Some Triangle Numbers

A positive whole number n is a triangle number if it is the area of a right triangle with rational side lengths. Thus n is a triangle number if there are rational numbers a, b, and c with a^{2} + b^{2} = c^{2} and n = ab/2. The following table lists every n ≤ 40, whether or not it is a triangle number, and, if so, the side lengths of the triangle, and a solution to the corresponding elliptic curve.

n  Side Lengths  Point on y^{2} = x^{3} − n^{2}x
1  −
2  −
3  −
4  −
5  a = 20/3, b = 3/2, c = 41/6  (−4, 6)
6  a = 3, b = 4, c = 5  (−3, 9)
7  a = 24/5, b = 35/12, c = 337/60  (25, 120)
8  −
9  −
10  −
11  −
12  −
13  a = 323/30, b = 780/323, c = 106921/9690  (−36/25, 1938/125)
14  a = 8/3, b = 21/2, c = 65/6  (18, 48)
15  a = 4, b = 15/2, c = 17/2  (−9, 36)
16  −
17  −
18  −
19  −
20  −
21  a = 12, b = 7/2, c = 25/2  (−3, 36)
22  a = 140/3, b = 33/35, c = 4901/105  (2178, 101640)
23  a = 41496/3485, b = 80155/20748, c = 905141617/72306780  (559728/17689, −287596140/2352637)
24  −
25  −
26  −
27  −
28  −
29  a = 99/910, b = 52780/99, c = 48029801/90090  (−4900/169, 6930/2197)
30  a = 5, b = 12, c = 13  (−20, 100)
31  a = 720/287, b = 8897/360, c = 2566561/103320  (1681/49, 29520/343)
32  −
33  −
34 a = 15/2, b = 136/15, c = 353/30 (−16, 120)
35  −
36  −
37 a = 450660/777923, b = 777923/6090, c = 605170417321/4737551070 (−1764/21025, −32672766/3048625)
38  a = 1700/279, b = 5301/425, c = 1646021/118575  (−1922/81, 105400/729)
39  a = 5/2, b = 156/5, c = 313/10  (−36, 90)
40  −