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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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Finding Solutions to a System of Diophantine Equations

I'm trying to find triplets of integer $(x, y)$ pairs - $(x_1, y_1), (x_2, y_2), (x_3, y_3)$ - that satisfy the following equations: $$ {x_1}^2 + {y_1}^2 = {x_2}^2 + {y_2}^2 = {x_3}^2 + {y_3}^2 \\ x_1 + x_2 + x_3 = 5 \\ y_1 + y_2 + y_3 = 0 \\ (x_1,…
GodOrGovern
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Number Of Solutions for the equation: $a^2 - b^2 = X$

Suppose there is an equation $a^2 - b^2 = X$ how many pairs of $(a,b)$ exist to find $X$, when $a,b$ and $X$ are positive integers. It would be great if you could direct to proof although not necessary. An example: $a = 13, b = 11 X = 48$. 13^2 -…
mstar
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How to find all solutions of the optic equation $\frac{1}a+\frac{1}b = \frac{1}c$

Optic equation says for $${\frac {1}{a}}+{\frac {1}{b}}={\frac {1}{c}}$$ All solutions in integers $a, b, c$ are given in terms of positive integer parameters $m, n, k$ by $$a=km(m+n) , \quad b=kn(m+n), \quad c=kmn$$ where $m$ and $n$ are…
simonzack
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Finding all integer solutions of $\frac{1}{x} + \frac{1}{y} - \frac{1}{z} = \frac12$

If we have integers $x,y,z$ such that $x,y,z \ge 3,$ find all solutions to $$\frac{1}{x} + \frac{1}{y} - \frac{1}{z} = \frac12.$$ I was thinking of first expanding out this, and then simplifying from there, but the equation got very messy. Is there…
questionasker
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Second Degree Diophantine Equation, When exists solutions?

In second degree diophantine equations, ax2+bxy+cy2+dx+ey+f=0, Is there a theorem or criterion that allows us to decide whether or not any second degree diophantine equation has solutions without resolution? Thank you, Javier
Javier Álvarez
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Values ​of the Diophantine equation in an interval

Set $a$ and $b$ so that for each $k$, with $ a < k \leq a + b $, there exist $ x $ and $ y $ with $ x \geq 2 $ and $ y \geq 1 $ such that $ k = 6xy + x -y $ Is it possible to proof that, set $a$, there is a value of $b$ for which this cannot…
user140242
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Finding integer solutions to $x^2-yx+ay+b = 0$

What's the best technique to find integer solutions for $x$ & $y$ to equations in the form $x^2-yx+ay+b = 0$ where $\space(a,b)\space$ are known integers (or even just to find the number of solutions that exist)?
user8333
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Why $x^4+x=y^2+y$ has only a finite number of integer solutions?

In an attempt to understand and solve this problem, I tried to play with some small finite example, one of which is $$x^4+x=y^2+y$$ Playing with Wolfram-Alpha indicates indeed equations of similar form, where a generic parametric solution (i.e.…
cr001
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Find out, with no resolution, whether or not the second degree diophantine equations with 2 unknowns have solutions

I want to find out the existence of the solutions in diophantine equations of the style: $$-259y ^2+ 2400yx + 1817y + 2122x = $$ $$1057364602723981500371957207036553770637547302056514367123547565680640946707606178926389130616$$ The point is that…
Javier Álvarez
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Solving second-degree diophantine equation

I'm interested in the following Diophantine equation: $y^2 - y - x^2 + x = 2xy$ I've managed to find some solutions like $(6, 15)$ and $(35, 85)$, but I need some general method of solving this type of equations. As I understood, it's a hyperbola,…
dajfufjrgfkdlw dgdvrbkwkslchrj
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proof that no non negative odd integer solution of this type of diophantine equation exists given even non negative integer solution

Given the equation $$ (2^n-1)(1+x+x^2+...x^{2k})(1+y+y^2+...+y^{2j})=2^nx^{2k}y^{2j}-1 $$ a non negative integer solution exists of the form $(2^n,2^{2kn+n}$). How can I (try to) show that (1) this is the only non negative solution and/or (2) that…
thestar
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Exponential Diophantine representation of the factorial

How can I use the identity $$\sum_{n=0}^\infty \frac{\tau^n}{n!} = \lim_{y \to \infty} \left( 1 + \frac{\tau}{y} \right)^y$$ to find an exponential Diophantine representation of the factorial? I was hoping to use a very large value of $y$ (so that…
quanta
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Suppose $x$, $y$, and $z$ are integers that satisfy the system equations

Suppose $x$, $y$, and $z$ are integers that satisfy the system equations : $x^2y$+ $y^2z$ + $z^2x$ $= 2186$ $y^2x$+ $z^2y$ + $x^2z$ $= 2188$ What is $x^2+y^2+z^2$ ?
mortimer-shaun
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Integer solutions to $x^3 + 3x^2y - 3xy^2 - 3y^3 = 1$

Prove that the only integer solutions to $x^3 + 3x^2y - 3xy^2 - 3y^3 = 1$ are $(1,0)$ and $(-2,3)$. My only idea for now is to try to represent this in the form $A^3 + B^3 = C^3$ and apply Fermat's Last Theorem. However, I cannot find suitable…
DesmondMiles
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