Questions on finding integer/rational solutions of polynomial equations.

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0
votes
2answers
42 views

Two diophantine equations with lots of unknowns

Is it possible (tractable) to determine if the following system of equations has any nontrivial solutions (ie, none of the unknowns are zero) in the domain of integers? $$A^2 + B^2=C^2 D^2$$ $$2 C^4 ...
4
votes
2answers
67 views

Need help with a diophantine expression

I'm faced with this problem. Under what conditions is this expression a positive odd integer: $$\frac{2^g(x^2+y^2-z^2)}{x+y-z}$$ where $g,x,y,z$ are nonnegative integers. x and z are odd, and y is ...
14
votes
2answers
232 views

Integer values of $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$?

What are the possible integer values of $$\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$$ where $x$, $y$, and $z$ are positive integers? My suspicion is the the only integer values are $3$ and $5$, the former ...
3
votes
0answers
75 views

Does the special Pell equation $X^2-dY^2=Z^2$ have a simple general parameterization?

In Carmichael's Diophantine Analysis ($\S8$), he notes that the equation $$X^2-dY^2=Z^2 \qquad(\dagger)$$ has a two-parameter solution $$x=m^2+dn^2, \quad y=2mn, \quad z=m^2-dn^2. \qquad(\star)$$ He ...
3
votes
3answers
121 views

Rational points on a surface

I am trying to find rational points on this surface $$ \left( \left( 1-x \right) ^{2}+{y}^{2} \right) \left( \left( 1+x \right) ^{2}+{y}^{2} \right) ={z}^{2}$$ I am actually only interested in ...
0
votes
1answer
27 views

Base convertion and equations

I am studying for an exam in my course, and I will certainly have a question of the kind: In what base is the equation right, for example: 42-3=36 Another ...
0
votes
1answer
36 views

Solutions to diophantine equation $m^2-12mn-3m+2=0$

I am trying to find all solutions to the relativly simple diophantine equation $m^2-12mn-3m+2=0$. I suspect that the only solutions are $n=0$, $m=1$ and $n=0$, $m=2$, but I am currently unable to show ...
1
vote
3answers
65 views

How to show $n^2+m^2 = a^2+b^2 = (n-a)^2+(m-b)^2$ has no nonzero integer solutions?

How do we prove that $$n^2+m^2 = a^2+b^2 = (n-a)^2+(m-b)^2$$ has no nonzero integer solutions? I know two ways to prove this by taking a geometric interpretation but I don't want such a version. How ...
1
vote
2answers
36 views

Equation over Z

Solve the equation $xy+1=3x+y$ over $\mathbb{Z}^2$ Indeed, $$ xy+1=3x+y \Longleftrightarrow (x-1)(y-3)=2 $$ or $ \textrm{Div}(2)=\{k \in \mathbb{Z}/ k|2 \}=\{-1;1;-2;2\}$ Then $(x-1)/2 \implies ...
0
votes
3answers
60 views

Solving different types of Diophantine equation [closed]

In each of the following three equations I need help in finding all solutions in positive integers : i) $\dfrac 1x+\dfrac 2y-\dfrac3z=1 $ ii) $\dfrac 1{x^2}+\dfrac 2{y^2}+\dfrac 3{z^2}=\dfrac 23$ ...
3
votes
1answer
87 views

motivation for talking about torsion points on an elliptic curve

Let's consider elliptic curves over a fixed field $k$. I understand that viewing the set of $k$-rational as an abelian group is interesting and useful, but I am confused about why the torsion points ...
3
votes
1answer
57 views

For what positive integers $p$ and $q$: $(p+1)!+(q+1)!=(pq)^2$

I tried this problem using brute force and got the answers as $(3,4)$ and $(4,3)$,but is there a way to solve this question?
3
votes
0answers
52 views

Solution of a equation in natural number nvolving reciprocal of prime

Let $p$ be a prime and $n$ a natural number . Solve in $\mathbb{N}$ the equation $$\sum_{k=1}^{n}\frac{1}{x^k_k}=\frac{1}{p}$$
1
vote
3answers
59 views

How can I show the complete symmetric quadratic form has no zeros?

The quadratic complete symmetric homogeneous polynomial in $n$ variables $t_1,\ldots,t_n$ is defined to be $$h_2(t_1,\ldots,t_n) := \sum_{1 \leq j \leq k \leq n} t_j t_k = \sum_{j=1}^n t_j^2 + ...
0
votes
1answer
72 views

How prove this diophantine equation $3^x-2^y=k$ have finitely many integral solutions

For any $\forall k\in N^{+}$ show that the diophantine equation $3^x-2^y=k$ have finitely many integral solutions. My try: if $k=2m$,then $$3^x=2^y+k=2^y+2m$$ It is clear there is no integer ...
0
votes
1answer
32 views

Solve equation. sum of negatie powers of two equal to one. Diaphantite.

Is the following correct? Let $\sum_{i=1}^n \frac{1}{2^{x_i}}=1$ where $x_i \in \mathbb{N}_0$ for $i \in \{1,\ldots,n\}$ than the only solutions is $$x_i=n-1, \quad \forall i \in \{1,\ldots,n\}.$$ ...
1
vote
0answers
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. ...
2
votes
1answer
40 views

Why are these the solutions of the diophantine equation?

According to my notes: $$ \begin{align} & \text{ Let } a,b \text{ not both } 0. \\ & ax+by \text{ has a solution iff } (a,b) \mid c \\ & \text{ If } d:=(a,b) \mid c \text{ and } a=d \cdot ...
0
votes
1answer
68 views

Is there any easy way to find the positive integer solutions $(x,y,z)$ from this linear equation?

The equation is like this: $3^x -2^y = 19^z$ It seems that no way to find the solution except using trial and error. I got only one solution: $x=3, y=3, and z= 1$ by using trial and error. But, ...
0
votes
0answers
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 ...
4
votes
1answer
82 views

Integer solutions of $x^y-z^3=2$

Is it an open question to solve $x^y-z^3=2$ in integers (both positive, zero and negative)? If not, what kind of methods the solution requires?
3
votes
1answer
57 views

Is it true that $f(x,y)=\dfrac{x^2+y^2}{xy-t}$ has only finitely many distinct integer values with $x,y$ positive integers?

Prove or disprove that if $t$ is a positive integer, $$f(x,y)=\dfrac{x^2+y^2}{xy-t},$$ then $f(x,y)$ has only finitely many distinct integer values with $x,y$ positive integers. In other words, ...
0
votes
1answer
57 views

The Diophantine equation $x^2 + py + a = 0$.

Let $p$ be an odd prime. Show that the Diophantine equation $$x^2 + py + a = 0$$ with $\gcd(a, p) = 1$ has an integral solution if and only if $\left(\frac{-a}{p}\right) = 1$
0
votes
5answers
44 views

Diophantine Equation (Hint or Help)

Solve the equation below: $19x+29y=1000$ My try: $19x=1000-29y \rightarrow x=\frac {1000-29y}{19} , x \in N \rightarrow 1000-29y\equiv 0\pmod {19} \rightarrow 29y\equiv 1000\pmod{19} \rightarrow ...
0
votes
1answer
29 views

Positive integer solutions of $0=10x³-(2y+5)x²+(y-4)x+76$

To practise my mathematical skills, I often solve some problems in my free time. In this case, one should find every positive integer solution $(x,y)\inℤ^+\timesℤ^+$ of $0=10x³-(2y+5)x²+(y-4)x+76$. ...
2
votes
1answer
90 views

Quadratic Diophantine Equation $x^2 + axy + y^2 = z^2$

I have been reading about this quadratic Diophantine equation of the form $x^2 + axy + y^2 = z^2$ where x, y, z are integers to be solved and a is a given integer. All integral solutions are given ...
-2
votes
1answer
71 views

Project Euler - task №390.

When this task was not clear what the equation to be solved. This equation? $x^2y^2+z^2y^2+x^2z^2=r^2$ in integers. It is not clear, because this equation is quite simple and I do not think that ...
2
votes
2answers
50 views

Solving $a^2-24 a=k^2$ across the integers

$$a^2-24 a=k^2$$ I was trying to solve another problem that essentially boiled down to finding all possible integer values of a such that $a^2-24 a$ was a perfect square, and I thus came up with the ...
1
vote
1answer
51 views

Diophantine equation in four variables

I would like to find a parametric solution for the following diophantine equation: $-4 (1-a_1^2)(1-a_2^2) + (1+a_3^2 -a_1^2 -a_2^2)^2 = a_4^2$ Does such a solution exist? How does one go about ...
0
votes
0answers
51 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 ...
1
vote
4answers
159 views

Solutions to the diophantine equation: $2a^2 + 2b^2- c^2- d^2 = 0$

As suggested on Mathoverflow (http://mathoverflow.net/questions/168536/solutions-to-the-diophantine-equation-2a2-2b2-c2-d2-0) I am transfering this question to math-stackexchange: I am looking for ...
2
votes
1answer
86 views

Diophantine equation: $2 a^2 + 2 b^2 = c^2 + d^2$ [duplicate]

I am looking for integer solutions of the following equation: $$ 2a^{2} + 2b^{2} - c^{2} - d^{2} = 0 $$ Preferentially the solutions should obey $a+b+c+d=0$. By inspection I found the solutions: ...
1
vote
2answers
87 views

Diophantine equation with cubes.

Interested in the solution in general Diophantine equations of the form: $X^3+Y^3+Z^3=3XYZ+q$ $q$ - what some integer. Solutions similar equations can be written. Since this equation is easy, as ...
0
votes
3answers
59 views

Prove the solutions of following equation exists [closed]

$x^2+y^2=z^n$ has a solution in $\mathbb{N}$, for all $n \in \mathbb{N}$. The problem is to show that for every natural number n there exists 3 integers which show the above relation for ...
1
vote
2answers
62 views

Find all integral solutions

Find all the integral solutions of $$x^4+y^4+z^4-w^4=1995.$$ Please elaborate the solution. I tried but can't understand what to do.
3
votes
3answers
92 views

Solve the following equation in positive integers $x$ and $y$

What are the solutions in positive integers of the equation: $${1+2^x+2^{2x+1}=y^2}$$ I tried to factorize the equation but it didn't help much. Clearly $y $ is an odd integer. Substituting $y ...
-1
votes
1answer
106 views

The equation $X^{n} + Y^{n} = Z^{n}$ , where $ n \geq 3$ is a natural number, has no solutions at all where $X,Y,Z$ are intergers.

The equation $X^{n} + Y^{n} = Z^{n}$ , where $n \geq 3$ is a natural number, has no solutions at all where $X,Y,Z$ are integers. My solution: False. Because if we let $X=0 ...
1
vote
0answers
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 ...
1
vote
1answer
33 views

Positive Integer points of $f(x)=\frac{1}{c-\frac{1}{x}}$, where c is fixed

So I am looking for the integer solutions of $f(x)=\frac{1}{c-\frac{1}{x}}$ for fixed $c\in \mathbb{Q}$ i.e. points $(x,f(x))\in \mathbb{N}\times \mathbb{N}$. (The c equals $\frac{4}{n}-\frac{1}{k}$ ...
1
vote
0answers
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 ...
2
votes
3answers
78 views

Finding integral solutions to the equation $x^4-ax^3-bx^2-cx-d=0$

How many integral solutions exist for the equation: $$x^4-ax^3-bx^2-cx-d=0\qquad a\ge b\ge c\ge d\qquad a,b,c,d\in\Bbb{N}$$ I have no idea where to begin even.Please help.
0
votes
0answers
32 views

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 $$ ...
0
votes
0answers
21 views

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 ...
1
vote
1answer
42 views

non-negative solutions with upper boundary, in diophantine equation

I wanted to find out in how many ways I can do something, but I don't know combinatorics enough. Can you help me and show or give advice, what we should do in such situations as presented below? Let's ...
0
votes
3answers
39 views

Equation involving power of two

I want to show that the equation $2^x - 1 = 3^y$ does not have any positive integer solutions except for $ x = 2 , y = 1$ . Is it possible to prove the assertion using binary representation of powers ...
0
votes
1answer
50 views

Is there any solution to this quadratic Diophantine equation?

Can one find all positive integer triplets $(x,y,z)$ satisfying this parametric equation : $$ax^2 + (1-a)x + by^2 + (1-b)y = cz^2 + (1-c)z$$ Here, $a, b, c$ are positive integers greater than $1$. ...
0
votes
1answer
31 views

Random nonegative solution of multivariate linear Diophantine equation

Consider a diophantine equation in n variables: $a_1x_1+a_2x_2+...+a_nx_n=k$ All $a_i$'s, $x_i$'s and $k$ are restricted to non-negative integers $\mathbb{Z^+}$. (note that because of domain ...
1
vote
3answers
90 views

Curves triangular numbers.

Sometimes you have to deal with this equation: $X^2+aX+Y^2+bY=Z^2+cZ$ $a,b,c$ - integer coefficients. I wrote below - to start a particular solution of Diophantine equations. To do this, use the ...
6
votes
0answers
121 views

Solution of $\large\binom{x}{n}+\binom{y}{n}=\binom{z}{n}$ with $n\geq 3$

I found this question in an old problem set. There's no hint or solution mentioned. For $n \geq 3$, prove or disprove the existence of $(x,y,z) \in \mathbb N^3, ...
4
votes
3answers
212 views

Positive integer solutions of $a^3 + b^3 = c$

Is there any fast way to solve for positive integer solutions of $$a^3 + b^3 = c$$ knowing $c$? My current method is checking if $c - a^3$ is a perfect cube for a range of numbers for $a$, but this ...