# Tag Info

## Hot answers tagged diophantine-equations

### Irreducible elements in $\mathbb{Z}[\sqrt{5}]$

Throughout this argument, we will use the fact that the norm is multiplicative, that is, $N(x)N(y)=N(xy)$ (I implore you to check this). A unit is an element $x$ with $N(x)=1$. An element $x$ is ...
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### The Diophantine equation $45x^4-42x^2y^2+5y^4=8$

Here is a practical answer, for the theory or algorithm description check the links, or someone else can elaborate if needed. This is a Thue equation and there are efficient algorithms for finding all ...
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### Integer solutions for $2abcd=(a+2)(b+2)(c+2)(d+2)$.

First divide through and rewrite the equation as $$\left(1+\frac{2}{a}\right)\left(1+\frac{2}{b}\right)\left(1+\frac{2}{c}\right)\left(1+\frac{2}{d}\right)=2$$ It's usually a good idea with these ...
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### Integer solutions for $2abcd=(a+2)(b+2)(c+2)(d+2)$.

If all $a,b,c,d \geq 11$, then we have that $$2abcd = \prod_{cyc}(a+(\sqrt[4]{2}a-a)) > \prod_{cyc}(a+2) =(a+2)(b+2)(c+2)(d+2)$$
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### Please help me solve this polynomial equation. I would like to know how many integer solution it has?

First note that no solution can be of the form $(-5,y)$ (you should check this). Since your polynomial is linear in $y$, we can solve for $y$ and get y=\frac{8x^3-x-13}{x+5}=8x^2-40x+199+\frac{982}{...
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