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Questions tagged [infinite-descent]

Infinite descent, also named Fermat's descent, is a technique to prove that a diophantine equation has no solutions by construction a 'smaller' solution from a given one.

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What is the algebraic intuition behind Vieta jumping in IMO1988 Problem 6?

Problem 6 of the 1988 International Mathematical Olympiad notoriously asked: Let $a$ and $b$ be positive integers and $k=\frac{a^2+b^2}{1+ab}$. Show that if $k$ is an integer then $k$ is a perfect ...
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7answers
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Solve $x^2+2=y^3$ using infinite descent?

just so this doesn't get deleted, I want to make it clear that I already know how to solve this using the UFD $\mathbb{Z}[\sqrt{-2}]$, and am in search for the infinite descent proof that Fermat ...
12
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1answer
346 views

How to get Fermat's descent working on the conic $x^4+y^4=2z^2$?

Fermat solved the Diophantine equation $(x^2)^2 + (y^2)^2 = z^2$ using descent, the key step was using the Pythagorean triples: $x^2 = u^2 - v^2$ $y^2 = 2 u v$ $z = u^2 + v^2$ but then it is seen ...
3
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5answers
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If $a,b,c$ are integers such that $4a^3+2b^3+c^3=6abc$, is $a=b=c=0$?

If $a,b,c$ are integers such that $4a^3+2b^3+c^3=6abc$ , then is it true that $a=b=c=0$ ? I was thinking of infinite descent but can't actually proceed , please help. Thanks in advance
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0answers
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Proofs by infinite descent on the number of prime factors of an integer

Many (most?) number theory proofs employing the method of infinite descent proceed something like this: Assume a given [Diophantine] equation (e.g., $x^3+y^3=z^3$) has solutions in positive integers. ...
3
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2answers
278 views

Show that $x^3+2y^3+4z^3=0 $ has no non trivial solutions without using infinite descent

The question is to show that the equation $$x^3+2y^3+4z^3=0 $$ has no (non trivial) integer solutions. I know it can be done using infinite descent, but how do you do it using only the modular ...
1
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0answers
76 views

Proof By Descent FLT

So I've been studying the FLT and I am able to prove by descent that there are no solutions with $x,y,z$ natural numbers to $x^4+y^4=z^2$ I am trying to now prove by descent that there are no ...