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interests / rec.puzzles / Re: primitive Pythagorean triples (a, b, c) --- 3 j , 12 k

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* primitive Pythagorean triples (a, b, c) --- 3 j , 12 khenh...@gmail.com
+- Re: primitive Pythagorean triples (a, b, c) --- 3 j , 12 kEdward Murphy
`- Re: primitive Pythagorean triples (a, b, c) --- 3 j , 12 khenh...@gmail.com

1
primitive Pythagorean triples (a, b, c) --- 3 j , 12 k

<b3d53acc-9c66-4816-abec-621f00f96dafn@googlegroups.com>

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Subject: primitive Pythagorean triples (a, b, c) --- 3 j , 12 k
From: henha...@gmail.com (henh...@gmail.com)
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 by: henh...@gmail.com - Sun, 14 Aug 2022 18:01 UTC

------- 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.

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

<tdc8ea$vov$1@gioia.aioe.org>

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From: emurph...@zoho.com (Edward Murphy)
Newsgroups: rec.puzzles
Subject: Re: primitive Pythagorean triples (a, b, c) --- 3 j , 12 k
Date: Sun, 14 Aug 2022 18:44:36 -0700
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 by: Edward Murphy - Mon, 15 Aug 2022 01:44 UTC

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.

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

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Subject: Re: primitive Pythagorean triples (a, b, c) --- 3 j , 12 k
From: henha...@gmail.com (henh...@gmail.com)
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 by: henh...@gmail.com - Thu, 18 Aug 2022 01:36 UTC

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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