# Tagged Questions

Questions on finding integer/rational solutions of equations.

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### Solutions to simultaneous Diophantine equations $2y^2-3x^2=-1$ and $z^2-2y^2= -1$

I am looking for integer solutions for the following set of equations: $2y^2-3x^2=-1$ $z^2-2y^2= -1$ I know that there are the solutions (1,1,1) and (-1,-1,-1) for this set of ...
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### $N=(x^2-1)(y^2-1)$ has more than one solution

Given that $N=(x^2-1)(y^2-1)$ where $N,x,y,a,b$ are positive integers, find with proof the smallest value of $N$ such that $N=(x^2-1)(y^2-1)=(a^2-1)(b^2-1)$, where $a$ is not equal to either $x$ or ...
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### If $\frac1x-\frac1y=\frac1z$, $d=\gcd(x,y,z)$ then $dxyz$ and $d(y-x)$ are squares

Let $x, y, z$ be three non negative integer such that $\dfrac{1}{x}-\dfrac{1}{y}=\dfrac{1}{z}$. Denote by $d$ the greatest common divisor of $x, y, z$. Prove that $dxyz$ and $d(y-x)$ are ...
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### Find minimum of the $n$ such $x+11y+11z=n$ has $16653$ triples of postive integers solution

I wish to solve following problem $$x+11y+11z=n(n\in N^{+})$$ has $16653$ triples $(x,y,z)$ of postive integers. Find $n_{\min}$ Of course, I can't solve it by Now, so there any solution? Problem ...
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### Find all integer solutions for $x*y = 5x+5y$

For this equation $x*y = 5x + 5y$ find all possible pairs. The way I did it was: $x=5y/(y-5)$ And for this I wrote a program to brute force a couple of solutions. If it helps, some possibilities ...
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### Integral solutions $(a,b,c)$ for $a^\pi + b^\pi = c^\pi$

We know that $a^n + b^n = c^n$ does not have a solution if $n > 2$ and $a,b,c,n \in \mathbb{N}$, but what if $n \in \mathbb{R}$? Do we have any statement for that? I was thinking about this but ...
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### Can only the middle school math knowlegde help to find solutions for $2013 y^2 -xy -4026 x=0$?

I found the following equation form an answer written for a question. $$2013 y^2 -xy -4026 x=0$$ But I'm confused that can I really learn how to find the positive integer solutions for $x,y$ with ...
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### On a remarkable system of fourth powers using $x^4+y^4+(x+y)^4=2z^4$

The problem is to find four integers $a,b,c,d$ such that, ...
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### Find the solutions of the diophantine equation $(x^2-y^2)(z^2-w^2)=2xyzw$

Let $x,y,z,w$ be postive integers. Find all solutions of: $$(x^2-y^2)(z^2-w^2)=2xyzw$$ This gives: $$\left(\dfrac{x}{y}-\dfrac{y}{x}\right)\left(\dfrac{z}{w}-\dfrac{w}{z}\right)=2$$ ...
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### Looking for the most elementary proof that $48X^4+12X^2+1=Y^2$ has no non-trivial integer solution.

As relayed in this question of mine (which is more general in scope), I believe I have found a relatively easy, and completely elementary, way to show that the equation $$48X^4 + 12X^2+1 = Y^2$$ has ...
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### Weight 2 Newforms of large level computations.

I am stuck with some weight $2$ newform computations of large level. For example I want to compute newforms of level $11520$. Can anyone suggest me a way to do it? I need it to solve some diophantine ...
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### $x^3 + 5x + 6 = 3\cdot 2^{1+x-k}$

Does anyone know how to solve $$n^3 + 5n + 6 = 3\cdot 2^{1+n-k}$$ where n,k are natural numbers? I was told that there are prime number arguments that can be used but I am totally stuck. It is a ...
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### proof - if $x^2 + y^2 + z^2 = 2xyz$ then $x = y = z = 0$ [duplicate]

Well, I have been trying to prove that: $$x^2 + y^2 + z^2 = 2xyz \implies x = y = z = 0$$ and have made little progress. Till now, I have only been able to prove that if this is to happen then $x$, ...
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### Show that there exist no $a, b, c \in \mathbb Z^+$ such that $a^3 + 2b^3 = 4c^3$

Find all positive integer solutions of $a^3 + 2b^3 = 4c^3$. Proof: There don't exist any integer solutions for the give equation. Proof by the Well Ordering Principle. Let $d$ be the set of all ...
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### Diophantine equation in the integers

Find all integers $a,b$ such that $6(a^2-ab+b^2) = 31(a+b)$ Ideas I have had so far: Move everything to the left side of the equation and try to solve a polynomial in $a$,whose discriminant in $b$ ...
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### Looking for a general method for this type/class of Diophantine equation

I have the following Conjecture: If $w$ and $z$ are non-negative integers satisfying the equation $$w(w+6) = z(16z^2+36z+27), \tag{\star}$$ then $w=z=0$. I believe it to be true for the ...
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### Natural numbers $a,b,c$ satisfaying $abc=2(a+b+c)$

How can one find all natural numbers such that: $a≤b≤c$ $$abc=2(a+b+c)$$ I tried this : $abc-2c=2a+2b$ so $c=\frac{2(a+b)}{ab-2}$
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### Equation $x^3+2x+1=2^n$ in positive integers

Determine all pairs of positive integers $(x,n)$ which satisfy the condition $$x^3+2x+1=2^n$$ My work so far: Obviously, $x$ is odd. We show that solutions exist only for $n \in\{1,2\}$. Suppose ...
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### Solve the equation for $x,y\in\mathbb{Z}$: $x^4-2x^3+x=y^4+3y^2+y$.

As in the title. I have no idea how to deal with such equations, I'm completely new to this topic.
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### Is the following theorem usefull for number theory?

Odd integer $N=6p+5; p=0,1,2,...$ is a prime number if and only if no one of two diophantine equation $y=(p+1-x)/(6x-1)$ $y=(p+1+x)/(6x+1)$ has solution. Odd integer $N=6p+7; p=0,1,2,...$ is ...
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### solve $ax^3+by^3+cz^3+dx^2y+ex^2z+fxy^2+gxz^2+hy^2z+iyz^2=0$ for all triplets $(x,y,z)$.

let $x,y,z$ be any 3 positive integers. If for all $x,y,z$, we have : $$ax^3+by^3+cz^3+dx^2y+ex^2z+fxy^2+gxz^2+hy^2z+iyz^2=0$$ What can be said about the integral coefficients $a,b,c,e,f,g,h,i$? I ...
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### Largest Erdős–Diophantine graphs

A Diophantine graph is a set of vertices in the plane with integer coordinates, all at integer distances from eachother. An Erdős–Diophantine graph is a maximal Diophantine graph, so that it cannot ...
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### Exponential diophantine equation of the form $x^x + y^y = 2 z^z$ [closed]

Suppose that $x,y,z$ are natural numbers and $$x^x + y^y = 2 z^z$$ Prove that $x=y=z$.
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### Find all $a,b,c\in\mathbb{Z}_{\neq0}$ with $\frac ab+\frac bc=\frac ca$

As the title implies, I'm looking for triples $(a,b,c)$, where $a,b,c$ are nonzero integers, with $$\frac ab+\frac bc=\frac ca$$ I checked the cases $-100<a,b,c<100$ where $a,b,c\neq 0$ (using ...
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### About 3-powerful numbers in an equation

In this page, I am not interested on the Beal conjecture itself. But I am interested on the following problem: The author claimed: Any solutions to the Beal conjecture will necessarily ...
How can I prove that the Diophantine equation $$\frac 1 x_1 +\frac 1 x_2 + ... + \frac 1x_n +\frac 1 {x_1 x_2 ... x_n} = 1$$ has at most one solution? All $x_i$ and $n$ are natural numbers. My ...
How many integer-sided right triangles exist whose sides are combinations of the form $\displaystyle \binom{x}{2},\displaystyle \binom{y}{2},\displaystyle \binom{z}{2}$? Attempt: This seems like ...