Puzzle Time - polynomials
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wrote on 11 Nov 2024, 13:50 last edited by
Find natural numbers x, y, z such that
x/(y+z) + y/(x+z) + z/(x+y) = 4 .
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wrote on 11 Nov 2024, 16:49 last edited by Klaus 11 Nov 2024, 17:21
35,132,627 seems to be close. I didn't think of course.
Edit: OK, that's not the solution. It's only close.
There are no solutions in the range 1 to 10,000, even though 411 812 4601 is really close!
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wrote on 11 Nov 2024, 17:28 last edited by jon-nyc 11 Nov 2024, 17:29
This puzzle is me being a dick. The smallest number in the smallest solution has more than 80 digits. Which is pretty amazing if you think about it.
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wrote on 11 Nov 2024, 17:34 last edited by
Wolfram Alpha quickly returns ‘Standard computation time exceeded’ and asks me to upgrade.
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wrote on 11 Nov 2024, 18:19 last edited by
I've discovered a remarkable way to calculate this, but this forum software is inadequate to express the required mathematical notation.
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This puzzle is me being a dick. The smallest number in the smallest solution has more than 80 digits. Which is pretty amazing if you think about it.
wrote on 11 Nov 2024, 19:54 last edited by Klaus 11 Nov 2024, 19:55@jon-nyc said in Puzzle Time - polynomials:
This puzzle is me being a dick. The smallest number in the smallest solution has more than 80 digits. Which is pretty amazing if you think about it.
But is there a way to come up with the solution analytically?
I know that Diophantine equations are undecidable in general, and analytical solutions only exist for a few special cases.
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wrote on 11 Nov 2024, 23:31 last edited by
Yes. In fact it has to involve some clever analysis before doing automated searches. Naive messing around on a computer is not going to find an 80-digit number.
https://drive.google.com/file/d/1NXYxl3tH7_mZMTBYL7pCeP8RY2bquVyF/view?usp=drivesdk
The search involved elliptic curves.
I just love how basic and innocent the problem seems. It looks like the solution should involve some one- or two-digit numbers.