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        riddles >> hard >> two pythagorean triples
        
(Message started by: Christine on Jul 16^th, 2013, 10:30am)

Title: two pythagorean triples
Post by Christine on Jul 16^th, 2013, 10:30am

Is it possible to find two pythagorean triples (a, b, c) and (d, e, f) such that

(1) a+d, b+e, and c+f are all squares
(2) a+d, b+e, and c+f are all cubes

If this (1) and/or (2) not possible, find the near misses

Title: Re: two pythagorean triples
Post by towr on Jul 16^th, 2013, 1:03pm

[hide]Being lazy, I let the computer do the work on (1), with the following solutions for a-f below 1000:
33 544 545 256 480 544
66 112 130 130 144 194
120 442 458 280 342 442
264 448 520 520 576 776
585 928 1097 640 672 928
No cubes so far, so perhaps applying some more intelligence is called for there.[/hide]

Title: Re: two pythagorean triples
Post by JohanC on Jul 19^th, 2013, 5:38pm

on 07/16/13 at 13:03:29, towr wrote:

[hide]Being lazy, I let the computer do the work on (1), with the following solutions for a-f below 1000:
....
585 928 1097 640 672 928
....[/hide]

Hi Towr,
Your last solution seems to have one number above 1000.
My list of squares with a-f below 1000 would be:
[hide]36 48 60 493 276 565
345 460 575 280 165 325
120 225 255 780 451 901
256 480 544 33 544 545
25 60 65 96 40 104
100 240 260 384 160 416
225 540 585 864 360 936
120 442 458 280 342 442
66 112 130 130 144 194
264 448 520 520 576 776[/hide]

Do you really think some non-trivial search optimization would be possible for this puzzle?

Title: Re: two pythagorean triples
Post by towr on Jul 20^th, 2013, 2:15am

on 07/19/13 at 17:38:50, JohanC wrote:

Your last solution seems to have one number above 1000.

Yeah, in truth I just calculated c and f and had no check on them. And I missed a few because I specified a<b and d<e (allowing d>e finds the missing ones).

Quote:

Do you really think some non-trivial search optimization would be possible for this puzzle?

Well, I suspect that with perhaps some modular arithmetic it's possible to show the cube case is impossible. There comes a point when it's smarter to see if what you're searching for exists than it is to keep searching.

Title: Re: two pythagorean triples
Post by JohanC on Jul 20^th, 2013, 6:43am

Hi Towr,

The cubes seem to need at least 6 digits:
[hide]4884 53613 53835 161491 131580 208309
67488 71891 98605 148512 94484 176020
186760 234117 299483 156240 15930 157050[/hide]
But so far, no solutions with d<e popped up.
Neither do primitive solutions to the squares' puzzle, but I imagine the smallest such solutions could be very hugh.

Once again you're right with your suggestion to involve modular arithmetic.

Title: Re: two pythagorean triples
Post by JohanC on Jul 24^th, 2013, 4:23am

Looking for cube solutions smaller than 500000, 2 more pop up, one of them in the "natural" order:
[hide]8120 80997 81403 334880 169050 375130
18500 44400 48100 147875 171600 226525[/hide]