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Christine
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Posts: 159
 two pythagorean triples   « on: Jul 16th, 2013, 10:30am » Quote Modify

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
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towr
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 Re: two pythagorean triples   « Reply #1 on: Jul 16th, 2013, 1:03pm » Quote Modify

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.
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JohanC
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Posts: 460
 Re: two pythagorean triples   « Reply #2 on: Jul 19th, 2013, 5:38pm » Quote Modify

on Jul 16th, 2013, 1:03pm, towr wrote:
 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 ....

Hi Towr,
Your last solution seems to have one number above 1000.
My list of squares with a-f below 1000 would be:
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

Do you really think some non-trivial search optimization would be possible for this puzzle?
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towr
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 Re: two pythagorean triples   « Reply #3 on: Jul 20th, 2013, 2:15am » Quote Modify

on Jul 19th, 2013, 5:38pm, 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.
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JohanC
Senior Riddler

Posts: 460
 Re: two pythagorean triples   « Reply #4 on: Jul 20th, 2013, 6:43am » Quote Modify

Hi Towr,

The cubes seem to need at least 6 digits:
4884    53613    53835    161491    131580    208309
67488    71891    98605    148512    94484    176020
186760    234117    299483    156240    15930    157050

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.

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JohanC
Senior Riddler

Posts: 460
 Re: two pythagorean triples   « Reply #5 on: Jul 24th, 2013, 4:23am » Quote Modify

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

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