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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How does this geometric application of proof by infinite descent work

In the book "The Moment of Proof" by Donald C. Benson, Chapter 4 page 51, he gave a geometric application of infinite descent. proposition 4.7: Let $S$ be a finite set of points. Suppose ...
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Show that there are infinitely many solutions $x, y, z \in\mathbb N$ of the following equation $x^2 + y^3 = z^7.$ [closed]

Show that there are infinitely many solutions $x, y, z \in\mathbb N$ of the following equation $x^2 + y^{3} = z^7$. I am thinking about using proof by infinite descent, but I am not too sure how to ...
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No positive integers $(a,b)$ satisfying $2^b -1\mid 2^a +1$ (AoPS Vol 2)

I posted this proof on AoPS but I'm having trouble understanding what's wrong with it. I was hoping this community could offer a different perspective? Suppose the claim is false. That is to say, ...
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Using Fermat's descent to find sum of squares for a prime

I am finding the sum of squares for 1973. I was given the initial expression: ${259}^2+1^2=\ 34\cdot1973$ and using descent have gotten it down to $\left(53\right)^2+\left(84\right)^2=5\cdot 1973$ I ...
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Elementary number theory contest problem: USAMO 2004/2

Here is a problem from the 2004 USA Math Olympiad. Suppose that $a_1,\dots,a_n$ are integers whose greatest common divisor is $1$. Let $S$ be a set of integers with the following properties: (i) For $...
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Prove that below equation has only one solution in integers, (0, 0, 0).

Prove that $9a^3 + 3b^3 + c^3 = 0$ has only one solution in integers, $(0, 0, 0)$.
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Can the Carmichael Function be used with Proof By Infinite Descent in the following way?

I was reading up on the Carmichael Function and I had a question about its use. Can it be used to show that if $x > 1$, then $n=1$ for: $$2^m(2^x - 1) = 3^n(3^y - 1)$$ Here's the argument that I ...
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Right angled triangles

Are right angled triangles the only triangles that can be divided into 2,4,8,16- - (2^n)smaller triangles, each being the same shape as the original? In other words is it the only one where infinite ...
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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!)...
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Let there be $2n+1$ integers such that when one of them is removed, the sum can be divided into $2$ groups each with $n$ integers.

Let $a_1,a_2,\cdots,a_{2n+1}$ be integers such that, if any one of them is removed, the remaining ones can be divided into two sets of $n$ integers with equal sum. Prove $a_1 = a_2 = \cdots = a_{2n+1}$...
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Does it follow that if gcd$(x,6)=1$ and $n \ge 4$, then $x[3^{n-1}a + \sum\limits_{i=1}^{n-1}(3^{n-1-i}2^{P_i})]$ cannot have the same form?

Let: $n\ge 4, x, a, b$ be integers gcd$(c,d)$ be the greatest common divisor of $c$ and $d$ gcd$(x,6)=1$ $p_1, p_2, \dots, p_n$ be positive integers $q_1, q_2, \dots, q_n$ be positive integers $P_i = ...
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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(\...
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Infinite descent argument in the case of $x^5+y^5=z^5$.

Euler proved that: $$2u(u^2+3v^2)=(2a(a^2+3b^2))^3$$ had no non-trivial solutions. Several other mathematicians have proven that: $$x^5+y^5=z^5$$ Using a similar argument, it is easy to see when $x$ ...
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Proof Writing For Pythagorean Triple “Reduction” Problem

You are given a primitive Pythagorean triple $(a,b,c)$ such that $a,b,c$ are integers and $a<b<c$. “Reduce” the triple via the transformation $(a,b,c)\to (a,\sqrt{b^2-a^2},b)$, rearranging if ...
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How to prove that the difference of squares of two natural numbers can't be a perfect square if their sum is a perfect square? [duplicate]

I came across the following problem and wrote up the following proof a few years ago - what do you think? It is required to demonstrate the truth of Statement S1: "If $a^2 + b^2$ is a perfect square, ...
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Structured Theorems(Programs) in Number Theory Proving Logical Conjunctions Using Infinite Descent [closed]

Any comments/answers concerning the block-quote (red) text would be appreciated. I've been examining proofs of the Fundamental Theorem of Arithmetic and constructing my own using the method of ...
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Given integer $b \gt 1$ use infinite descent to prove that every integer $m \gt 0$ has one and only one Base-$b$ representation.

My motivation for looking into this was sparked by the 'big-list' question What are the principal (different) mechanisms of infinite descent proof? Is the following proof valid? If an integer $m \gt ...
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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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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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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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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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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 $p\equiv 3 \pmod 4$? For such a prime we have lemma: $$p\mid a^2+b^2 \implies p\mid a\;\; {\rm and}\;\; p\mid b$$ Let'...
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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 \...
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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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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}=...
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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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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 ...
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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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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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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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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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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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3 votes
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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 ...
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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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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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9 answers
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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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4 votes
6 answers
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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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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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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 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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3 votes
5 answers
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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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6 votes
2 answers
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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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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=...
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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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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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13 votes
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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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