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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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$abx^2+bcy^2+acz^2=(xyz)^2+2abc$ has no integral solutions if $a,b,c,x,y,z >1$?

let $a,b,c,x,y,z$ be all pairwise coprime integers . Show that: $$abx^2+bcy^2+acz^2=(xyz)^2+2abc$$ has no integral solutions if $a,b,c,x,y,z >1$. I tried to confirm the results in wolfram but I am totally clueless as to how to prove this. Any hints?
user97615
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Show that a cubic Diophantine equation has no solutions

Show that the Diophantine equation $x^3+117y^3 = 5$ has no solutions. I tried using like an odd and even argument for $x$ but it doesn't seem to work because it doesn't matter if $x$ is odd or even. Any help would be great.
user19405892
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Proving a power Diophantine equation has no solutions

Show that the Diophantine equation $2^n-x^m = 1$ with $x,n,m > 1$ has no solutions. How do I show that $x^m$ can never be $1$ less than a power of $2$? I tried factoring it but it doesn't seem like I can.
user19405892
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solution of Diofantine equation

Find number of non-negative integer solutions of equation: $x+y^2+z=x^2z+y$. I have tried to rearrange it like $x^2z-y^2=x-y+z$, but I don't have an idea what to do in the next step. Thanks for any advice.
Pls2
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$x^2+(a^2+b^2)x+ab=0$ where $a,b,x\in \mathbb Z$

Trivial solutions include $a=0, x=-b^2$ and $b=0,x=-a^2$. Then $a=b=1,x=-1$ or $a=b=-1, x=-1$. Is there any other? How could you prove there aren't? $(a^2+b^2)^2-4ab=(a^2+b^2)+2ab(ab-1)$ needs to be a square. I tried to look at it $\mod 4$ but it…
chx
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Positive integer solutions to $5a^2 + 8a + 4 = 4b^2$

I'm convinced $5a^2 + 8a + 4 = 4b^2$ can be somehow turned into Pell's equation. My first steps: Rewrite as $5(a + 4/5)^2 + 4/5 = 4b^2$. Rewrite $B = 2b, A = 5a + 4$ to get $A^2 + 4 = 5B^2$. I'm almost there, I just need to know how to solve this…
qwr
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Determining the highest value of c in a linear diophantine equation for which there exists three positive solutions

Given $$5x+7y=c$$What is the largest value of c for which there exists exactly 3 solutions (x,y)? I've tried researching how to find the exact number of positive integer solutions for linear diophantine equations but didn't find it much help. How…
Jonathan
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Diophantine equations - how to go about it?

What are the methods of solving such tasks? For example: $$ \begin{cases} x^2+y-z=100\\ y^2+x-z=124 \end{cases} $$ What first?
JWa
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Number of solutions of a diophantine equation using the rounding function

With the equation defined as: $$\left\lfloor \frac {(x-1)(a_n-a_1)}{n-1}\right\rceil+\left\lfloor \frac {(y-1)(a_n-a_1)}{n-1}\right\rceil=N$$ How many integer values can $x$ and $y$ take as a $f(a_n,a_1,n,N)$ if $1\le x,y \le n$? I have tried…
GuPe
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Solve equation $m^2=n^5-4$ in $\mathbb{N}$

I have an exercise to solve an equation like in the title. My goal until now is that both $m$ and $n$ are odd, but then I can not continue. Can you help me? Thanks a lot!
mapping
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Prove that for any integers $x,y,z$ there exist $a,b,c$ such that $ax+by+cz=0$

It is rather obvious that for any 3 coprime integers $x,y,z$ there exist 3 non-zero integers $a,b,c$ such that: $$ax+by+cz=0$$ Any simple argument to prove it?
user97615
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Diophantine sets

I am reading the following part: Diophantine sets A subset of a power $\mathbb{Z}^n$ of the set $\mathbb{Z}$ of integers is diophantine if it can be written as $$\{\overline{x} \in \mathbb{Z}^n : (\exists \overline{y} \in \mathbb{Z}^m )…
Mary Star
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Write $x(a^2+b^2)+(2ab)y$ as a product of factors.

Let $a,b,c,x,y \in\mathbb{Z}>1$ and $\gcd(a,b)=\gcd(x,y)=\gcd(a,b,x,y)=1$, Can $$x(a^2+b^2)+(2ab)y$$be factorized ?
user97615
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How do I approach these Diophantine equations?

I'm a high school student, so I think my question will be an easy one. I would like to know if there is an easy way of approaching these Diophantine…
RoDeLoPo
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Diophantine solutions to a large geometric figure

I have a related question to one I've read today, see: Integer solutions to $2x^2+5x+y^2=19$ The integers solution are part of an ellipse, with an obvious finite number of $x$. What I would like to know is, when we have a large ellipse so that…
user244037
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