Counting in Pythagorean numbers: 3, 4 and 5 etc

Rein · **Posted:** Tue Oct 25, 2011 06:57:31 am

How to solve a few ones I missed!

33² = 1089; 1089-1² = 1088; 1088/(2*1) = 544 hence 33, 544 and 545 (545=544+1)

33² = 1089; 1089-3² = 1080; 1080/(2*3) = 180 hence 33, 180 and 183 (180+3)

Using 2, 4, 6 or 8 does not make sense as after the substraction (resulting in an odd number) we can not divide by 4, 8, 12 or 16!!!

33² = 1089; 1089-5² = 1064; 1064/(2*5) = 106.4 No good!

33² = 1089; 1089-7² = 1040; 1040/(2*7) = 74 2/7 No good

33² = 1089; 1089-9² = 1008; 1008/(2*9) = hence 33, 56 and 65 (56+9)

to be continued ....

drvvr · **Posted:** Tue Oct 25, 2011 14:21:40 pm

Rein wrote:

Maybe 36, 77 and 85???

Absolutely

Rein · **Posted:** Tue Oct 25, 2011 18:48:19 pm

37:

37²+684² = 1369+467856=469225=685²

Rein · **Posted:** Tue Oct 25, 2011 18:59:01 pm

38:

multiple of 19: 19, 180, 181

38, 360, 362

Rein · **Posted:** Tue Oct 25, 2011 19:04:15 pm

39:

multiple of 3: 3, 4 and 5

39, 52 and 65

multiple of 13: 13, 84 and 85

39, 252 and 255

And yet: 39, 760 and 761

39²+760² = 1521+577600=579121=761²

petiauda · **Posted:** Tue Oct 25, 2011 19:42:15 pm

Rein wrote:

I can take 10203 and then 104101209 must be split up in 52050604 and 52050605

Yes, but

10203*10203 + 13604*13604 = 17005*17005

is simpler (1+0+2+0+3=6, so 10203 is divisible by 3).

Oh, and

10203*10203 + 14396*14396 = 17645*17645

is a primitive?

drvvr · **Posted:** Tue Oct 25, 2011 21:17:27 pm

petiauda wrote:

Rein wrote:

I can take 10203 and then 104101209 must be split up in 52050604 and 52050605

Yes, but

10203*10203 + 13604*13604 = 17005*17005

is simpler (1+0+2+0+3=6, so 10203 is divisible by 3).

Oh, and

10203*10203 + 14396*14396 = 17645*17645

is a primitive?

No it just a method for finding for ODD not primitive

Rein · **Posted:** Tue Oct 25, 2011 21:25:14 pm

petiauda wrote:

Rein wrote:

I can take 10203 and then 104101209 must be split up in 52050604 and 52050605

Yes, but

10203*10203 + 13604*13604 = 17005*17005

is simpler (1+0+2+0+3=6, so 10203 is divisible by 3).

Oh, and

10203*10203 + 14396*14396 = 17645*17645

is a primitive?

Petiauda,

YES, it seems so

But can you prove it????

your numbers are certainly smaller! But that was not what I tried to show. Take an odd integer and you can produce two other integers that comply - not bothering about smaller ones

Indeed as 10203 is a multiple of 3: 3*3401

and hence 4*3401 = 13604 and 5*3401 = 17005 are OK!

And still using the blunt strategy:

3401 gives 5783400 and 5783401 !

So 10203, 17350200 and 17350203 are another (still bigger) triplet!

10203 = 3*19*179 so using 19, 180 and 181 you will get: 10203, 96660 and 97197

Rein · **Posted:** Tue Oct 25, 2011 21:27:26 pm

drvvr wrote:

petiauda wrote:

Rein wrote:

I can take 10203 and then 104101209 must be split up in 52050604 and 52050605

Yes, but

10203*10203 + 13604*13604 = 17005*17005

is simpler (1+0+2+0+3=6, so 10203 is divisible by 3).

Oh, and

10203*10203 + 14396*14396 = 17645*17645

is a primitive?

No it just a method for finding for ODD not primitive

drvvr,

which method led to 10203*10203 + 14396*14396 = 17645*17645 ???

Rein · **Posted:** Tue Oct 25, 2011 22:34:43 pm

10203 is a multiple of 179 !

Take 179 and its square 32041 !

use the crude "method" for odd integers:

Calculate (32041-x²); then divide by 2*x - as 10203 is 3*19*179 you will go on multiplying with 3*19 (57) - and you will get the 14396 by taking x = 57 !

The third number you get by adding 57² (x*57) to 14396 => 17645

Similarly I got along the road:

10203, 913140 and 913197 (x=1)
10203, 304304 and 304475 (x=3)
10203, 47530 and 48603 (x=19)

and at last:

10203, 14396, 17645 (x=57)

to be continued ...

petiauda · **Posted:** Tue Oct 25, 2011 22:38:40 pm

Rein wrote:

petiauda wrote:

Oh, and

10203*10203 + 14396*14396 = 17645*17645

is a primitive?

Petiauda,

YES, it seems so

But can you prove it????

Well, I could say that 10203 and 14396 are relative prime, that their greatest common divisor is 1, and then I could wave my hands around a lot. Some people could take that for a proof, yes? :lol:

Rein · **Posted:** Tue Oct 25, 2011 22:47:33 pm

petiauda wrote:

Rein wrote:

petiauda wrote:

Oh, and

10203*10203 + 14396*14396 = 17645*17645

is a primitive?

Petiauda,

YES, it seems so

But can you prove it????

Well, I could say that 10203 and 14396 are relative prime, that their greatest common divisor is 1, and then I could wave my hands around a lot. Some people could take that for a proof, yes? :lol:

Petiauda,

Fine with me! Using the crude method above and taking x>57 led to nowhere! x = 73 was just above 10203 and x = 75 just below....

Rein · **Posted:** Wed Oct 26, 2011 00:30:43 am

40:

Multiple of 5 and 8

5, 12 and 15 give 40, 96 and 120

8, 15 and 17 give 40, 75 and 85

to be continued ....

Rein · **Posted:** Wed Oct 26, 2011 00:34:05 am

41:

41, 840 and 841 as 41² + 840² = 1681 + 705600 = 707281 = 841²

to be continued ....

Rein · **Posted:** Wed Oct 26, 2011 06:45:09 am

42:

42 = 2*3*7

hence

14* 3, 4 and 5 -> 42, 56 and 70

6* 7, 24 and 25 -> 42, 144 and 150

2* 21, 220 and 221 -> 42, 440 and 442

to be continued ...

Rein · **Posted:** Thu Oct 27, 2011 00:53:56 am

43:

43² + 924² = 1849 + 853776 = 855625 = 925²

to be continued ....

Rein · **Posted:** Fri Oct 28, 2011 07:30:29 am

44:

Multiple of 4 and 11...

4 times 11, 60 and 61 -> 44, 240 and 244

a bit more complicated:

44² + 483² = 1936 + 233289 = 235225 = 485²

or

44² + 117² = 1936 + 13689 = 15625 = 125²

to be continued ....


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Counting in Pythagorean numbers: 3, 4 and 5 etc

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