# Questions tagged [diophantine-equations]

Use for questions about finding integer or rational solutions to polynomial equations.

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### solve in N the equation $x^2-2^y=2021$

I have this equation to solve $x^2-2^y=2021, x,y \in N$ I was thinking of seeing it as a Diophantine equation, but it doesn't seem very logical
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### Find all integer solutions for $x^3+1=y^2$.

Find all integer solutions for $$x^3+1=y^2.$$ Attempt: By guessing, I found five pairs of integer solutions for the equation: $(2, \pm 3)$, $(0, 1)$, $(-1, 0)$ and $(0, -1)$, but really I don't know ...
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### Value of m for which the function will give integers as an output. [closed]

$F(m)=(2m^3+2m)/(m^2+1)$ and $g(m)=(m^4+1)/(m^2+1)$ What are the values of $m$ other than $1$ for which solution of both function will be integers. Please tell if there is any formula to find so or ...
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### If $\frac{2m^3+2m}{m^2+1}$ and $\frac{m^4+1}{m^2+1}$ are integers then $m=?$ [closed]

$\frac{2m^3+2m}{m^2+1}$ and $\frac{m^4+1}{m^2+1}$ Find value of $m$ other than $1$ for which solution of both equations are in integers. Please tell if there is any formula to find so or any technique....
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### Show that for any positive integer $n>1$ there exist positive integers $x$ and $y$ such that $n^3+x^3=(x-1)^3+y^3$.

A quick check shows that $x=y=n+1$ work pretty good. Since, $n^3+(n+1)^3=\{(n+1)-1\}^3+(n+1)^3.$ How to check whether there exist any other type of solutions or not? Please suggest.. Thanks in ...
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### Find $a \in \mathbb N$ such that $x^2+ax-1 = y^2$ has a solution in positive integers

Question: Find all the positive integers $a$ such that $x^2+ax-1 = y^2$ has a solution in positive integers $(x,y)$. Comments: It's easy to see that this equation rarely has a solution (in the sense ...
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### An example of a polynomial function $f$ where the equation has no solution or has a finite number of solutions

I am asking about the following Diophantine equation: $$32(d+1)²a²-cf⁴(b)=0$$ where $f$ is a polynomial function. From this question (A Diophantine equation with infinitely many positive integer ...
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### A Diophantine equation with infinitely many positive integer solutions

I am asking if the following Diophantine equation: $$32(d+1)²a²-cb⁴=0$$ has infinitely many positive integer solutions.
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### Fermat's last theorem for case of n=3

I'm aware that one can use a method of infinite descent, or even just refer to the more general case which has been proven by Andrew Wiles, but I was thinking about it the other day, and I remember ...
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### Find all positive integers which are representable uniquely as $\frac{x^2+y}{xy+1}$ with $x,y$ positive integers.

$\textbf{Question:}$ Find all positive integers,which are representable uniquely as $\frac{x^2+y}{xy+1}$ where x and y are positive integers. I think this question maybe has something to do with ...
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### Find all integer solutions for Mordell's Equation $x^2=y^3+k$, where $k=-35$.

I know that there are definitely solutions to this equation, two of which are $(\pm 36,11)$. However, I now need some guidance on where to begin finding the remaining solutions for this case of ...
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### Number Theory and Square Number Problem [closed]

Find the least positive integer $n >1$ such that the arithmetic mean of the first $n$ non zero perfect squares is again a perfect square. Please help. Hope it gives me an idea of equations with ...
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### $\left(3z\right)^3\ne 3\left(x+y\right)\left(3z-x\right)\left(3z-y\right)$

Prove that $$\left(3z\right)^3\ne 3\left(x+y\right)\left(3z-x\right)\left(3z-y\right)$$ Is true for $x$, $y$ and $z$ being positive integers, with $x$ and $y$ being co-prime and $3z<x<y$. ...
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### Prove $3(x+y)(x+z)(y+z)\neq a$ cubic when $x,y,z$ are different co-primal positive integers

Prove $3(x+y)(x+z)(y+z)\neq a$ cubic when $x,y,z$ are different co-primal positive integers. I believe you can look at the prime factors of $x,y$ and $z$. As for the equation to equal a cubic, the ...
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### On a homogeneous Diophantine equation

I want to solve the diophantine equation $c_1x_1 + c_2 x_2 + c_3 x_3 + c_4 x_4 = 0$ (I), when $c_1+c_2+c_3+c_4=0$. I first consider $x_1 = t$ and $(c_2+c_3+c_4)t = c_2x_2+ c_3 x_3 + c_4 x_4$; then I ...
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### Diophantine equations that involve Gregory coefficients: a computational exercise

In this post, for integers $k\geq 1$, we denote the Gregory coefficients as $G_k$. Wikipedia has an article for Gregory coefficients, are known as reciprocal logarithmic numbers (I add this as ...
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### Generalized Fermat equation for signature [4,5,7]

The generalized Fermat equation has been solved for many signatures. But, I can't find a determination that the signature $[p,q,r]=[4,5,7]$ has no solutions. Is this signature still an open problem? ...
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### Method to solve factored quadratic diophantine equations?

Is there a method that can solve all quadratic diophantine equations of the following type $$X (X + a) = Y (Y + b)$$ where $a,b$ are given integers?
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### proof that no non negative odd integer solution of this type of diophantine equation exists given even non negative integer solution

Given the equation $$(2^n-1)(1+x+x^2+...x^{2k})(1+y+y^2+...+y^{2j})=2^nx^{2k}y^{2j}-1$$ a non negative integer solution exists of the form $(2^n,2^{2kn+n}$). How can I (try to) show that (1) this ...
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### find integer solutions under square root [duplicate]

I have a equation $y = \sqrt{5x^2+2x+1}$ and I'm trying to generate integer solutions. I've tried Vieta jumping but it failed. So I generated by brute force few solutions and find these: x=2, 15, ...
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### Can my short proof of Fermat´s Last Theorem be true or is there a serious flaw?

A proof of Fermat´s Last Theorem using only Gauss´s Lemma for the roots of monic polynomials with integer coefficients.** Writing the Fermat equation $$a^n + b^n - c^n = (c-p)^n + (c-q)^n - c^n = 0$$...
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### Are there infinitely many primitive Pythagorean $n$-tuples?

Let's define a Pythagorean $n$-tuple ($n \geq 3$) as a tuple of distinct natural numbers $(x_1, ... , x_{n-1}, y)$, such that $x_1^{n-1} + ... + x_{n-1}^{n-1} = y^{n-1}$. Let's call a Pythagorean $n$-...
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### Fundamental Discriminant

Let $d=2\bmod4$ and $D=d^3-1$. We will show that the elliptic curve $E_D: y^2=x^3+D$ has no integer solutions $x,y$. At first we show that $x$ is odd which is simple. Then we have the following ...
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### How to find infinitely many positive integer solutions to $x^2+y^2+z^2=w^2$

How to find infinitely many positive integer solutions to this equation? $$x^2+y^2+z^2=w^2$$ Why my answer makes no sense: $x=n$ $y=(n+1)$ $z=\sqrt{ 2n(n+1) }$ then $w^2=(n+(n+1))^2$ for some ...
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### What are enough conditions for $x,y,z$ to have $x^3+y^3+z^3$ a perfect square?

In my question here I want to know if there are enough conditions for $x, y,z$ to have $x^3+y^3+z^3$ a perfect square , The necessary condition is clear that $x,y,z$ must be different than $4$ or $5$ ...
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### Criteria for checking if points are the vertices of a hypercube

I asked a question over at Code Golf Stack Exchange which essentially asked folks to write a program to determine if a collection of $2^n$ points in $\mathbb{Z}^m$ is the vertex set of some $n$-...
### Solutions to Diophantine equation $\frac{1}{n}+\frac{1}{p}=\frac{1}{N}$
For each prime $p$ there seems to be a uniqe solution $n=(p-1)p$ to the Diophantic equation $\frac{1}{n}+\frac{1}{p}=\frac{1}{N}$. Is that right and if so, how to prove the unicity? In spite of my ...
### Why $c$ closed to $-2\times10^n$ in $(1-c^2)^3+(c^3+10^nc^2-1)^3+(10^n c^2-1)^3=n$ for $n >1$?
I have tried many times to evaluate $(1-c^2)^3+(c^3+10^nc^2-1)^3+(10^n c^2-1)^3=n$ for $n >1$ as polynomial for some values of integer $n$ which are greater than $1$ for the solution of the titled ...