A278711 Triangle T read by rows: T(n, m), for n >= 2, and m=1, 2, ..., n-1, equals the positive integer solution x of y^2 = x^3 - A(n, m)^2*x with the area A(n, m) = A249869(n, m) of the primitive Pythagorean triangle characterized by (n, m) or 0 if no such triangle exists.
12, 0, 45, 240, 0, 112, 0, 525, 0, 225, 1260, 0, 0, 0, 396, 0, 2205, 0, 1617, 0, 637, 4032, 0, 3520, 0, 2496, 0, 960, 0, 6237, 0, 5265, 0, 0, 0, 1377, 9900, 0, 9100, 0, 0, 0, 5100, 0, 1900, 0, 14157, 0, 12705, 0, 10285, 0, 6897, 0, 2541, 20592, 0, 0, 0, 17136, 0, 13680, 0, 0, 0, 3312, 0, 27885, 0, 25857, 0, 22477, 0, 17745, 0, 11661, 0, 4225, 38220, 0, 36652, 0, 33516, 0, 0, 0, 22540, 0, 14700, 0, 5292, 0, 49725, 0, 47025, 0, 0, 0, 36225, 0, 0, 0, 0, 0, 6525
Offset: 2
Examples
The triangle T(n, m) begins: n\m 1 2 3 4 5 6 7 8 2: 12 3: 0 45 4: 240 0 112 5: 0 525 0 225 6: 1260 0 0 0 396 7: 0 2205 0 1617 0 637 8: 4032 0 3520 0 2496 0 960 9 0 6237 0 5265 0 0 0 1377 ........................................... n = 10: 9900 0 9100 0 0 0 5100 0 1900, n = 11: 0 14157 0 12705 0 10285 0 6897 0 2541, n = 12: 20592 0 0 0 17136 0 13680 0 0 0 3312, n = 13: 0 27885 0 25857 0 22477 0 17745 0 11661 0 4225, n = 14: 38220 0 36652 0 33516 0 0 0 22540 0 14700 0 5292, n = 15: 0 49725 0 47025 0 0 0 36225 0 0 0 0 0 6525. ... The triangle of solutions [x,y] begins ([0,0] if there is no primitive Pythagorean): n\m 1 2 3 4 2: [12,36] 3: [0,0] [45,225] 4:[240,3600] [0,0] [112,784] 5: [0,0] [525,11025] [0,0] [225, 2025] ... n=6: [1260,44100] [0,0] [0,0] [0,0] [396,4356], n=7: [0,0] [2205,99225] [0,0] [1617,53361] [0.0] [637,8281], n=8: [4032,254016] [0,0] [3520,193600] [0,0] [2496,97344] [0,0] [960,14400], n=9: [0,0] [6237,480249] [0,0] [5265,342225] [0,0] [0,0] [0,0] [1377,23409], n=10: [9900,980100] [0,0] [9100,828100] [0,0] [0,0] [0,0] [5100,260100] [0,0] [1900, 36100]. ...
Links
- Avner Ash and Robert Gross, Elliptic tales: curves, counting, and number theory, Princeton University Press, 2012.
- Keith Conrad, The Congruent Number Problem, The Harvard College Mathematics Review, 2008.
Formula
T(n, m) = (n^2 - m^2)*n^2 if n > m >= 1, gcd(n, m) = 1 and n+m is odd, and T(n, m) = 0 otherwise.
Comments