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Vieta's Jumping technique

i tried to find about vieta's jumping but everywhere they have shown its application in some of the IMO problems. I want to know the basic principle and and where it comes into use . Any reference or books??please share. And anything about other concepts related to vieta's jumping will also be welcomed.

Note by Tejas Kasetty
4 years, 2 months ago

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The main reason why you have only found applications in olympiad problems, is because this technique's name arose from application in an olympiad problem.

At it's heart, the idea of Vieta's jumping, is about finding integer (or rational) points on a conic, where we proceed via the method of descent, generally through reflection using the condition on \( x + x^* \) obtained from Vieta's formulas. It is a special (simplified) case of various results on Pell's equation, continued fractions, and other results in quadratic fields.

For an interesting use that is often not mentioned, you can read up on Markov numbers, which are solutions to \(x^2 + y^2 + z^2 = 3 xyz \). If you continue reading through to the Markov Spectrum, you will start to see non-trivial uses of Vieta's jumping.

Calvin Lin Staff - 4 years, 2 months ago

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This would be very useful to know for a certain 230 point number theory problem! I will add this to my solution, thank you.

Michael Tong - 4 years, 2 months ago

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thank you very much calvin . But is there any specific book related to the concept, so that i can know more about it.

Tejas Kasetty - 4 years, 2 months ago

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Tejas - To go further, you need to write a post that demonstrates what you do understand about this idea, or why you are unable to follow up on what Calvin wrote. From your post it sounds like you are aware of Olympiad problems using this idea, but it is unclear if you have read the solutions and/or understand them.

Trying to interpret what Calvin said for you:

Calvin's second paragraph is giving you the other names that you can follow up on in more 'standard' literature, which mostly explains why you can't find a book titled "Vieta's Jumping" but you can find a book titled "Pell's Equation".

Doc Mo - 4 years, 2 months ago

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@Doc Mo k i get it. thanks

Tejas Kasetty - 4 years, 2 months ago

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