# 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.

29 questions
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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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### 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 ...
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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 ...
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### What is an example of a proof by minimal counterexample?

I was reading about proof by infinite descent, and proof by minimal counterexample. My understanding of it is that we assume the existance of some smallest counterexample $A$ that disproves some ...
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### Geometric proof of number theory result: If the geometric and quadratic means of integers $a$ and $b$ are themselves integers, then $a=b$

We start with the following problem: Let $a$ and $b$ be positive integers such that their geometric and quadratic means are integers. Show that $a=b$. One possible approach is to write down the ...
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### Proving that there does not exist an infinite descending sequence of naturals using minimal counterexample

This might be a long-winded way of proving something so obvious, but I want to have it checked if it holds up. Claim: There does not exist a sequence $\{{a}_{n \in \mathbb{N}}\}$ of natural numbers ...
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### Fermat's Challenge of composition of numbers

In his letter to Carcavi (August 1659), Fermat mentions the following challenge There is no number, one less than a multiple of $3$, composed of a square and the triple of another square. He ...
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### Why is this proof that $\sqrt{2}$ is irrational titled as “Proof by infinite descent”?

I am reading this wikipedea article on the proof of irrationality of $\sqrt{2}$. It uses the principle of infinite descent. I understand it as: We assume $\sqrt{2}=\dfrac pq$, where $p$ and $q$ are ...
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### Fermat's infinite descent for finding the squares that sum to a prime

Fermat's theorem on sum of two squares states that an odd prime $p = x^2 + y^2 \iff p \equiv 1 \pmod 4$ Applying the descent procedure I can get to $a^2 + b^2 = pc$ where $c \in \mathbb{Z} \gt 1$ I ...
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### Find all solutions to the equation $x^2 + y^2 + xy = (xy)^2$

Can anyone help me find the number of solutions to the equation: $$x^2 + y^2 + xy = (xy)^2$$ Let me give a brief account of what I've tried to proceed with: Case 1: One of $x$ and $y$ is odd. ...
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### Solving the homogeneous Diophantine equation $x^3 + 2y^3 = 7z^3$ for $x,y,z \in \mathbb{Q}$

I want to solve the homogeneous Diophantine equation $x^3 + 2y^3 = 7z^3$ for $x,y,z \in \mathbb{Q}$. First note that $(x,y,z) = (0,0,0)$ is a solution. For further solutions it suffices to search ...
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### find all natural solution that satisfy $x^2+y^2 = 3z^2$

I need to find all natural $x,y,z$ that satisfy the following $x^2+y^2 = 3z^2$ $(0,0,0)$ is an answer of course. What I tried: I tried solving with congruences. I know that every square number is ...
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### Show that $\sqrt{2} \notin \mathbb{Q}(i)$ using infinite descent

Standard exercise is to show $\sqrt{2} \notin \mathbb{Q}$ (e.g. Wikipedia). There are examples on Math.SE such as [1, 2, 3, 4]. If we adjoin an element to $\mathbb{Q}$ does the same proof by ...
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### What are the principal (different) mechanisms of infinite descent proof?

I’m interested in building a list (including, where possible, links to proofs/papers/examples) which presents all known mechanisms of infinite descent (ID). I think this list would best be presented ...
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### Show $x^2$ + $y^2$ + $z^2$ = $2xyz$ has no non-trivial solutions using infinite descent.

I want to show that the equation $x^2$ + $y^2$ + $z^2$ = $2xyz$ has no non-trivial solutions using infinite descent. This question has been solved with infinite descent by showing that $x,y,z$ are ...
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### 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 ...
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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. ...
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### Can Fermat's descent for $x^4+y^4=z^2$ be interpreted on a conic?

Fermat proved the Diophantine equation $$(x^2)^2 + (y^2)^2 = z^2$$ has only solutions $(0,0,0)$, $(0,\pm 1,\pm 1)$ and $(\pm 1,0,\pm 1)$ using "infinite descent". The conic $C : X^2 + Y^2 - 1$ has a ...
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### Proof of irrationality

I am trying to prove that $\,\sqrt[3]{2} \cdot (\sqrt[3]{2} + q)\,$ is irrational for all rational choice of $q$. However, I am completely stuck. I tried cubing it and trying the trick that you use ...
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### Combinatorics problems that can be solved via infinite descent

I'm looking for high school problems that can be solved with the method of infinite descent. Usually, those problems are from number theory, but I would be very happy if someone could provide a ...
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### How to understand the two definitions of finite descent are logical equivalent?

I find two definitions of finite descent principle. The first is in the book "A beginner's guide to mathematical logic", Ch4, P40: Suppose a property P is such that for any natural number n, if P ...
### How to solve $x^2+y^2=pz^2$ in $x,y,z\in\mathbb{N}$ if $p$ is such a prime that $p\equiv 3 \pmod 4$?
How to solve $x^2+y^2=pz^2$ in $x,y,z\in\mathbb{N}$ if $p$ is such a prime that $p\equiv 3 \pmod 4$? Is the following proof OK? For such a prime we have lemma: p\mid a^2+b^2 \implies p\mid a\;\; {...
### Find all $x$ such that $4^{27}+4^{1000}+4^{x}$ is a perfect square.
Let,$4^{27}+4^{1000}+4^{x}=n^2$ $4^{2}(4^{25}+4^{998}+4^{x-2})=n^2$ LHS is multiple of 4.$n$ is also multiple of 4 Let $n=4a _1$. $4^{2}(4^{25}+4^{998}+4^{x-2})=16.a_1^2$ \$4^{25}+4^{998}+4^{x-2}=...