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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 ...
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 ...
9
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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 ...
8
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4answers
2k views

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 ...
7
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1answer
250 views

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 ...
6
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5answers
837 views

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 ...
6
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3answers
107 views

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 ...
6
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2answers
565 views

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 ...
5
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3answers
599 views

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 ...
4
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4answers
164 views

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. ...
4
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2answers
98 views

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 ...
3
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6answers
117 views

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 ...
3
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4answers
238 views

If $\gcd(a,b)=1$, then every odd factor of $a^2 + 3b^2$ has this same form

EDIT: Please see EDIT(2) below, thanks very much. I want to prove by infinite descent that the positive divisors of integers of the form $a^2+3b^2$ have the same form. For example, $1^2+3\cdot 4^2=49=...
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 ...
3
votes
2answers
63 views

Proving $x^2-y^2 = 2xyz$ has no integer solutions greater than 0.

"Prove that there are no integers $x,y, z >0$ such that $x^2-y^2=2xyz$. Solve by infinite descent." I've been able to solve this problem myself just by playing around with the algebra, but I have ...
3
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5answers
100 views

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
3
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1answer
137 views

Lamé's proof of Fermat Last Theorem for n=3

Among the great mathematicians who struggled to try and figure out a proof of Fermat's last theorem in 19th century, Lamé was probably one of the most convinced in having succeded. I first came across ...
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3answers
245 views

Show that $x^2 + y^2 = 3$ has no rational points [duplicate]

Are there rational numbers such that $x^2 + y^2 = 3$ ? If I want to find a rational paramterizatio of $x^2 + y^2 = 1$ could start with the point $(1,0)$ and find lines $\ell$ of slope $m \in \...
2
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2answers
131 views

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 ...
1
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4answers
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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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2answers
144 views

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 ...
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 ...
1
vote
0answers
97 views

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. ...
1
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0answers
109 views

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 ...
0
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3answers
40 views

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 ...
0
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1answer
74 views

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 ...
0
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2answers
144 views

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 ...
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1answer
63 views

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\;\; {...
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3answers
112 views

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}=...