Questions on finding integer/rational solutions of equations.

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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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27 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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60 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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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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60 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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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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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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117 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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126 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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310 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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191 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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80 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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Diophantine equation $ax + by = c$ has an integer solution $x_0, y_0$ if and only if $\gcd(a,b)|c$

Let $a,b,c$ be positive integers. Verify that Diophantine equation $ax + by = c$ has integer solution $x_0, y_0$ if and only if $GCD(a,b)|c$. Attempt Diophantine $ax + by = c$ has integer solution ...
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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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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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29 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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45 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 ...
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49 views

System of quadratic diophantine equations 2

I am looking for a way to simultaneously transform the following four expressions into perfect squares, $1+x_1^2, 1+x_2^2, 1+x_3^2, x_1^2+x_2^2+x_3^2$, i.e. I want to find a rational parametrization ...
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Least squares over the integers (diophantine least squares?)

I have the following problem and I do not even know under which mathematical field I should look for an answer, so any hint is highly appreciated: Let S be the ellipsoid $$ ...
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Explanation for a simple comparison

Ok, Yesterday I started to learn how to solve problems with comparisons, but I couldn't understand one thing of the "solve algotithm". Here is a part from a solve from a simple example problem ...
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Diophantine like philosophy for computing trigonometric functions with approximation around intervals

I noticed that diophantine expressions are great to approximate constants or simple functions, as far as I know, they are not so great when it comes to approximate and compute transcendental functions ...
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32 views

Nature of roots of a quadratic equation

If $L+M+N=0$ and $L, M, N$ are rationals the roots of the equation $( M+N-L) x^2 +(N+L-M)x + (L+M-N) =0$ are $a)$ Real and irrational $b)$ Real and rational $c)$ Imaginary and equal ...
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20 views

Matrices and diophantine equations

Let A be $mxn$ matrix with integral elements, and let r denote the rank of A. For $1 \leq k \leq r$, let $d_k (A)$ be the gcd of the determinants of all $kxk$ matrices. This is called determinantal ...
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Diophantine equation with 6 variables.

In this equation: $aX^2+bY^2+cZ^2=abc+2XYZ+F$ $F$ - integer number given by the condition of the problem. A rather Tran decision: $a=(2pk-p^2+p-k^2)((t-s)^2-1)+2tsk+p(1-t^2)-(2k-p+1)s^2+F$ ...
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Find the general solution to diophantine equation $-221x + 187y - 493 = 0$

I have to find the general solution to $$-221x + 187y - 493 = 0$$ The main issue, I'm figuring out if I have found the general solution or not. Below, are my steps: The $\gcd{(-221,187)} = 17$ and ...
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25 views

prove that $ax+by=b+c$ has solution in $\mathbb Z$ if and only if $ax+by=c$ has solution in $\mathbb Z$

Let $a,b,c,d\in \mathbb Z$ prove that: a)$ax+by=b+c$ has solution in $\mathbb Z$ if and only if $ax+by=c$ has solution in $\mathbb Z$ b)$ax+by=c$ has solution in $\mathbb Z$ if and only if ...
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33 views

Evaluation of certain trigonometric sums

In trying to approximate the number of solutions to the equation $3^n - 2 = k^2$ for positive integers $n, k$, I tried to use the circle method. In doing so, I had to bound the trigonometric sum for ...
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39 views

Divisors of Pell Equation Solutions

Let $d > 0$ be square-free. Let $\epsilon = x_0 + y_0 \sqrt{d}$ be the minimal solution to the Pell's equation $x ^ 2 - d y ^ 2 = 1$. Let $x + y \sqrt{d} = \epsilon ^ l, l \geq 1$ be a solution. ...
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57 views

Finding all integral solutions of a positive definite quadratic equation

Let $q(x_1,\ldots,x_n)$ be an integral positive definite quadratic form. For $d\in\mathbb{N}$ the equation $$q(x_1,\ldots,x_n)=d$$ has a finite number of integral solutions. Is there an algorithm to ...
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61 views

Diophantine equation using Pell's equation

I asked this question some days ago: Is there a way to find for which A the system $X^2+Y^2=Z^2+T^2+1$ $XZ−YT=A$ has only one solution in positive integers? Looking for the solution of the ...
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72 views

Diophantine equation: x^2+2=y^3

just so this doesn't get deleted, I want to make it clear that I already know how to solve this using the UFD $\mathbb{Z}[\sqrt{-2}]$, and am in search for the infinite descent proof that Fermat ...