VOOZH about

URL: https://oeis.org/A278711

⇱ A278711 - OEIS

login
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.
2
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
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
KEYWORD
nonn,tabl,easy
AUTHOR
Wolfdieter Lang, Nov 27 2016
STATUS
approved