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

2,1

Comments

The corresponding triangle with the square root of the positive integer solutions y is A278712.

A primitive Pythagorean triangle is characterized by two integers n > m >= 1, gcd(n, m) = 1 and n+m odd. See A249866, also for references.

For the one-to-one correspondence between rational Pythagorean triangles with area A > 0 and rational points on the elliptic curve y^2 = x^3 - A^2*x with y not vanishing see Theorem 4.1 of the Keith Conrad link or Theorem 15.6, p. 212, of the Ash-Gross reference.

Links

Table of n, a(n) for n=2..106.

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.

Example

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].
...
-------------------------------------------

Crossrefs

Cf. A249866, A249869, A278712.

Sequence in context: A332053 A225951 A333577 * A331911 A307841 A257949

Adjacent sequences: A278708 A278709 A278710 * A278712 A278713 A278714

Keyword

nonn,tabl,easy

Author

Wolfdieter Lang, Nov 27 2016

Status

approved