Puzzle Time - polynomials

[jon-nyc]

Find natural numbers x, y, z such that

x/(y+z) + y/(x+z) + z/(x+y) = 4 .

[Klaus]

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!

[jon-nyc]

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.

[jon-nyc]

Wolfram Alpha quickly returns ‘Standard computation time exceeded’ and asks me to upgrade.

[Horace]

I've discovered a remarkable way to calculate this, but this forum software is inadequate to express the required mathematical notation.

[Klaus]

@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.

[jon-nyc]

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.