# 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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### Mordell's Theorem and Fermat's Last Theorem for $n=4$: A general method?

In his book Elliptic Curves, A. Knapp illustrates the close relationship between the proof of Mordell's Theorem and Fermat's proof (both via infinite descent) that the equation $u^4 + v^4 = w^2$ has ...
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### Prove that below equation has only one solution in integers, (0, 0, 0). [closed]

Prove that $9a^3 + 3b^3 + c^3 = 0$ has only one solution in integers, $(0, 0, 0)$.
1 vote
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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 ...
1 vote
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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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1 vote
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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 ...
1 vote
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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 ...
1 vote
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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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### 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 ...
### 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 ...