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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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Solving for natural numbers: $x^{y}=y^{x-y}$

Solve for natural numbers: $$x^{y}=y^{x-y}$$ What I have tried: \begin{equation} x>y \geqslant 2 \end{equation} \begin{equation} \left(\frac{x}{y}\right)^{y}=y^{x-2 y} \end{equation} \begin{equation} \frac{x}{y}=k, k \in…
user910220
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Is it possible to solve for n unknowns from one equation?

We can solve for 2 unknowns from one equation: Is it possible to solve for two unknowns from one equation? But is it possible to solve for n unknowns from one equation?
sbh
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Are there integers nonzero integers $a,b,c,d,x,y,m,n,p$ such that $(a^2-mb^2)(c^2-nd^2)=x^2-py^2$? ($m,n,p)$ are square free non equal integers.

We are all familiar with Fibonacci-Brahmagupta's identity: $$(a^2-mb^2)(c^2-md^2)=(ac+ mbd)^2-m(ad+bc)^2$$ I am trying to find whether there is a similar identity: $$(a^2-mb^2)(c^2-nd^2)=x^2-py^2$$ where $p,m,n$ are not all equal. If this problem…
NumThcurious
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Solve for $x$ and $y$ in integers.

In the equation $x^3+y^3=(x+y)^2$, solve for $x$ and $y$ in integers. So far, I factorized and cancelled out a copy of $(x+y)$ on both sides, leaving me with $x+y=x^2-xy+y^2$. Then, I added $xy+1$ on both sides, and got $(x+1)(y+1)=x^2+y^2+1$. I…
xrider1000
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Solving a Diophantine equation involving fourth powers and squares

I am trying to show that $x^4 - 3y^4=z^2$ has no solutions over the positive integers. I tried working on $\mathbb{Z}[\sqrt{3}]$ but couldn't go far. Factorizing didn't do much either.
rmdnusr
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Diophantine Equation with 3 Variables example

$15x+72y+21z=9 (1)$ $15+72y = gcd(15,72).r1$ (1) can be written as : 15x+72y=gcd(15,72).r1 (2) gcd(15,72).r1 + 21z = 9 (3) $15x+72y=gcd(15,72).r1$ Calculating $GCD(15,72)$ gives: $72 = 4*15 + 12$ $15 = 1*12 + 3$ $12 = 4*3 + 0$ from (2) and…
gav
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Do Diophantine solution bounds themselves require a secondary solving operation?

With a Diophantine equation like $$889a + 90b - 90c - 2000d = 0$$ Sympy provides the following solution: $$a = 10 t_0$$ $$b = t_1$$ $$c = 79121 t_0 + 1001 t_1 + 200 t_2$$ $$d = -3556 t_0 - 45 t_1 - 9 t_2$$ where the only known constraints on all $t$…
Reinderien
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Solve a linear equation with 3 unknowns and 1 parameter

$$(a+1)x+y+3z=1$$$$8x+2y+(a+3)z=2$$$$3x+y+2z=-1$$ This question can be calculated with Gauss-elimination and I want to take away the y by taking $-2$ from the middle and $-1$ from the top and keep the third one as it is, my question is how do I…
user1838781
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Closed form of solutions of $xy-zv=±1$

Let us consider a real dynamical system $s′=h(s)$. In order to study the stability of the central manifold, we reformulate the problem as follows: I am asking if this equation $$xy-zv=±1$$ has integer solutions $x,y,z,v$. We can find some special…
Safwane
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About the solutions of the diophantine equation $ x×(2a-x)=b$

Let us consider a real dynamical system $s′=g(s)$. In order to study the stability of the central manifold, we reformulate the problem as follows: For given fixed positive integers $a,b$, I am asking if this equation $$ x(2a-x)=b$$ has positive…
Safwane
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Finding all nonzero integers $k$ so that $\sqrt{k^2 - pk}$ is a positive integer

Let $p$ be a prime. Find all nonzero integers $k$ such that $\sqrt{k^2 − pk}$ is a positive integer. I first let $k^2 - pk = x^2,$ where $x$ is a positive integer. However, I got stuck from here as I wasn't quite sure what to do with the condition…
questionasker
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What is the genral solution to $x+3y=8+6k$ in this diophantine equation?

I am doing maths for fun and stumbled upon this amazing worksheet. The second last question is a Diophantine equation with three variables and the solution ends with the general solution to the equation. I understand everything except how they go…
linker
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The Diophantine Equation $x^2 - y^2 - z^2 = d$

Is there any literature / reference available for solving the Diophantine Equation $x^2 - y^2 - z^2 = d$? I looked for it and found the following that were close, but not exactly the equation given here: This MathOverflow article: The Diophantine…
vvg
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Multi variable Diophantine equation

Consider this equation $$z^3\sqrt{x^2+y^2-z^2}=w(x^2+y^2)$$ where $w$, $z$ are positive integers, $x$, $y$ are any integers. Squaring both sides we get the following Diophantine equation $$z^6(x^2+y^2-z^2)=w^2(x^2+y^2)^2$$ Some solutions in…
user125368
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The general Ramanujan–Nagell equation $x^2+D=y^n$

It is known that the general Ramanujan–Nagell equation $x^2+D=y^n$ Have no solution for $ D<100 $ see the table in the paper CLASSICAL AND MODULAR APPROACHES TO EXPONENTIAL DIOPHANTINE EQUATIONS II. THE LEBESGUE–NAGELL EQUATION by YANN BUGEAUD,…
Aster Phoenix
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