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Questions tagged [diophantine-equations]

Use for questions about finding integer or rational solutions to polynomial equations.

Use this tag for questions about finding integer, or perhaps rational, solutions to polynomial equations.

Diophantine equations are named after Diophantus of Alexandria, a third century Greek mathematician.

An example of a Diophantine equation is to find all quadruples of integers $(w,x,y,z)$ such that $$w^2+x^2=3(y^2+z^2).$$

Solving Diophantine equations often involves other areas of mathematics such as congruences, linear algebra, inequalities, forms (e.g., binary quadratic forms), and elliptic curves. Special solution methods include comparing divisors, considering orders of magnitude, Fermat's method of descent, and finding intersections of curves with lines of rational slope through a known rational point.

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Second degree Diophantine equations

I found a question whether there are general methods to solve second degree Diophantine equations. I was unable to find an answer so is this known? In particular, the original writer wants to know whether one can find all integers satisfying $x^2 +…
Jaakko Seppälä
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diophantine equations $x^3-2y^3=1$

I'm not familiar with diophantine equations. At most my approaches doesn't give results. I need to solve the following equation $$x^3-2y^3=1$$ Where $x,y,z\in\mathbb{Z}$ I know $x=-1,y=-1.x=1,y=0,$ Aware of any other integer solutions. Prove
tianzhidaosunyouyu
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Pythagorean quadruples

Another Project Euler problem has me checking the internet again. Among other conditions, four of my variables satisfy: $$a^2+b^2+c^2=d^2 .$$ According to Wikipedia, this is known as a Pythagorean Quadruple. It goes on to say all quadruples can be…
Mike
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Prove that the equation $x^2-y^{10}+z^5=6$ has no integer solutions

I have a nice diophantine equation which I tried to solve since march but no solution. Tried modulo 11, tried to write it in some way to figure out a solution... I posted this a few months ago, but it was removed. The problem: Prove that the…
user85046
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Math contest integer triplet problem: $x + y + z + xy + yz + xz = xyz − 1$

Can any one help me with this? Determine all integer triples $(x,y,z)$ such that $1 ≤ x ≤ y ≤ z$ and $x + y + z + xy + yz + xz = xyz − 1$. I thought of Vieta's formula but don't let me lead you into a dead end.
user87611
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Quadratic Diophantine Equations

I note that the Diophantine equation, $x^2 + y^2 = z^2$, with $x, y, z \in \mathbb{N}$, has infinitely many solutions. Indeed, $(x, y, z) = (3,4,5)$ provides a solution, and for any $k \in \mathbb{N}$ : $(kx, ky, kz ) = (3k, 4k, 5k)$ provides a…
Harry Williams
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Quintic diophantine equation

How can I find non trivial primitive integer solutions, to the Diophantine equation $$a^4+b^4+c^4=d^5$$ Can anyone find me solutions to this equation? Or if possible a parametric equation that generates solutions? I would appreciate any help Ive…
Ethan Splaver
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The diophantine equation $a^{a+2b}=b^{b+2a}$

While working on the recent inequality question $\qquad$Find the $least$ number $N$ such that $N=a^{a+2b} = b^{b+2a}, a \neq b$. posted by Nirbhay, I decided to fool with the associated diophantine equation, and I did manage to solve it. Here is…
quasi
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$x^4+ry^4=z^4$: find primitive Diophantine solutions with prime $r$.

$x^4+ry^4=z^4$: find primitive Diophantine solutions with prime $r$. Background. I tried to find an answer to this question Diophantine equation $x^4+5y^4=z^4$ that gives a prime primitive solution $(x,y,z,r)=(1,2,3,5)$, but I was unable to make…
Old Peter
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Let $a,b,c$ positive integers such that $\left(1+\frac{1}{a}\right)\left(1+\frac{1}{b}\right)\left(1+\frac{1}{c}\right) = 3$. Find those triples.

Full question: Let $a$,$b$,$c$ be three positive integers such that $\left(1+\frac{1}{a}\right)\left(1+\frac{1}{b}\right)\left(1+\frac{1}{c}\right) = 3$. Find those triples. This is actually a national competition question in Vietnam (Violympic),…
William Phoenix
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How prove this diophantine equation $(x^2-y)(y^2-x)=(x+y)^2$ have only three integer solution?

HAPPY NEW YEAR To Everyone! (Now Beijing time 00:00 (2015)) Let $x,y$ are integer numbers,and such $xy\neq 0$, Find this diophantine equation all solution $$(x^2-y)(y^2-x)=(x+y)^2$$ I use Wolf found this equation only have two nonzero integer…
math110
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Solve equation $ 1+2^x=3^y$

Find integers $x$ and $y$ such that$$ 1+2^x=3^y.$$ It is obvious that $x = y = 1$ and $x = 3, y = 2$ are solutions. I think others are not. How to show that?
medicu
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The Diophantine equation $x_1^6+x_2^6+x_3^6=z^2$ where exactly one $(x_i)\equiv 0{\pmod 7}$.

Apart from the trivial $a^6+0^6+0^6=(a^3)^2$, primitive solutions seem difficult to find. That’s primitive as in not of the form $(kx_1,kx_2,kx_3,k^3z)$ where $(x_1,x_2,x_3,z)$ is a smaller solution. As far as I can see, either one or two of…
Old Peter
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Diophantine equation question concerning squares

How many squares there are of the form $d^2={b}^{2}-4ac$ if $a ,b ,c ,d$ are natural numbers between $1$ and $n$ such that $0\le{b}^{2}-4ac$? My first approach is for $d$, that must be an square of the form ${(b-2k)}^{2}$ for some natural $k$, but…
dot dot
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All integer solutions for $x^4-y^4=15$

I'm trying to find all the integer solutions for $x^4-y^4=15$. I know that the options are $x^2-y^2=5, x^2+y^2=3$, or $x^2-y^2=1, x^2+y^2=15$, or $x^2-y^2=15, x^2+y^2=1$, and the last one $x^2-y^2=3, x^2+y^2=5$. Only the last one is valid.…
Jozef
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