# Tagged Questions

Questions on finding integer/rational solutions of equations.

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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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### Solutions to a quadratic diophantine equation $x^2 + xy + y^2 = 3r^2$.

Let $k,i,r \in\Bbb Z$, $r$ constant. How to compute the number of solutions to $3(k^2+ki+i^2)=r^2$, perhaps by generating all of them?
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### What is the best time complexity for this case?

I only want to know if the following system has any integer solution or not. Actually, I do not need to know the solution(s), and only need to know the answer of question "Does the system have any ...
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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?
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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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### 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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### Find all natural roots of: $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=d$ given that: $a<b<c$

Find all natural roots of: $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=d$ given that: $a<b<c$ Rearranging the equation gives: $$ab+bc+ac=abcd$$ What can we do with this?
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### 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!!
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### Find all positive integer roots of : $5xy=19x+96y$

Find all positive integer roots of : $5xy=19x+96y$ I tried using decomposition technique but no success...,it seems suitable factorization of this equation is IMPOSSIBLE!! Handy calculations show ...
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### Find all integer solutions to $7595x + 1023y=124$

Find all integer solutions to $7595x + 1023y=124$ Using the Euclidean algorithm I have found the $\gcd(7595,1023)=31$ and found the Bezout identity $31=52\cdot1023-7\cdot7595$ but I'm not really sure ...
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### Determine if quadratic diophantine equation in two variables will generate perfect squares

I have come across two equations with variables $x,y$ \begin{align*} (x+ay)^2+ 4 x y\\ (x-y)^2-4 c x y \end{align*} where $a,c\in \mathbb{Z}_+$ are some constants. I would like to determine the ...
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### Failure of an elementary 'proof' of Fermat's Last Theorem?

Can someone explain to me why this does not constitute a proof of Fermat's Last Theorem, please? Basically, using something I've read online, it appears you can write an equation for $(a, b, c)$ to ...
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### find the number of tuples of positive integers [closed]

find the number of tuples (a,b,c,d) of positive integers \begin{array}{l} {a^3} = {b^2}\\ {c^3} = {d^2}\\ c - a = 64 \end{array} answer should be one of 0 , 1 , 2 , 4
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### Solving Diophantine equation $1/x^2+1/y^2=1/z^2$

How can we find positive integers solutions $(x,y,z)$, where $\gcd(x,y,z)=1$ for the equation: $$1/x^2+1/y^2=1/z^2$$ Can we conclude that $x$ and $y$ are not coprimes for it to have solution?
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### Number of solutions of: $3x+y=5702$

Find the number of ordered pairs $(x,y)$ satisfying $3x+y=5702$ in natural numbers restricted by: $x+y\le2003$ I don't know any method for counting number of solutions of such equations...
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### If $x$ and $y$ are non-negative integers for which $(xy-7)^2=x^2+y^2$. Find the sum of all possible values of $x$.

I am not able to reach to the answer. I have used discriminant as $x$ and $y$ are both integers but it didn't give any hint to reach to answer. I am not able to understand how should I deal with these ...
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### What is the easiest way to solve diophantine equation with three unknowns?

Suppose we have a diophantine equation of the form: $$ax + by + cz = d$$ What is the best (simplest, easiest) way to find the solution(s)? Should I apply extended Euclidean algorithm?
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### Diophantine equations using Euclidean algorithm

I solved two systems of Diophantine equations using the Euclidean algorithm and I can't figure out where I went wrong because the solutions I test aren't working but I have rechecked my work several ...
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### Diophantine equation with binomial coefficient

Suppose that $p$ is a prime number and $p \le q \le p^2$ is an integer. How many solutions are there to the following equation? $$\binom{p^2}{q}-\binom{q}{p}=1$$ This question was proposed ...
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### How many generators needed for Pell-equation-related group

Let $d$ be a positive integer which is not a perfect square. We have the norm multiplicative group homomorphism, $N:{\mathbb Q}[\sqrt{d}] \to {\mathbb Q}$ defined by $N(x+y\sqrt{d})=x^2-dy^2$. It ...
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### Showing that there are infinitely many integer solutions for the hyperbolic formula $|a^2 - 26 b^2| = 1$

I want to show that the formula $$| a^2 - 26\cdot b^2| = 1$$ has infinitely many solutions $(a, b) \in \mathbb{Z}^2$. First I tried to solve the formula for one of the two variables, to get ...