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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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Equation in rational numbers

I can't seem to find the way to solve the following equation so help would be much appreciated.. $x^2+y^2=x^3+y^3$ over $\mathbb{Q}$
user129426
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Linear recurrence solution to Diophantine equation

I have a Diophantine equation of the form: $$ax^2 + bx + c = y^2, \quad x, y \in \mathbb{Z^+}$$ Is it true that there will always be a linear recurrence formula that generates all the solutions for $x$, of the form: $$x_n = \alpha_1x_{n-1} +…
jamaicanworm
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For what values of $c$ does $8x + 5y = c$ have exactly one strictly positive integer solution?

We know that any Diophantine equation of the form $ax + by = c$ has either no solutions, or infinite solutions of the form: $$x = x_0 + n\frac{b}{(a, b)}$$ $$y = y_0 - n\frac{a}{(a, b)}$$ Where $n$ is any integer, $(x_0, y_0)$ is one solution, and…
Cisplatin
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Solutions to the equation $xk = x^k$

The equation, $xk = x^k$ (where $x$ and $k$ are both integers). Are there any solutions other than $\{ (1,1), (2,2) \}$ ?
tmj
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Help solving a partial differential equation by separation of variables

Here is thtial x} - 5y\fal u}{\partial x} - 5y\frac{\partial u}{\partial y} + 4u=0$$ I have isolated tial x} - 5y\f $$\frac{x^2}F\tial x} - 5y\fl x} - 5y\fG}{dy}$$where we assume $u$ is separable i.e. $u(x,y)=F(x)G(y)$ I am stumped on where to go…
yellowsmoke
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Diophantine Equations: Factoring

I am looking to factor a diophantine equation. However, I am unable to figure out the steps in order to reach the simplification: Find all integral solutions to the equation: $(x^2 + 1)(y^2 + 1) + 2(x − y)(1 − xy) = 4(1 + xy)$ However, I am unsure…
zenithh
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Positive Integer solutions for $(x-y)^2 = \frac{4xy}{x+y-1}$

I was solving a question which asked for the number of positive integer solutions to the problem, $(x-y)^2 = \frac{4xy}{x+y-1}$. I started by finding solutions using hit and trial. First I found $(1,3)$ (and hence $(3,1)$ as the equation is…
Param_1729
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How to solve parameters given 3 different types of information?

How Do I solve this eqns? $$x+y+z = A$$ $$xyz = B$$ $$x^2+y^2+z^2 = C$$ I have tried it in this way,,, $$yz = B/x = P$$ $$y+z = A-x = Q$$ $$y(Q-y) = P$$ $$\implies y^2-Qy+p = 0$$ I can't figure out what to do next., Solving the this quadratic…
Tamim Addari
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A diophantine equation

I want to understand why the equation $U^2-(m^2-4)V^2=-4$ (when $U,V,m$ are odd number and $m > 3$) is impossible. (This came from a post I was reading here)
A A
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Equation representing all numbers

Joe Roberts writes, in Lure of the Integers, that Matijasevič showed that "every integer has a representation in the form $a^2+b^2+c^2+c+1$". The citation he gives is Ju. V. Matijasevič, A Diophantine representation of the set of prime numbers…
Charles
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How do you find all the answers to this Diophantus equation: $462x + 273y = 63$?

How do you find all the answers to this Diophantus equation (only whole numbers): $462x + 273y = 63$ ? I started with finding the gcd using the Euclidean…
Peter
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Solving $x^2-d(y+1)^2=1$

I am reading a Wikipedia article http://en.wikipedia.org/wiki/Diophantine_set. They say the diophantine equation $x^2-d(y+1)^2=1$ has a solution in the unknows $x, y$ precisely when the parameter is $0$ or not a perfect square. $1$ is a perfect…
Dávid Natingga
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Number of integer distance grid points in a cubic grid

Given an $n$-dimensional integer lattice $\mathbb Z^n$, how does the number of grid points $N(k)$ at integer distance $\leq k$ from a given grid point scale asymptotically? I'm mostly interested in $n = 3$ (I'm a physicist :) ), but having also the…
Johannes
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Solving a first-order diophantine equation with many terms

Given a linear Diophantine equation with many terms, for example $aw + bx + cy + dz = e$ How do you work out $w, x, y, z$, without brute force? $a, b, c, d, e$ are given; they are also natural numbers. $a, b, c, d$ are co-prime. $w, x, y, z$ can be…
Nick ODell
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Integer solutions to $2^{^{11}} a + 2^{^{11}} b + ab = 1$

$2^{^{11}} a + 2^{^{11}} b + ab = 1$ By guessing that $a+b = 0$, I was able to find the solutions (a, b) = (-63, 65), (65, -63). Is there any practical way of finding other solutions to the equation?
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