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  • primitive Pythagorean triples (a, b, c) --- 3 j , 12 k

    From henhanna@gmail.com@21:1/5 to All on Sun Aug 14 11:01:27 2022
    ------- pls (pls, Please... PLEASE ! ) wait a few days before posting answers or hints.



    primitive Pythagorean triples (a, b, c)

    Why is it that ...
    (1) C is never a multiple of 3
    (2) A*B is always a multiple of 12




    There are 16 primitive Pythagorean triples of numbers up to 100:

    (3, 4, 5) (5, 12, 13) (8, 15, 17) (7, 24, 25) (20, 21, 29) (12, 35, 37) (9, 40, 41) (28, 45, 53)
    (11, 60, 61) (16, 63, 65) (33, 56, 65) (48, 55, 73)
    (13, 84, 85) (36, 77, 85) (39, 80, 89) (65, 72, 97)

    Each of these points forms a radiating line in the scatter plot. Other small Pythagorean triples such as (6, 8, 10) are not listed because they are not primitive; for instance (6, 8, 10) is a multiple of (3, 4, 5).

    .......... primitive Pythagorean triples of numbers up to 300:

    (20, 99, 101) (60, 91, 109) (15, 112, 113) (44, 117, 125)
    (88, 105, 137) (17, 144, 145) (24, 143, 145) (51, 140, 149)
    (85, 132, 157) (119, 120, 169) (52, 165, 173) (19, 180, 181)
    (57, 176, 185) (104, 153, 185) (95, 168, 193) (28, 195, 197)
    (84, 187, 205) (133, 156, 205) (21, 220, 221) (140, 171, 221)
    (60, 221, 229) (105, 208, 233) (120, 209, 241) (32, 255, 257)
    (23, 264, 265) (96, 247, 265) (69, 260, 269) (115, 252, 277)
    (160, 231, 281) (161, 240, 289) (68, 285, 293)





    ------- pls (pls, Please... PLEASE ! ) wait a few days before posting answers or hints.

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  • From Edward Murphy@21:1/5 to henh...@gmail.com on Sun Aug 14 18:44:36 2022
    On 8/14/2022 11:01 AM, henh...@gmail.com wrote:

    primitive Pythagorean triples (a, b, c)

    Why is it that ...
    (1) C is never a multiple of 3
    (2) A*B is always a multiple of 12

    Easily derived from two of the things listed here:

    https://en.wikipedia.org/wiki/Pythagorean_triple#Elementary_properties_of_primitive_Pythagorean_triples

    Anyone know how to prove those things in the first place, though? I
    skimmed the references, but neither one was obviously spelled out.

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  • From henhanna@gmail.com@21:1/5 to henh...@gmail.com on Wed Aug 17 18:36:54 2022
    hint. A few weeks ago, we saw that A and B can't BOTH be odd.


    (1) C is never a multiple of 3

    (any integer) is 0,1,2 (mod 3)
    (any integer)^2 is 0,1 (mod 3)

    if C was 0 (mod 3), then both A, B would have to be 0 (mod 3)
    Contradiction.



    On Sunday, August 14, 2022 at 11:01:29 AM UTC-7, henh...@gmail.com wrote:
    ------- pls (pls, Please... PLEASE ! ) wait a few days before posting answers or hints.



    primitive Pythagorean triples (a, b, c)

    Why is it that ...
    (1) C is never a multiple of 3
    (2) A*B is always a multiple of 12




    There are 16 primitive Pythagorean triples of numbers up to 100:

    (3, 4, 5) (5, 12, 13) (8, 15, 17) (7, 24, 25)
    (20, 21, 29) (12, 35, 37) (9, 40, 41) (28, 45, 53)
    (11, 60, 61) (16, 63, 65) (33, 56, 65) (48, 55, 73)
    (13, 84, 85) (36, 77, 85) (39, 80, 89) (65, 72, 97)

    Each of these points forms a radiating line in the scatter plot. Other small Pythagorean triples such as (6, 8, 10) are not listed because they are not primitive; for instance (6, 8, 10) is a multiple of (3, 4, 5).

    .......... primitive Pythagorean triples of numbers up to 300:

    (20, 99, 101) (60, 91, 109) (15, 112, 113) (44, 117, 125)
    (88, 105, 137) (17, 144, 145) (24, 143, 145) (51, 140, 149)
    (85, 132, 157) (119, 120, 169) (52, 165, 173) (19, 180, 181)
    (57, 176, 185) (104, 153, 185) (95, 168, 193) (28, 195, 197)
    (84, 187, 205) (133, 156, 205) (21, 220, 221) (140, 171, 221)
    (60, 221, 229) (105, 208, 233) (120, 209, 241) (32, 255, 257)
    (23, 264, 265) (96, 247, 265) (69, 260, 269) (115, 252, 277)
    (160, 231, 281) (161, 240, 289) (68, 285, 293)





    ------- pls (pls, Please... PLEASE ! ) wait a few days before posting answers or hints.

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)