# Questions tagged [diophantine-equations]

Use for questions about finding integer or rational solutions to polynomial equations.

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### Rational solutions to $\{x^2+y^2=2,x^2+z^2=4\}$

I am working through some examples from Silverman's The Arithmetic of Elliptic Curves and I've run into the following system of equations, $$\left\{x^2+y^2=2,x^2+z^2=4\right\}$$ I am quite confident ...
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### All natural number solutions to the equation $a^2+b^2=c^2+d^2=2x^2$

Yesterday, I posted this question, and got that if $a$, $b$ and $c$ are in the form $$a=k(m^2-n^2+2mn)$$ $$b=k(n^2-m^2+2mn)$$ $$c=k(m^2+n^2)$$ where $m$ and $n$ are natural numbers, $a$, $b$ and $c$ ...
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### Fast algorithm for solving diophantine equation $x^4=a^4+b^4+c^4+d^4$

This problem was posed to an acquaintance of mine (at their university) and piqued my interest so I tried to solve it. The description goes as follows: Write a program that finds a solution to the ...
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### All natural number solutions for the equation $a^2+b^2=2c^2$

$a$, $b$ and $c$ of all Pythagorean triplets can be written in the form $$\begin{split} a &= 2mn\\ b &= m^2-n^2 \\ c &= m^2+n^2 \end{split}$$ where $m$ and $n$ are natural numbers. For ...
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### Proving no integer solution exists that makes a polynomial a perfect square

The context for this is the following coding problem on Hackerrank. I'm trying to understand why one of their sample inputs (Sample Input 4) has no solution. After a bit of math, it comes down to ...
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### Solve Diophantine equation $a^2+5ab+3b^2-c^2=0$

Solve Diophantine equation $a^2+5ab+3b^2-c^2=0$ My thoughts are to express it as $(pa+qb)^2 = c^2$and then solve it as Pell's equation. One solution is $(1,9,17)$.I don't know whether it is a ...
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### Solve the diphantine equation: xy + 114x - 81y = 9245 [duplicate]

Im not very experienced in this type of maths as I have only solved simple diophantine equations with euclides, so please explain the whole process used to get to the solution. The exercise which this ...
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### Parametrizations of $x^4+y^4+z^4=9t^2$ integer solutions

I would like to derive all the parametrizations for the nontrivial solutions of this Diophantine equation: $x^4+y^4+z^4=9t^2$ I already know that with the Fauquembergue's parametrization I can find ...
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