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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.

5324 questions
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How find this diophantine equation integer solution $a^3+b^3=(2ab+1)^2$

Find this following diophantine equation integer solution $$a^3+b^3=(2ab+1)^2$$ I think this equation only have two following solution $$(a,b)=(1,0),(0,1)$$ maybe this equation have no other solution? because can see wolframalpha.com I think…
math110
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integer solutions of the equation $a^3-5b^2$=2?

Has the equation $a^3-5b^2$=2 any integer solutions ? With brute force, I checked that a must be greater than 10^9.
Peter
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Cubic Diophantine

I cannot seem to find a solution to the following Diophantine Equation: $x^2-y^3=2$, where $x,y \in \mathbb{Z}.$ I thought that I could maybe reduce it to a simpler equation , maybe check for the extension $\mathbb{Q}\left[\sqrt{2}\right],$ but…
Peter Allen
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What is the simplest Diophantine equation equivalent to N is not zero?

Given an integer N. What is the simplest Diophantine equation equivalent to the statement that integer $N\neq0$ ? I can do one in 5 variables. Using the fact that any integer can be written as the sum of four…
zooby
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On integer solutions to $a^3+b^3+c^3+d^3=3$

I was challenged to prove that there are infinitely many solutions to the equation$$a^3+b^3+c^3+d^3=3\ \ \text{ with }(a,b,c,d)\in\mathbb Z^4$$ That was easy: elementary algebra is enough to prove that $\forall z\in \mathbb Z$ then…
fgrieu
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Solve the diophantine equation $3\,{a}^{3}b-13\,{b}^{3}-26\,a-24\,b=0.$

Solve the diophantine equation $3\,{a}^{3}b-13\,{b}^{3}-26\,a-24\,b=0.$ I have found two obvious solutions $a=b=0$ and $a=b=5.$ Are there another solutions?
Leox
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Diophantine equation $a^3+a=b^2+1$

I have this Diophantine: $$ a^3+a=b^2+1 $$ I found $a=2$, $b=3$ works. Also $a=13$ , $b=47$ works. How can I find all the integer solutions?
Problemsolved
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Find all $x,y$ so that $\dfrac{x+y+2}{xy-1}$ is an integer.

I am trying to find the integers $x,y$ so that $\dfrac{x+y+2}{xy-1}$ is an integer. What I have done: I suppose there exists $t$ such that $$t=\dfrac{x+y+2}{xy-1}$$ where $xy\neq 1$ then consider the following scenarios: $$x=y$$ $$x>y>0$$ …
user97615
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the system of diophantine equations: $x+y=a^3$; $xy=\dfrac{a^6-b^3}{3}$ has only trivial solutions.

Without using Fermat's Last Theorem, how can one prove that the following system of diophantine equations has only trivial solutions: $$x+y=a^3$$ $$xy=\dfrac{a^6-b^3}{3}$$ We suppose of course that $\gcd(x,y)=\gcd(a,x)=\gcd(a,y)=\gcd(a,b)=1$
user97615
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Does $4x^2+1=5^y$ have a solution in integers with $y>1$?

Consider the following equation : $4x^2+1=5^y~$ with $y>1$ Has this equation solutions in integers ? I wrote small Maple program in order to find solutions but couldn't find anyone . for x from 1 to 6000 do for y from 2 to 2000 do if 4*x^2+1 = 5^y…
Pedja
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Special kind of a linear Linear Diophantine equation

Could any one help me to point out some literature/ papers which solves a homogenous linear Diophantine equation (one equation) of the form $a_1 \times x_1+a_2 \times x_2 + a_3 \times x_3+....+a_n \times x_n=0$, where $a_1,a_2,...,a_n$ are positive…
Raj
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Rational points on a surface

I am trying to find rational points on this surface $$ \left( \left( 1-x \right) ^{2}+{y}^{2} \right) \left( \left( 1+x \right) ^{2}+{y}^{2} \right) ={z}^{2}$$ I am actually only interested in points where both $x$ and $y$ are in the range of…
Mark
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Integer solutions of $x^y-z^3=2$

Is it an open question to solve $x^y-z^3=2$ in integers (both positive, zero and negative)? If not, what kind of methods the solution requires?
guest
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Find all pairs of positive integers $(x,y)$ such that $x^2-y^2=a^3$ and $x^3-y^3=b^2$ for integral $a, b$?

How to find ALL pairs of positive integers $(x,y)$ such that the difference in their squares is a perfect cube and the difference in their cubes is a perfect square. i.e., Positive integers $(x,y)$ such that $x^2-y^2=a^3$ and $x^3-y^3=b^2$ for…
ScottT
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Show that $x(x+1) = y^4+y^3+ay^2+by+c$ has a finite number of positive integral solutions.

More precisely, If $a$, $b$, and $c$ are integers, show that there are only a finite number of positive integers $x$ and $y$ such that $x(x+1) = y^4+y^3+ay^2+by+c$. I have a solution, which I will show in two days if no better one is found.
marty cohen
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