|
%I #19 Nov 28 2016 14:26:28
%S 12,0,45,240,0,112,0,525,0,225,1260,0,0,0,396,0,2205,0,1617,0,637,
%T 4032,0,3520,0,2496,0,960,0,6237,0,5265,0,0,0,1377,9900,0,9100,0,0,0,
%U 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
%N 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.
%C The corresponding triangle with the square root of the positive integer solutions y is A278712.
%C 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.
%C 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.
%H Avner Ash and Robert Gross, Elliptic tales : curves, counting, and number theory, Princeton University Press, 2012
%H Keith Conrad, <a href="http://www.math.uconn.edu/~kconrad/articles/congruentnumber.pdf">The Congruent Number Problem</a>, The Harvard College Mathematics Review, 2008
%F 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.
%e The triangle T(n, m) begins:
%e n\m 1 2 3 4 5 6 7 8
%e 2: 12
%e 3: 0 45
%e 4: 240 0 112
%e 5: 0 525 0 225
%e 6: 1260 0 0 0 396
%e 7: 0 2205 0 1617 0 637
%e 8: 4032 0 3520 0 2496 0 960
%e 9 0 6237 0 5265 0 0 0 1377
%e ...........................................
%e n = 10: 9900 0 9100 0 0 0 5100 0 1900,
%e n = 11: 0 14157 0 12705 0 10285 0 6897 0 2541,
%e n = 12: 20592 0 0 0 17136 0 13680 0 0 0 3312,
%e n = 13: 0 27885 0 25857 0 22477 0 17745 0 11661 0 4225,
%e n = 14: 38220 0 36652 0 33516 0 0 0 22540 0 14700 0 5292,
%e n = 15: 0 49725 0 47025 0 0 0 36225 0 0 0 0 0 6525.
%e ...
%e -------------------------------------------
%e The triangle of solutions [x,y] begins ([0,0] if there is no primitive Pythagorean):
%e n\m 1 2 3 4
%e 2: [12,36]
%e 3: [0,0] [45,225]
%e 4:[240,3600] [0,0] [112,784]
%e 5: [0,0] [525,11025] [0,0] [225, 2025]
%e ...
%e n=6: [1260,44100] [0,0] [0,0] [0,0] [396,4356],
%e n=7: [0,0] [2205,99225] [0,0] [1617,53361]
%e [0.0] [637,8281],
%e n=8: [4032,254016] [0,0] [3520,193600] [0,0] [2496,97344] [0,0] [960,14400],
%e n=9: [0,0] [6237,480249] [0,0] [5265,342225]
%e [0,0] [0,0] [0,0] [1377,23409],
%e n=10: [9900,980100] [0,0] [9100,828100] [0,0]
%e [0,0] [0,0] [5100,260100] [0,0]
%e [1900, 36100].
%e ...
%e -------------------------------------------
%Y Cf. A249866, A249869, A278712.
%K nonn,tabl,easy
%O 2,1
%A _Wolfdieter Lang_, Nov 27 2016
|
