Two solutions found:
3x^2 - 2y^2 - 4z^2 + 54 = 0 (1)
5x^2 - 3y^2 - 7z^2 + 74 = 0 (2)
8x^2 - 5y^2 - 11z^2 + 128 = 0 (1) + (2)
2x^2 - y^2 - 3z^2 + 20 = 0 (2) - (1)
21x^2 - 14y^2 - 28z^2 + 378 = 0 (1) x 7
20x^2 - 12y^2 - 28z^2 + 296 = 0 (2) x 4
x^2 - 2y^2 + 82 = 0
x^2 + 82 = 2y^2 (3)
x is even, let x = 2*a; a can be odd or even
4a^2 + 82 = 2y^2
2a^2 + 41 = y^2
3x^2 - 2y^2 - 4z^2 + 54 = 0 (1)
4z^2 = 3x^2 - 2y^2 + 54
z^2 = (3/4)*x^2 - (1/2)y^2 + 27/2
z^2 = (3x^2 - 2y^2 + 54)/4
5x^2 - 3y^2 - 7z^2 + 74 = 0 (2)
7z^2 = 5x^2 - 3y^2 + 74
z^2 = (5x^2 - 3y^2 + 74)/7
Values for x and y which solve equation (3):
[[4, 7],
[16, 13],
[40, 29],
[100, 71],
[236, 167],
[584, 413],
[1376, 973],
[3404, 2407],
[8020, 5671]]
Plugging in these x,y values into either equations (1) or (2) finds only 2 integer solutions:
(x, y, z)
(4, 7, 1)
(16, 13, 11)
On second thought, a brute force search would have been much less work, although slower for large numbers
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Posted by Larry
on 2025-04-08 10:46:28 |
Solution
