dt) and (A+Et)*(B+Ft) are each triangular numbers (= T(r) = r(r+1))/2) for all integer t. Also, the above example gave a interesting result in that when the 12 results for r (= the integer part of the square root of 2T(r)) are sorted by value the difference between adjacent r values when t = 1 was one of the even factors of 666. It would seem to me that the easiest way to prove this conjecture would be to derive and prove a formula for r. I will give the corresponding r values below for t = 1 1. 72,110 2. 54,184 3. 48,258 4. 42,480 5. 40,702 6. 38,1368 Also I found that r follows an arithmetic series as t varies, so it is important to find the pattern for the above values, since the difference between them and 36 is the difference value of the arithmetic series. Anyone see a pattern?