Questions on finding integer/rational solutions of polynomial equations.

learn more… | top users | synonyms

2
votes
0answers
33 views

Integer solutions of $a^3+2a+1=2^b$

What are the solutions in integers of $a^3+2a+1=2^b$? [Source: Serbian competition problem]
2
votes
1answer
19 views

Coprime numbers and equations

Suppose $~m~$ and $~n~$ are coprime and both of them are greater than one. Is it right that equation $~mx + ny = (m-1)(n-1)~$ has solutions over non-negative integers? For example $~ (x,y) = (6,0) ...
0
votes
1answer
31 views

Parametrization of $ax^2+bxy+c=0$

Can I just fix $y=t$ and use quadratic formula to get the rational points of the diophantine $$ax^2+bxy+c=0?$$ or is there another method? I feel like I am turning in circles with the quadratic ...
0
votes
1answer
18 views

Rational points of $ax^2+by^2=z^r$, $r $ odd integer.

I am trying to find the rational points of:$$ax^2+by^2=z^r$$ I am aware that:$$(u^r-2^{r-2}v^r)^2+(2uv)^r=(u^r+2^{r-2}v^r)^2$$ How can I deduct the results?
5
votes
1answer
154 views

$2 \times 3 = 5+1$ and $2+3 = 5 \times 1$. When else can we switch the operators like this? [duplicate]

I noticed the following: $$2 \times 3 = 5+1$$ If you switch the operators, it is still true: $$2+3 = 5 \times 1$$ There is another obvious/trivial example where you can swap the operators: $$2\times 2 ...
2
votes
0answers
46 views

integral solutions of $ ax^2+by^2=c$ [on hold]

Let $a,b,c,x,y$ be all non-zero positive integers, $\gcd(a,b,c)=1$, find the integral solutions of:$$ ax^2+by^2=c$$ Any hint?
3
votes
2answers
52 views

If $(a^2+b^2) \mid (c^2+d^2)$ and $\gcd(a,b)=\gcd(c,d)=1$ and $\gcd(a,c)>1$, what can be said about the components?

While working on a divisibility problem in integers $a,b,c,d$, with $\gcd(a,b)=\gcd(c,d)=1$, I've come up against the hypothetical condition $$ (a^2+b^2) \mid (c^2+d^2), \tag{$\star$} $$ where, also ...
5
votes
0answers
61 views

Positive integer solutions to $x^4+y^7=z^9$

A while ago, a maths teacher gave me this problem: find solutions to $x^4+y^7=z^9$ with $x,y,z>0$. I found $(2^{56})^4+(2^{32})^7=(2^{25})^9$. In general, if $k=8+9l$ then ...
0
votes
2answers
66 views

Can Legendre's theorem really help solve this equation: $ax^2+by^2=cz^2$?

let $a,b,c,x,y$ be non-zero positive integers such that $$\gcd(x,y,z)=1$$ find all the non-trivial integral solutions of the diophantine equation:$$ax^2+by^2=cz^2$$ I know that the Legendre's theorem ...
1
vote
2answers
43 views

How many natural solutions does the equation $x^2 - c y^2 = 1$ have?

How many natural solutions has equation $x^2-cy^2=1$ depends on value of $c$ . I think I've seen this problem somewhere as a theorem but I can't remember where .
0
votes
3answers
46 views

List one of the ways in which Mario could buy the stars and comets. Note: Mario needs to spend all of his gold coins

Mario has 773500 gold coins to purchase a number of stars and comets. Each star costs 299 gold coins, and each comet costs 208 gold coins. If the number of stars that Mario buys is at least twice the ...
0
votes
1answer
32 views

Linear Diophantine Equation Question

Mario has $773500$ gold coins to purchase a number of stars and comets. Each star costs $299$ gold coins, and each comet costs $208$ gold coins. If the number of stars that Mario buys is at least ...
1
vote
1answer
77 views

Find all positive integers s.t. $10^m-8^n=2m^2$

Find all pairs of positive integers $(m,n)$ such that $10^m-8^n=2m^2$
1
vote
1answer
62 views

Triplets of distinct integers > 1 that return integer values.

If $(A, B, C)$ are distinct integers $> 1$, and $$f(A, B, C) = \frac{\frac{A^2-1}{A} + \frac{B^2-1}{B}}{\frac{C^2-1}{C}},$$ then for what (if any) triplets $(A, B, C)$ is $f(A, B, C)$ an integer? ...
0
votes
0answers
26 views

Can you write a variable as the sum of two variables?

I was seeing this question and, in the develop of an answer, a question arised: I have a variable $k\in [1,40]_{\Bbb N}$ and I want write it as the sum of one variable and something more with the ...
1
vote
1answer
42 views

How to solve the system $\frac{35-12b}{a-b}= \frac bk,\frac{12a-35}{a-b}= \frac {a}{1-k}$

I am trying to solve the system of diophantine equations: $$ \begin{align*} \frac{35-12b}{a-b} &= \frac bk \\[6pt] \frac{12a-35}{a-b} &= \frac {a}{1-k} \end{align*} $$ Where $a-b\ne 0,$ and ...
1
vote
2answers
38 views

Equation with tangent and powers

I need to solve this equation for x: $$2000 \sigma = 1 - \frac{20x}{\pi^2x^2 + 100} - \frac{2 \arctan(\frac{\pi x}{10})}{\pi} $$ $\sigma$ is a known value. I need to solve this for $ \sigma = ...
0
votes
1answer
36 views

Return to sum of powers question.

Previously I had asked a question about a Diophantine equation linked here. I have come back to think about this question but in a different manner. So here is the set up: Let A and $a_i$ be ...
0
votes
3answers
71 views

Prove that $\gcd(abc + abd + acd + bcd, abcd) = 1$

Let $a, b, c, d \in \mathbb Z$. Prove that $\gcd(abc + abd + acd + bcd, abcd) = 1$ if and only if $a, b, c, d$ are pairwise relatively prime. I am very confused as to how I should even start this ...
1
vote
2answers
50 views

Efficient software implementation of $x^2+3y^2=N$

I would like to implement a solver (in C) for the Diophantine equation $x^2+3y^2=N$ for non-negative integers $\{x,y\}$ and positive integer $N$. I have read online that one has to prime factorize N ...
1
vote
1answer
60 views

When is the equation $x^2-d^n y^2 = -1$ solvable?

My goal is to prove or disprove that if $x^2-dy^2=-1$ is solvable, then $x^2-d^ny^2 = -1$ is solvable for every odd $n \geq 1$. I do know that the former is solvable if and only if the continued ...
0
votes
2answers
79 views

Solving homogeneous quaternary quadratic Diophantine equation

Given the equation $w^2+x^2+y^2+z^2=wx+wy+wz+xy+xz+yz$, how does one systematically enumerate all non-negative integer solutions $\{w,x,y,z\}$?
2
votes
1answer
73 views

Diophantine Equation: $x^2 + 3y^2 = 11z^2$

I am having difficulty solving the following problem: Prove rigorously that there is no integer solution for the Diophantine Equation $x^2 + 3y^2 = 11z^2$ except when $x = y=z = 0$. ...
0
votes
1answer
31 views

Prove a system of simultaneous Diophantine equations has no solution.

I've been asked to show that the system of simultaneous Diophantine equations has no solutions: $3x+6y+z=3$ $12x+3y+2z=5$ I don't even know how to approach this problem, any help would be ...
1
vote
3answers
42 views

Diophantine Equation with 3 Variables

Find all solutions to $2x + 3y + 4z = 5$. I know how to do it with two variables, but I'm confused on how to start this with three variables.
0
votes
1answer
30 views

Diophantine equation $(x^2-1)^2-4y^2=0$

We have the diophantine equation $(x^2-1)^2-4y^2=0$, where $x$ and $y$ are positive integers. Is the only solution $x=1$, $y=0$ or can there be infinitely many solutions?
1
vote
0answers
30 views

Siegel's theorem and singular curves

I notice that often Siegel's theorem (there are only finitely many integral points on a curve of genus greater than 0) is stated with the requirement that the curve be smooth. Other times the ...
0
votes
1answer
32 views

Diophantus problem

I was given following problem as an example of early mathematics with the solutions. But it seems i can't understand from where they are getting the 35z^2 = 5 from in the solutions. Could someone ...
-2
votes
0answers
32 views

how to find GENUS of a given curve [duplicate]

Could you please help me to find GENUS of $x^2 -x + y- y^5 = 0$ hyper-elliptic curve. Also, explain how to apply FALTINGS theorem to find rational roots. high regards and many many thanks...
0
votes
1answer
32 views

Can the fundamental solution of a Pell equation be “triangulated” given multiple known solutions?

In this question about “descent” given a single Pell solution, Will Jagy gave the [accepted] answer that, for a Pell equation $$ U^2 - dV^2 = \pm 1, \tag{$\star$} $$ there is no way to determine ...
-1
votes
0answers
71 views

The curve with irreducible and has genus 2

Sketch the curve $f(x,y) = x^2 -x + y- y^5$, and prove that it is irreducible and has genus $2$. And, how to conclude that, $f(x,y)$ has only finitely many rational points as well as only finitely ...
0
votes
2answers
21 views

Solve this non-linear diophantine equation?

How do you go about systematically solving a Diophantine equation of this form : $217x^2 + 496y^2 = 15872$ ? I found that $\gcd(217, 496) = 31$ and reduced that equation to $7x^2 + 16y^2 = 512$ ...
-2
votes
1answer
54 views

Solve $ (u^n-v^n)=p(u-v)^2$ [closed]

Let $n, u, v,p $ be non-zero positive integers, if $\gcd(u,v)=1$ and $u-v>1$, find all the nontrivial solutions of the Diophantine equation:$$ (u^n-v^n)=p(u-v)^2$$
0
votes
0answers
44 views

Woking Heron's Formula In Reverse

I'm writing a program to generate randomized Heron's Formula word problems. I need to figure out how to work the problem in reverse so that the answer will come out to an integer. As an example, if I ...
0
votes
1answer
34 views

Buying dogs and Cats and mice

This problem is only considering positive integer solutions: You must spend exactly 100 and purchase exactly 100 animals. Each dog costs 15 and each cat costs 1 and each mouse costs .25. How many of ...
18
votes
1answer
663 views

Does an elementary solution exist to $x^2+1=y^3$?

Prove that there are no positive integer solutions to $$x^2+1=y^3$$ This problem is easy if you apply Catalans conjecture and still doable talking about Gaussian integers and UFD's. However, can this ...
0
votes
1answer
17 views

FLT and non-maximal orders

The ring of integers of a cyclotomic field $\mathbb{Q}(\zeta_n)$ is the unique maximal order of that field. Kummer's attempt at proving FLT fails for prime exponents which are irregular, i.e. divide ...
0
votes
2answers
28 views

Diophantine Equation Related To Triangles

a,b and c are the sides of a triangle and a, b, c are integers. I need to solve the following Diophantine equation for positive integral values of k. $bc(b+c-a) = k^{2}(a+b+c)$ I think some ...
1
vote
2answers
27 views

Find integer solutions of $(1) xy=2x+2y$ and $ (2)xy=2x+y.$

I've tried this for the first one:$xy=2(x+y).$ Therefore either x or y is divisible by 2. And I'm totally stuck on the second. How to solve these?
5
votes
2answers
235 views

Prove that there are infinitely many integer solutions to a diophantine equation

Prove that there are infinitely many integer solutions to the diophantine equation: $(x-y)^7 = x^3y^3$
0
votes
2answers
45 views

Number of integer solutions of two similar equations

Find the number of integer solutions of: (a) $${1\over\sqrt{x}}+{1\over\sqrt{y}} = {1\over\sqrt{20}}$$ (b) $${1\over\sqrt{x}}+{1\over\sqrt{y}} = {1\over\sqrt{2014}}$$ I know the ...
2
votes
0answers
48 views

Quadruples of integers with $20^x + 14^{2y} = (x + 2y + z)^{zt}.$

Determine all quadruples $(x,y,z,t)$ of positive integers such that $$20^x + 14^{2y} = (x + 2y + z)^{zt}.$$ We can check that $20+14^2=216=(1+2+3)^3$. But how can we check if there are other ones?
0
votes
3answers
42 views

Show that there are only trivial solutions

How can I show that the only solutions of the diophantine equation $x^2+y^2=1$ are the trivial ones: $(x,y)=(0,1), (0,-1), (1,0), (-1,0)$ ? That's what I thought: $$x \equiv 0,1 \pmod 2 \Rightarrow ...
0
votes
1answer
24 views

Is there also an other way, to prove that the diophantine equation has no solution?

I am looking at the following exercise: The diophantine equation $y^2=x^3+7$ has no solution. If the equation would have a solution, let $(x_0,y_0)$,then $x_0$ is odd. ( If $x_0$ is even, $x_0=2k ...
0
votes
0answers
18 views

Diophantine equation with three variables

I'm trying to solve a diophantine equation with $3$ variables. The problem can be written as a system of equations: $\begin{cases} 0.5x + 4y + 9z &= 97 \\ x+y+z &= 34 \end{cases} \implies ...
2
votes
2answers
100 views

What is the density of solutions for Pythagoras' and Fermat's equation $x^2+y^2=z^2$

It is now proved that, for integer $n\geq 2$, the equation $x^n+y^n=z^n$ has integer solution only when $n=2$. When $n=2$, this equation has an infinity of solutions. My question is whether there is ...
2
votes
1answer
80 views

Homogeneous diophantine equation

Here I have a diophantine equation featuring a homogeneous polynomial: $$x^2+5y^2+34z^2+2xy-10xz-22yz=0; x, y, z\in\mathbb{Z}$$ I have no idea how to approach this, I've tried various substitutions ...
7
votes
2answers
182 views

3 Variable Diophantine Equation

Find all integer solutions to $$x^4 + y^4 + z^3 = 5$$ I don't know how to proceed, since it has a p-adic and real solution for all $p$. I think that the only one is (2, 2, -3) and the trivial ones ...
2
votes
0answers
52 views

Is Legendre’s solution of the general quadratic equation the only one?

Legendre famously solved the general quadratic equation $$ ax^2+bxy+cy^2+dx+ey+f=0 $$ by noting that \begin{equation*} 4a(b^2-4ac)(ax^2+bxy+cy^2+dx+ey+f) = 0 \tag{$\star$} \end{equation*} along with ...
1
vote
3answers
97 views

Non-linear diophantine equation

Let $k$ and $n$ be positive integers and $y(n-x)=(k+nx)$. What is the condition of $k$ and $n$ such that there exist positive integers $x, y$ as the solution of $y(n-x)=(k+nx)$?