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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)





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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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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.

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