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.

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 square. The usual way to show this involves a…
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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 that $x,v,u$ is another Pythagorean triple, $x =…
quanta
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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 claimed to have found. I've alaways been fascinated by…
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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 proposition $P$, then go onto show that there is 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 corresponding diophantine equations and to do…
Phira
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Does the equation $y^2=3x^4-3x^2+1$ have an elementary solution?

In this answer, I give an elementary solution of the Diophantine equation $$y^2=3x^4+3x^2+1.$$ In this post, the related equation $$y^2=3x^4-3x^2+1 \tag{$\star$}$$ is solved (in two different answers!) using elliptic curve theory. Is there an…
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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 such that $a_{n+1} < a_{n}$ for every $n$. Proof:…
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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 says that he has solved it using infinite descent.…
Henry
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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 some positive integers. We have $2q^2 = p^2…
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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 want $c = 1$, so how do I proceed from here? How do…
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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, $1^2+3\cdot 4^2=49=7^2$, and indeed $7,49$ can both be written in…
Is Ne
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Collatz Conjecture: Is there a straightforward argument showing that there are no nontrivial 2 "step" repeats (where each "step" is an odd number)

Let: $C(x) = \dfrac{3x+1}{2^w}$ where $w$ is the highest power of $2$ that divides $3x+1$ $C_n(x) = C_1(C_2(\dots(C_n(x)\dots)) = \dfrac{3^n x + 3^{n-1} + \sum\limits_{i=1}^{n-1}3^{n-1-i}2^{\left(\sum\limits_{k=1}^i…
Larry Freeman
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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 this topic when reading the book…
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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 for solutions in $\mathbb{Z}^3$, because if we have…
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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. This results into a contradiction where the parity…
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