# Tagged Questions

Questions on finding integer/rational solutions of equations.

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### A divisibility conjecture related to the Ramanujan-Nagell equation

The Ramanujan-Nagell equation is $$x^2+7=2^n,$$ where it has been proven (using non-elementary methods) that the complete solution is $n \in \{3, 4, 5, 7, 15\}$. I've found an elementary way to ...
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### Diophantine relations using an equation with polynomials of degree at most 4

I'm completely stuck at exercise 5.8.5 of Mathematical Logic, Chiswell & Hodges: Here are the mentioned definition and theorem: I'm stuck because I failed to use the hint given in the ...
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### Number of integer solutions (ordered and unordered)

$$\frac1 a + \frac 1 b +\frac 1 c = \frac 34$$ Find number of triplets of $a\ , b\ , c\in \mathbb{Z}^+$ Should it not be infinite since it can be $\frac 34$ or $\frac38$ or $\frac9{12}$ etc. ...
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### Find all natural roots of $\sqrt{x}+\sqrt{y}=\sqrt{1376}$ given that $x\leq y$

Find all natural roots of $\sqrt{x}+\sqrt{y}=\sqrt{1376}$ given that $x\leq y$ I'm confused of this equation because $1376$ is not a square!! So maybe it has no natural root! Am I right??
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### Integer solutions to $210y^2=(x)(x+1)(2x+1)$

I'm looking to find integer solutions for large positive $y$ values (say over 1000) to the following equation: $210y^2=(x)(x+1)(2x+1)$ What I know so far: Integer solutions include (0,0) and (7,2) ...
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### Given some arbitrary roots of a polynomial p(x,y,z,…) with integer coefficients, is it possible to tell if p has a root in the Gaussian integers?

I'm trying to find if p(x,y,z,...)=0 has a Gaussian integer root (more specifically, I want to find if p has a Gaussian integer root where the imaginary components are even, but if that can't be done, ...
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### How many integer solutions are there of the equation $|x_{1}|+|x_{2}|+\cdots +|x_{k}|=n$?

How many solutions are there to the equation $$|x_{1}|+|x_{2}|+\cdots +|x_{k}|=n$$ for $n,k\in \mathbb N$ and $\forall\ 1\leq i\leq k,\ x_{i}\in \mathbb Z$? Any ideas? I don't know how to ...
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### solve $x^y-y^x=xy^2-19,$ $x,y\in\mathbb{Z}$

I have been struggling to solve this exercise but with no result: $$x^y-y^x=xy^2-19,$$ $x,y\in{\mathbb Z}$ I have started to think it has no solutions at all. I have no idea how to solve it so I was ...
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### Find the integer $x$ such $x^6+x^5+x^4+x^3+x^2+x+1=y^3$

Find the equation integer solution $$\color{red}{y^3=x^6+x^5+x^4+x^3+x^2+x+1}$$ It is obvious $x=0,y=1$ or $x=-1,y=1$ are solutions. How to find all solutions?
### Solve $z^3=kx+ny$ , ($k\neq{n},k,n\in \mathbb{N}$)
Solve $z^3=kx+ny$ , ($k\neq{n},k,n\in \mathbb{N}$) for positive integer unknowns $x,y,z$ I have really no idea for this!!