# Tagged Questions

Questions on finding integer/rational solutions of equations.

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### Diophantine equation $n^2+n+1=m^3$

Is there an elementary method for solving Diophantine equation $n^2+n+1=m^3$ for integers $m$ and $n$? There is a similar one, which I could solve:$$p^2-p+1=q^3,$$where $p$ and $q$ are prime numbers. ...
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### Link between the negative pell equation $x^2-dy^2=-1$ and a certain continued fraction

Consider the generalized continued fraction $$F(x)=(x-1)-\cfrac{(x+1)}{x+\cfrac{(-1)(5)} {3x+\cfrac{(1)(7)}{5x+\cfrac{(3)(9)}{7x+\cfrac{(5)(11)}{9x+\ddots}}}}}$$ I experimentally discovered that at ...
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### Show that $x^2 + y^2$ and $x^2 - y^2$ cannot both be perfect squares at the same time where $x, y \in \mathbb{Z}^+$.

Show that $x^2 + y^2$ and $x^2 - y^2$ cannot both be perfect squares at the same time where $x, y \in \mathbb{Z}^+$. I think that $x^2 + 2xy + y^2$ and $x^2 + y^2$ are not consecutive squares since ...
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### Preserving modulus residue under division

Modulus residue is preserved or honored (sorry, I don't know the correct term. Is it homomorphism?) under addition and multiplication. For example: 2 + 4 = 6 2 * 4 = 8 Then, making those values ...
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### Solving the equation $x^3+y^2=4x^2y$ over integers.

$$x^3+y^2=4x^2y$$ This is a quadratic in $y$, the discriminant of which must be $>0$ $$\implies 16x^4-4x^3>0$$ $$\implies x \in (-\infty,0) \cup (1,\infty)$$ (So we have nothing new up to this ...
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### Diophantine System Solution

Could you please help with finding of general solution of diophantine system for rational a, b, c, d $(a^2+b^2)(c^2+d^2)=A^2$ $(a^2-b^2)(c^2-d^2)=B^2$ for some rational A and B. This is related ...
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### Solving Diophantine Equation $xB=(2^N)-1$

If given a value for $x$, does anyone have a way to solve the diophantine equation below? $xB=(2^N)-1$ where $x,B,N\in\mathbb Z$ Where presumably a smaller $N$ is better, but any way to find a ...
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### Linear Diophantine equations of several variables

I know how to solve Diophantine equations of the form $ax+by=c$ but how can I solve linear Diophantine equations having more variables. Like what are the integer solutions of $43x+23y-435z+1324w=1$? I ...
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### Help answering Pell Equation questions

I understand the Pell equation is $$x^{2}-dy^{2}=1$$ However I don't understand how to use this to get $(x,y)$ for these questions. 1) Find a nontrivial solution of $x^{2} − 3y^{2} = 1.$ 2) Find ...
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### How do I count the solutions of $m^2 + m n + n^2 = T$?

I've come across this problem in my studies. I was wondering if there is a better algorithm for it: Given a fixed positive integer $T$, count the solutions of $$n^2 + n m + m^2 = T$$ where $m$ and $n$ ...
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### Solve $x^2 = 2^n + 3^n + 6^n$ over positive integers.

Solve $x^2 = 2^n + 3^n + 6^n$ over positive integers. I have found the solution $(x, n) = (7, 2)$. I have tried all $n$'s till $6$ and no other seem to be there. Taking $\pmod{10}$, I have been ...
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### How to enumerate 2D integer coordinates ordered by Euclidean distance?

The square of Euclidean distance between $(x, y)\in\mathbb{Z}^2$ and origin is $d = x^2+y^2$. How to enumerate the coordinates $(x, y)$ in ascending order of $d$? For example, the first 14 sets of ...
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### Numbers expressible as sum of 2 squares in 2 distinct ways

I was trying this question here which goes like: Find numbers which are squares and can be expressed as $x^2y^2-x^2-y^2+2$ for non-consecutive positive integers only. Let the number be $a$ \...
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### Solutions to the diophantine equation $6x^2 - 6x - y^2 + y=0$?

Are there any positive integer solutions to the diophantine equation in the title other than $(1,1)$? This equation looks easy enough so it could be that there is some simple argument that shows ...
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### Find all $x,y$ so that $\dfrac{x+y+2}{xy-1}$ is an integer.

I am trying to find the integers $x,y$ so that $\dfrac{x+y+2}{xy-1}$ is an integer. What I have done: I suppose there exists $t$ such that $$t=\dfrac{x+y+2}{xy-1}$$ where $xy\neq 1$ then consider ...
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### Coprime - Irreducibility - Natural numbers

In reference to this question, is anyone could deduce that if $x^2+2=y^3$ and $x,y \in \mathbb{N}$, then $x=5$ and $y=3$. I already prove that the only natural number $x$ for which $x+\sqrt{-2}$ is a ...
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### Non-linear Diophantine equation on integer quadruples

Find all integer quadruples $\{a,b,c,d\}$ such that $$ad = b + c$$ $$bc = a^2 - d$$ Working $\bmod 8$ (very messy) gives $d = 3 - 8k \quad \forall k \in \mathbb{N}$. Numerical searching has so ...
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### Solve equation over $\mathbb{N}\setminus\{0\}$ [closed]

I wonder whether there are any solutions besides considering $c=2^{5k+1}$ for this equation: $a^5+b^5=c^{2016}$, where $a,b,c\in\mathbb{N}\setminus\{0\}$
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### Quadratic Diophantine Equation $x^2 + 2y^2 = 2013$ [closed]

Find integer values of $x$ and $y$ (if any) such that $x^2 + 2y^2 = 2013$.
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### Parametrization of $a^2+b^2+c^2=d^2+e^2+f^2$

Is there an existing parametrization of the equation above that is similar to Brahmagupta's identity for $a^2+b^2=c^2+d^2$? I need either a reference to look it up or a hint to solve it. Thanks.
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### Why there isn't any solution in positive integers for $z^3 = 3(x^3 +y^3+2xyz)$?

Consider the following Diophantine equation $$z^3 = 3(x^3 +y^3+2xyz)$$ Is there any elementary proof for the non solubility in positive integers for this Diophantine equation, where $x, y$ and $z$ ...
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### Find all nonnegative integer solutions to $x^3 + 8x^2 − 6x + 8 = y^3$.

Find all nonnegative integer solutions to $x^3 + 8x^2 − 6x + 8 = y^3$. The only solution I have found is $x=0$. I have tried proving it by congruences and have had no success. I don't know how to ...
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### Help Project Euler Problem 269

I am stuck on prob 269 Project Euler. I've just tried brute force method to attempt this problem the example provided by PE For example, $P_{5703}(x)$ = $5x^3 + 7x^2 + 3$. We can see that: P$_n(0)$...
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### Transforming Diophantine quadratic equation to Pell's equation

I have been discussing the fastest and most efficient ways of solving QDEs in a separate question record (Alternative method to solve quadratic Diophantine equations). However, as suggested by individ,...
### Quartic Diophantine equation $2 x^4 - 2 x^2 = 3 (y^2 - 1)$
About the quartic Diophantine equation: $$2 x^4 - 2 x^2 = 3 (y^2 - 1)$$ On oeis.org/A180445 it says that all positive solutions $(x,y)$ are: $$(1,1)\ \ (2,3)\ \ (3,7) \ \ (6,29)\ \ (91,6761)$$ ...