Mathematics Stack Exchange
Mathematics Stack Exchange

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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Quintic diophantine equation $x^5+y^5=7z^5$

Are there any non-zero integer solutions to the equation $x^5+y^5=7z^5$? I am unsure how to approach this.
pre-kidney
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Find all integer solutions of $1+x+x^2+x^3=y^2$

I need some help on solving this problem: Find all integer solutions for this following equation: $1+x+x^2+x^3=y^2$ My attempt: Clearly $y^2 = (1+x)(1+x^2)$, assuming the GCD[$(1+x), (1+x^2)] = d$, then if $d>1$, $d$ has to be power of 2. This…
Nguyễn Hoài Nam
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Sum of two odd cubes plus four times a cube is zero

Let $x,y$ be odd integers and let $z$ be an integer. The question is to find all solutions to the equation, $$x^3+y^3+4z^3=0$$ Of course we have the trivial solution $(x,-x,0)$. Are there any others? By considering the equation modulo $4$ we see…
John Donne
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Diophantine Equation: $xy+ax+by+c=0$

How to find integer solutions $x,y$ of $xy+ax+by+c=0$ for given $a,b,c \in \mathbb{Z}$? Is there somewhere a treatise on this kind of equations?
siddhadev
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A Diophantine equation: $ -2a^3 + b^3 + c^3 = 36650$, where $a, b, c > 0$ are all positive integers, and $a, b, c \notin \{2, 9, 15, 16, 33, 34\}$

For a problem that I'm working on, I need to solve this Diophantine equation:- $ -2a^3 + b^3 + c^3 = 36650$, where $a, b, c > 0$ are all DISTINCT positive integers, and $a, b, c \notin$ { 2, 9, 15, 16, 33, 34} How does one go about solving this?…
Train Heartnet
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Find all $x,y$ such that $x^3+5y^3=(a^3+5b^3)^3$.

Let $a,b$ be coprime integers. I am trying to find all integers $x,y$ such that: $$x^3+5y^3=(a^3+5b^3)^3$$ What I have tried: $$5y^3=(a^3+5b^3-x)[(a^3+5b^3)^2+(a^2+5b^3)x+x^2 ]$$ There are 2 Cases depending which factor of $5y^3$ is…
NumThcurious
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Prove that $2c^3+1$ is never a perfect cube for $c \in \mathbb{N}^*$

I am working on proving that the equation: \begin{align} a^2-b^3 = 1 \end{align} where $a$ and $b$ are positive integers has only one solution $(a,b) = (3,2)$. The equation can be rewritten: \begin{align} (a-1)(a+1) = b^3. \end{align} When $b$ is…
Alfred F.
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Strange Cubic Diophantine Equations

Does anyone have any ideas towards solving these four equations one at a time? $a^3 - 3a^2b + b^3 = \pm 1$ $a^3 + 3a^2b - 6 ab^2 + b^3 = \pm 1$ I am guessing that the $1$ might mean we can use units in some algebraic number field to solve these…
quanta
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Diophantine equation to characterize natural numbers

Let's consider $$\Bbb N=\{0,1,2,3,\ldots\},$$ and, for each $k\in\{1,2,3,\ldots,\}$, let $$o_k=2k-1$$ be the sequence of odd natural numbers. Given that for each $m\in\Bbb N$, if $a$ is odd, the number $$m(m+a)$$ is even, it is easy to see…
leo
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$2^{x}+ 5^{y}= 7^{z}$

Help me solve this integers exponents equation: $$2^{x}+ 5^{y}= 7^{z}$$ We have: $z= \frac{i(2\,\pi\,n-i\,\log(2^{x}+ 5^{y}))}{\log(7)}$ I ask because I read that a slightly complication of the equation occuring in Fermat's last theorem can lead to…
user548665
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Diophantine equation-positive solutions

Consider the equation $(x+1)(y+1)(z+1)=3xyz$. I want to find all positive integer solutions such that $x\le y\le z$ to this equation using only pen and paper and mathematical techniques. How can I do this rigorously(ie also showing that the…
user48756
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Trying to solve $x^2 – 11x = y^2$ knowing that $x$ and $y$ are integers.

Once upon a time, I knew how to solve this equation. Yesterday, I tried (and failed) to solve it again. I remember the solutions, but I can't figure out how to find them…
Jean
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Find the integer $x$ such $x^6+x^5+x^4+x^3+x^2+x+1=y^3$

Find the equation integer solution $$\color{red}{y^3=x^6+x^5+x^4+x^3+x^2+x+1}$$ It is obvious $x=0,y=1$ or $x=-1,y=1$ are solutions. How to find all solutions?
math110
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Finding solutions to $(4x^2+1)(4y^2+1) = (4z^2+1)$

Consider the following equation with integral, nonzero $x,y,z$ $$(4x^2+1)(4y^2+1) = (4z^2+1)$$ What are some general strategies to find solutions to this Diophantine? If it helps, this can also be rewritten as $z^2 = x^2(4y^2+1) + y^2$ I've already…
MyNameIsKhan
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Is there a systematic way to solve in $\bf Z$: $x_1^2+x_2^3+...+x_{n}^{n+1}=z^{n+2}$ for all $n$?

Is there a systematic way to solve in $\bf Z$ $$x_1^2+x_2^3+...+x_{n}^{n+1}=z^{n+2}$$ For all $n$? It's evident that $\vec 0$ is a solution for all $n$. But finding more solutions becomes harder even for small $n$: When $n=2$, $$ x^2+y^3=z^4 $$ I'm…
YoTengoUnLCD
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