Questions on finding integer/rational solutions of polynomial equations.

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179 views

Upper bound for the quality of an $abc$-triple

A triple of positive integers $(a,b,c)$ is an $abc$-triple if $a$ and $b$ are coprime and $c = a + b$. Define the quality or power of an $abc$-triple as $P(a,b,c) = \frac{\log c}{\log ...
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29 views

coin problem with two coins, inductive proof

I want to ask something about the coin problem with two coins. Let $a,b$ be to numbers in $\mathbb{N} \setminus \{0\}$ (elsewhere I include zero) which have no prime factors in common. I will write $$ ...
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22 views

Powerful numbers in Pell solutions (or, more generally, any Lucas sequence)

There are several definitive results regarding perfect powers in the Pell numbers — e.g., the only perfect power is $P_7=169=13^2$. On the other hand, when it comes to powerful numbers, I've only ...
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40 views

Given a Pell “solution” in [integer] polynomials, what can be said about the components?

Let $x,y$ be integers, and $f(x,y)$, $g(x,y)$, and $h(x,y)$ be polynomials in $x$ and $y$ with integer coefficients such that $$ f(x,y)^2 - g(x,y)h(x,y)^2 = 1. \qquad(\star) $$ Furthermore, assume it ...
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26 views

Characterizing Coprimes

Here's a question about coprimes that I stumbled upon while doing some research. Providing insight into this question would prove quite helpful to me. Choose a pair of coprimes $x, y \in \mathbb Z$. ...
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19 views

Diophantine equations,is that what I have done right?

I have solved the following diophantine equations: $14x+35y=93$ $56x+72y=40$ That's what I have tried: $gcd(35,14)=7$ , but $7 \nmid 93,$ so the first diophantine equation has no solution. ...
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22 views

Why the proof of Catalan's conjecture is not easily generalizable?

Let $x,y>0$, $u,v>1$ be integers. Why is it easier to solve $x^u-y^v=1$ than $x^u-y^v=2$? Is there possible some group behind the first equation which has some nice property that the group made ...
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117 views

Integral points on varieties and solutions to Diophantine equations

I am looking for a book (or article, or notes...) explaining details about the link between integral points on varieties defined as complement of certain divisors and integral solutions to the ...
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31 views

To solve $ \dfrac1m+\dfrac1n-\dfrac1{mn^2}=\dfrac34$ on all integers

Refering To solve $ \dfrac1m+\dfrac1n-\dfrac1{mn^2}=\dfrac34$ , I think it is an interesting question, if the possible solution are integers, thus How do we find all integers $(m,n)$ such that $ ...
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28 views

Another triple.

Solving the equation. $X^2+Y^2+Z^2=X^3$ got some solutions, but still the question remains. Below are all the decisions or not? $X=5t^2+2t+2$ $Y=11t^3+5t^2+2t$ $Z=2t^3+10t^2+4t+2$ And more. ...
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51 views

Diophantine Equation $2x^2+25=y^3$

I'm trying to find integer solutions to: $2x^2+25=y^3$. Here's what I've managed to do so far: y is odd. y and x are co-prime. In $\mathbb{Q}(\sqrt{2},i)$ we can write: ...
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101 views

15th power Diophantine equation

I'd appreciate some help (a hint) for the following. If $x,y>1$ are so that $2x^2-1=y^{15}$ then $x$ is a multiple of $5$. Don't know if this helps but the equation can be rewritten as ...
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58 views

How to solve diophantine equation $\frac{x^p-y^p}{x-y}=n$

$$\frac{x^p-y^p}{x-y}=n$$ whit $p$ a prime greater than or equal to $3$,for what value to $n$, it's solvable and how to solve,and whether $\frac{x^p-y^p}{x-y}=q_1$ $\frac{x^p-y^p}{x-y}=q_2$ is ...
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34 views

Particular case of the n-degree equation.

It's know that there isn't a general formula for solve the general n-degree equation for $n>4$, but there is any formula to solve the case? $A^x+A=C$ where $A$ is the variable and $A,C$ are ...
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19 views

Is there an analogue of the 15 theorem for cubic forms?

The 15 theorem states that if an integral quadratic form with integral matrix represents the numbers 1, 2, 3, 5, 6, 7, 10, 14, 15, then it represents all numbers. Is there an analogue of this theorem ...
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87 views

Need a proofreading why all the units are satisfied $a^2-2b^2 =\pm1$ for $\mathbf{Z}[\sqrt{2}]$

All the units are satisfied Pell's equation $a^2-2b^2=\pm1$ for $\mathbf{Z}[\sqrt{2}]$, $a,b\in\mathbf{Z}$. Here is my proof: Let $a+b\sqrt{2}$ be a unit $\in\mathbf{Z}[\sqrt{2}]$. This implies ...
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62 views

How to solve the diophantine equation:$ xa^3+yb^3=c^3$

Let $a,b,c,x,y \in \mathbb{Z}> 1$. Any hint on how to solve of the diophantine equation $ xa^3+yb^3=c^3$?
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18 views

What are some techniques for reducing the dimension of an arbitrary Diophantine polynomial?

A set $S \subset \mathbb{N}^k$ is Diophantine if $$(x_1, \dots, x_k) \in S \iff \exists y_1, \dots, y_d \, p(x_1, \dots, x_k, y_1, \dots, y_d) = 0$$ for some Diophantine (integer coefficients) ...
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108 views

How many solutions to $x^3+y^3 = z^3\pm 1$ for $z$ less than a bound?

Assume $a,b,c, N$ as positive integers, let primitive be $\gcd(a,b,c) = 1$ and, $$a^2+b^2 = c^2\tag{1}$$ Supposing you want to know how many solutions there are with $c$ less than a bound $N$. ...
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96 views

Trying to prove that a triangular number equation has no solutions.

I want to prove -- using elementary math only -- that the following equation has no integer solutions for $t \ge 1$: $$ 6a^2(16a^2+1) = \frac{t(t+1)}{2}. \qquad(1)$$ I know it doesn't (or at least ...
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61 views

Parametric Equation solving over integers

I have a question on my mind I am trying to solve. However I am stuck at a point. If you could help, I would be very pleased. $$\frac {x_2y_2z_2}{x_2y_2+y_2z_2+x_2z_2}=\frac ...
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50 views

A diophantine definition of the Kleene star

Let $f(x \, | \, y_1, \dots, y_n)$ be a Diophantine polynomial that generates the Diophantine set $F$. By Matiyasevich, the set $F^*$ (Kleene star of $F$) is also Diophantine. My question: how can ...
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37 views

Gap:$\;\;L(90,28) := {\{n90 - m28 ∈ N, n, m ∈ N, n < 28\}}$

Which elements of the sets Gap:$$L(90,28) := {\{n90 - m28 ∈ N, n, m ∈ N, n < 28\}}$$ $$$$What would be a quick way to resolve?
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46 views

Is this a fruitless approach to solving diophantine equations?

Let $P(X, Y, Z)$ be a polynomial over $Q$. Let's be concerned with integer solutions. Namely that there are no solutions $(X,Y,Z)$ such that $\gcd(X,Y) = 1$. So let $X,Y$ be coprime and arbitrarily ...
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45 views

A question about the solutions of a diophantine equation

I would like to know if it's possible to find the solution of the following equation: $$x^k+y^h=z^{kh}$$ in which: $$\{x,y,z\}\subset\mathbb{N}$$ given $k$ and $h$ with: $$\{k,h\}\subset\mathbb{N}$$
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72 views

Find integer solutions to $x^2+xy+11y^2=p$ using Ring identities

Let $\theta = (1+\sqrt{-43})/2$ and consider $\mathbb{Z}[\theta]$, a principal ideal domain, with the multiplicative map $\psi (a+b\theta)=a^2+ab+11b^2$. Show there exists an integer solution to ...
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49 views

Wrong answer on elementary diophantine equation - why?

Solve the equation and show all possible, non-negative values for X and Y: $5X+4Y=60$ So I wanted to do it like that: $$5X+4Y=60\leftrightarrow0X+4Y=0 \pmod5$$, thus $4Y=5k$ where $k\in Z$. ...
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94 views

Solve the equation $x^4+y^4=d*z^2$

Solve the equation:$$x^4+y^4=d*z^2,$$ where $x,y,z$ are positive integers,and $d>1$ is a given square-free integer. I know if $p$ is an odd prime and $p|d,$ then $t^4\equiv -1 \pmod p$ is ...
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96 views

On the elliptic curve $x^4+y^4 =193z^2$

Given the simultaneous Diophantine equations, $$u^2+v^2=w^2\tag{1}$$ $$x^4+y^4 = (u^6+v^6)t^2\tag{2}$$ the only solutions seem to be for the first Pythagorean triple $u,v,w = 3,4,5$ which yield the ...
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118 views

Week of the problem on Diophantine equation

S.E board! This is a Diophantine equations problem, which is so interesting one can do by plugging the suitable values in unknown. When it comes for finding set of all solutions is may be tough. I ...
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83 views

Diophantine Equation: $f(x)f(y) = f(z^2)$ where $f$ is quadratic

In the study of the Diophantine Equation $f(x)f(y) = f(z^2)$ where $f$ is quadratic, the computational proofs I have seen (for specific $f$) rely on Pell's Equation. For example, if $f(t) = t^2+t+1$, ...
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24 views

Can one determine in finite time whether a point is $S$-integral

Let $x$ be a $\mathbf{Q}$-rational point of $\mathbf{P}^1-\{0,1,\infty\}$. Let $S$ be a finite set of primes. How do I check in finite time whether $x$ is $S$-integral or not? I know how to do this ...
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127 views

diophantine equation with squares over 3 variables

I am trying to find solutions for this diophantine equation $$x^2+y^2+x^2y^2=4z^2$$ I am looking for advice on a procedure to find all positive integer solutions for this equations.
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314 views

Is this Diophantine equation proof correct?

It's probably a good thing I decided to try working on problems from the book. This section seems to be proving difficult. In any case, I'm asked to prove that $x^4+4y^4=z^2$ has no non-trivial ...
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192 views

Quadratic fields and solving Diophantine equations

I would like to learn to solve Diophantine equations and I think my next step would be quadratic fields or number fields. What are kind of methods there are to use those on solving equations? And what ...
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82 views

Exponential Diophantine representation of the factorial

How can I use the identity $$\sum_{n=0}^\infty \frac{\tau^n}{n!} = \lim_{y \to \infty} \left( 1 + \frac{\tau}{y} \right)^y$$ to find an exponential Diophantine representation of the factorial? I was ...
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23 views

What are some [mostly trivial] Pell transformations?

Euler looked at some transformations which turned one Pell[-type] equation into another. Example 1: $$u^2-av^2=-1 \quad\iff\quad (2u^2+1)^2-a(2uv)^2=1.$$ Example 2: $$u^2-av^2=-2 \quad\iff\quad ...
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25 views

Superelliptic curves

I'm trying to find information on superelliptic curves and how to solve them over the integers. The equation is $$y^k = f(x)$$ where $k=3$ and $f$ has degree $d=3$. Does anyone know any ...
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40 views

The Diophantine equation $x^2 - y^2 = 4z^n$

If $n \geq 3$ is an integer, then under what conditions on $x, y, z$ does the equation $$x^2 - y^2 = 4z^n$$ have no solution in integers? (If there is any known result, please feel free to share with ...
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44 views

Diophantine Equations involving cubes

I'm doing some number theory research and I came across these two Diophantine equations (created under my own transformations): $$y^3 = ax^3 + bx$$ (where $a$ and $b$ are parameters) $$z^3 = x^2 + ...
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36 views

Hard Diophantine: $ xy-\frac{(x+y)^2}{n}=n-4 $

Solve in positive integers $x,y$: $ xy-\frac{(x+y)^2}{n}=n-4 $ $n>4$ is a given positive integer. I cannot even solve in the case $n=5$. I have been able to find $x,y$ and construct $n$ using ...
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29 views

How to solve this class of diophantine forms

I found a class of equations with the following form. $$A (Bm)^k | (Cm^2 + Dm + E)^n$$ $ m \ge 12$ can be any rational number, $n > k$ are natural numbers. $ 0 < A < 1$ is fixed and the ...
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32 views

Solving a general Diophantine Equation

For "normal" equations in one variable we have several techniques for solving equations, such as $\sin(5x) = 5\pi\cos(5x)$ or $\ln(x + 2) = 4$. However, imagine we have the following equations: ...
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43 views

Looking for solutions to $xy^2 = (1 + z)^2 (5 + 8z)$ in integers

I've been reading about Weierstrass equations and shifted Weierstrass equations and Mordell curve and elliptic curves, but so far I haven't been able to transform my equation to any of this type. ...
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29 views

Polynomials that represent a function

Let $D(x,n_1,\dots,n_k) \in \mathbb{Z}[x,n_1,\dots,n_k]$ be a polynomial. Every such polynomial represents a semi-decidable property of natural numbers by $$P(x) :\equiv (\exists n_1,\dots,n_k)\ ...
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35 views

Matrix Multiplication Integer Solution

Given a matrix multiplication and a vector addition. (A,b has rational entries) $$Ax+b$$ how do i get an $x$ for that $Ax+b$ is integer or show that there is not such a solution? $x$ has no ...
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21 views

Diophantine equations (Mordell theorem)

I have a really serious problem with this exercise, I don`t know how I can resolve it. Could you help me? I study in Spanish, so if you don't understand my translation, please ask me... We have the ...
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30 views

Differential Diophantine Equations?

So this is both a question on its own as well as a request for where I can find information on a given topic. Consider Differential Equations in two variables of the form: $$P(Z,Z', Z'' ... Z^{[n]}, ...
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48 views

Diophantine eqn, general solution?

Here's the equation: $$ 4 \left( x^2+y^2-z^2 \right)=\left( 2k+1 \right) \left( x+y-z \right) $$ Is there a nontrivial solution for this in integers? If not, why not? If there is, can a general ...
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12 views

Polynomial Roots of Bivariates

I've got a few polynomials that I am trying to get some results for (shown below). They come from the characteristic equation of a matrix. I have two variables in the polynomials, $\eta$ (which is ...