Questions on finding integer/rational solutions of polynomial equations.

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-3
votes
2answers
48 views

All natural numbers $m, n$ such that $m = \sqrt{\frac{1}3A^2 - 3n^2}$ [on hold]

I have $m = \sqrt{\frac{1}3A^2 - 3n^2}$. A is a known integer. How do I find all solutions for what m and n are if both m and n are naturals (round positive numbers)
2
votes
1answer
33 views

Generalization of Erdos-Selfridge

Consider the equation $P(x)=y^d$ where $d \geq 2$ is an integer and $P$ can be written $P(x)=c(x-r_1)(x-r_2)\ldots (x-r_t)$ where $c$ and all the $r_i$ are integers not all equal (some of them can be ...
0
votes
0answers
30 views

Exponential Diophantine equation to solve Project Euler problem

I am currently trying to solve problem 321 on project euler I know that each $n$ must exist such that $$8n^2+16n+1$$ is a perfect square. This is derived from the equation for the swapping of ...
1
vote
3answers
57 views

Multivariable Equation: $4ab=5(a+b)$

Find all Natural roots of the following multi-variable equation : $4ab=5(a+b)$ I have tried many handy candidate solutions and it seems there is no SOLUTION! Indeed we should show that it has no root ...
0
votes
3answers
33 views

Find more integral points on a hyperbola

Let $\mathcal H$ be a hyperbola (in the affine plane) whose defining equation has integers coefficients. Assume that one knows 2 points of $\mathcal H$ with integral coordinates. Is there a way to ...
3
votes
3answers
54 views

Nonnegative Integer solutions of $x+y-xy=0$

I would like to see other methods, besides the one I use here to find all the nonnegative integer solutions of an equation like $$x+y-xy=0$$. This is the one I used: First we note that for $x=1$ ...
39
votes
2answers
2k views
+200

Decomposing polynomials with integer coefficients

Can every quadratic with integer coefficients be written as a sum of two polynomials with integer roots? (Any constant $k \in \mathbb{Z}$, including $0$, is also allowed as a term for simplicity's ...
2
votes
3answers
89 views

Solution of a quadratic diophantine equation

I try to solve the Diophantine quadratic equation: $$X^2+Y^2+Z^2=3W^2.$$ Obviously, there is a non-trivial solution: $(1,1,1,1)$. So I tried to apply Jagy's method: Solutions to $ax^2 + by^2 = cz^2$ . ...
1
vote
0answers
35 views

On Catalan's complete solution to the equation $T^2=U^2+V^2+W^2$

Catalan proved the following: If $t,u,v,w$ are coprime integers such that \begin{equation*} t^2 = u^2 + v^2 + w^2, \end{equation*} then there exist integers $\alpha,\beta,\gamma,\delta$ such that ...
1
vote
1answer
31 views

Diophantine sets

I am reading the following part: Diophantine sets A subset of a power $\mathbb{Z}^n$ of the set $\mathbb{Z}$ of integers is diophantine if it can be written as $$\{\overline{x} \in \mathbb{Z}^n : ...
0
votes
2answers
48 views

A tricky diophantine equation with factorials

I am being unable to solve this diophantine equation. Does anyone have any suggestions. Let $n$ and $m$ both be non-negative integers. Find all solutions to $$n(nm - 2)! = (n!)^m$$ How would one ...
3
votes
0answers
121 views
+50

Integer solutions of a cubic equation

With $\mathrm {gcd}(x,y)=1$ I have the following equation: $$x^3-xy^2+1=N$$ I want to find the integer solutions, given an N, of the variables $x$ and $y$. I have tried factoring the equation into ...
0
votes
1answer
24 views

How do I put up a table of conditions so that $\frac{ab}{cd} \in \mathbb{Z}$ with $abcd \neq 0$

Let $a,b,c,d \in \mathbb{Z}\neq 0$, I am trying to put up an organized method to find the exhaustive list of conditions such that:$$\frac{ab}{bc} \in \mathbb{Z}$$ like a table. How can I do that?
0
votes
1answer
45 views

Diophantine equation - Special form (quadratic)

I am dealing with a series of quadratic diophantine equations that all have the same form: $$A^2x^2 - C^2y^2 + Dx - Ey + F = 0$$ ($A,C,D,E >0$ | $A$ and $C$ have a common factor (or $C=1$) | ...
0
votes
1answer
60 views

prove if $xyk \neq 0$, then: $x^3=3(k+xy)(k-xy-y^3)$ has no integral solutions.

Let $\gcd(x,y)=1, k \in \mathbb{Z}$ and $x \equiv 0 \pmod 3$. Show that if $xyk \neq 0$, then: $$x^3=3(k+xy)(k-xy-y^3)$$ has no integral solutions. Any hints? I keep getting lost in my reasoning.
17
votes
6answers
2k views

Linear diophantine equation $100x - 23y = -19$

I need help with this equation: $$100x - 23y = -19.$$ When I plug this into Wolfram|Alpha, one of the integer solutions is $x = 23n + 12$ where $n$ is a subset of all the integers, but I can't seem ...
3
votes
4answers
50 views

How do I show that we can't write $N=114^n-1$ as sum of $3$ squares for all natural number $n>2$?

I run some computations in wolfram alpha, I see that we can't write :$$N=114^n-1$$ as sum of $3$ squares, then Hop someone who can show me how I do prove that we can't write $N=114^n-1$ as sum of $3$ ...
4
votes
0answers
68 views

Exponential diophantine equation: $2p^2-6p+7=3^n$.

I'm trying to prove that the only integer positive solutions are $(n=1,\ p=1)$ and $(n=3,\ p=5)$. Is there a simple way to do that?
10
votes
4answers
257 views

When is $8x^2-4$ a square number?

I asked an earlier question on when $32x+32$ is a square number (here) and I got a very clear answer. Now I am looking to solve for which $x$ the equation $8x^2-4$ results in a square number. When I ...
0
votes
1answer
75 views

Prove quadratic diophantine has no solutions?

I am trying to prove a quadratic diophantine equation has no integer solutions. Any input would be great, I am interested in the general method for this type of equation so any explanation / link to ...
0
votes
0answers
23 views

The Relationship Between Euler Bricks and Pythagorean Quadruples

I've recently been studying the open problem of finding a Perfect Cuboid (a cube with integral sides and integral face and space diagonals). After doing some research, I came up with a conjecture that ...
2
votes
0answers
19 views

$L$-existential and $L$-diophantine

Could you explain to me the last sentence: "Whenever we want to stress dependence on the language, we will use the self-explanatory terms and $L$-existential and $L$-diophantine" ? What does ...
2
votes
1answer
396 views

What are the necessary and sufficient conditions for a cubic equation to have integers roots

Let's start with Fermat equation with the lowest power, $x^3 + y^3 = z^3$. Now let's set $y = x + a, z = x + b$ with $b > a$ and $a,b$ integers. then the equation becomes $$x^3 + (3a-3b)x^2 + ...
4
votes
3answers
111 views

How can $p^{q+1}+q^{p+1}$ be a perfect square?

How can one find all primes $(p,q)$ such that $p^{q+1}+q^{p+1}$ is a perfect square I considered it $\mod 2$ and found a trival solution . Im curious about an eventual answer Diophantine equations ...
1
vote
2answers
65 views

Finding a Solution to a Equation that Ends up as a Weird Repeating Series

I need to find the solution to this equation that ends up in a weird repeating series. The equation in question is: $$ \ln(y)=\frac{K}{\alpha}+\frac {x^{2}}{2\alpha\sigma}+\frac{\ln(\ln(y))}{2\alpha} ...
0
votes
2answers
48 views

Can I solve for all integer solutions of this diophantine equation?

I do not know much about this subject, but this problem is bothering me. $$ x + 33y = 2399 $$ How can I find the possible integer values of x and y? I know there are two solutions, which I ...
0
votes
0answers
18 views

What is the best time complexity for this case?

I only want to know if the following system has any integer solution or not. Actually, I do not need to know the solution(s), and only need to know the answer of question "Does the system have any ...
0
votes
0answers
27 views

Unique integer solutions for $\frac{1}{x} + \frac{1}{y} = \frac{1}{z} $, with $x,y,z \ne 0 $, given $z$ [duplicate]

What are the unique integer solutions for a given integer $z$ for $\frac{1}{x} + \frac{1}{y} = \frac{1}{z} $, with $x,y,z \ne 0 $? From what I can tell, $x|yz,\ y|xz,\ z|xy$, so $x,y,z$ must ...
1
vote
2answers
67 views

How to solve a bivariate quadratic (not necessarily Pell-type) equation?

Simple Pell equations often have solutions that can be found with little work given certain conditions. These are of the form $x_{n}^{2} - A y_{n}^{2} = \pm 1$. There are harder equations that involve ...
5
votes
2answers
207 views

Pythagorean Quadruples:

Consider the set of integers $x_1, x_2, x_3, x_4$ Such that: $$x_1^2 + x_2^2 + x_3^2 = x_4^2$$ How does one compute all the solutions to this system? I have the following method in place for ...
1
vote
2answers
40 views

Find all natural values n, that $\sqrt{P_{2}(n)}$ is also a natural number

I have a polynomial of the second degree $a\cdot n^2 + b \cdot n + c$ and I need to find out natural numbers $n$, such that $\sqrt{a\cdot n^2 + b \cdot n + c}$ is also a natural number. After ...
3
votes
3answers
82 views

Integer solutions for $x^2+y^2=208$

Which steps I can follow to find the integer solutions for the equation $x^2 + y^2 = 208$?
-3
votes
1answer
48 views

Example of a diophantine polynomial

A diophantine set is a subset of a power $\mathbb{Z}^k$ of the set $\mathbb{Z}$ of integers which can be written as $$\{x \in \mathbb{Z}^k : \exists y \in \mathbb{Z}^m : P(x, y)=0\}$$ where $P$ is a ...
0
votes
0answers
29 views

Single variable diophantine equations with many solutions.

Do significantly non-trivial (read on for clarification), many solution, single variable diophantine equations exist? Diophantine equations are equations where all variables must be integers. The ...
2
votes
6answers
64 views

find all natural solution that satisfy $x^2+y^2 = 3z^2$

I need to find all natural $x,y,z$ that satisfy the following $x^2+y^2 = 3z^2$ $(0,0,0)$ is an answer of course. What I tried: I tried solving with congruences. I know that every square number ...
0
votes
1answer
44 views

question on quadratic equations.

Let $p, q , r$ be distinct real nos such that $ap^2 + bp + c = (\sin(\theta))p^2 +(\cos(\theta))p$ similarly we get a total of three equations if we replace $q$ and $r$ in place of p. where $a, b, c$ ...
3
votes
1answer
52 views

Solve $2b(b-1) = t(t-1)$ as Pell's equation

I know the method of continued fractions to solve the Pell's equation. I need help turning $2b(b-1) = t(t-1)$, with $b, t$ as integers, into the form $x^2 - ny^2 = 1$, if possible. This is a Project ...
1
vote
5answers
123 views

I need Integer Solution to this Equation [duplicate]

I need to know how to solve this equation where x and y are both variables Find integer Solutions. $$ \frac{1}{x} + \frac{1}{y} = \frac{1}{2} $$ from what I know I need at least 2 equations to solve ...
2
votes
1answer
122 views

Show that the equation $y^2 = x^3 + 3$ has infinitely many rational solutions in $x$ and $y$.

Show that the equation $y^2 = x^3 + 3$ has infinitely many rational solutions in $x$ and $y$. I'm really not sure how to go about this question. I've been using trial and error and have not got ...
2
votes
1answer
30 views

System of Congruences with Special Symmetry

Show that the following system of congruences \begin{align} \begin{cases} 3 x^4 - 7 x^2 y^2 - 7 x^2 z^2 - 35 y^2 z^2 \equiv 0 \pmod{p} \\ 3 y^4 - 7 x^2 y^2 - 7 y^2 z^2 - 35 x^2 z^2 \equiv 0 \pmod{p} ...
4
votes
2answers
75 views

Substitutions that transform Fermat Equations to Elliptic Curves

I was reading Chapter 1 of Elliptic Curves - Number Theory and Cryptography by Lawrence C Washington. He was considering Fermat equations $$a^4+b^4=c^4\text{ and }a^3+b^3=c^3.$$ For the 1st equation, ...
4
votes
2answers
14k views

Is it possible to solve for two unknowns from one equation?

Is it possible to solve for two unknowns using only one equation? For example: $x+3y=32$ Where $x$ and $y$ are integers. Thanks :)
0
votes
1answer
35 views

Simple Linear Diophantine Equation - problem with proof.

I'm reading up on diophantine equations and one of the theorems is that "if $x,y$ is any solution of $ax + by = c$, then it is of the form $x_0 +\dfrac{b}{d}t ,\, y_0 - \dfrac{a}{d}t$ where $d = ...
8
votes
2answers
346 views

For which values of $\theta$ does this equation $x^{\cos\theta} +y^{\sin\theta }=1$ have solutions in integers?

For which values of $\theta$ does this equation $$x^{\cos\theta} +y^{\sin\theta}=1$$ have solutions in integers ? Note : $x, y$ integers, $\theta$ is real number. Thank you for your help.
2
votes
1answer
43 views

How do you calculate variables as exponents in a polynomial without a calculator?

Good day The problem is as follow: Find all solutions $(x, y)$, where $x, y \in \mathbb {Z^+}$ to the equation: $$1+3^x=2^y$$ Two solutions are $(0,1)$ and $(1,2)$ but how do you go about ...
3
votes
1answer
71 views

A golden trigonometric diophantine equation

After answering this question I reflected on the identity $$\cos\frac{\pi}{5}=\phi\cos\frac{\pi}{3}$$ and thought of looking for all the quadruplets of positive integers $(a,b,c,d)$ satisfying $$\cos ...
7
votes
2answers
104 views

Does the sum of the reciprocals of composites that are $ \le $ 1

The sum itself: $$ \frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}+\frac{1}{12}+\frac{1}{14}+ \frac{1}{15}+ \frac{1}{39}... \le 1 $$ These are all sums of reciprocals of composites that ...
7
votes
1answer
99 views

Distribution of the sum reciprocal of primes $\le 1$

$$\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{43}+\cdots \le 1 $$ This is an interesting infinite summation. This is very closely resembling my other problem with has to do with the distribution of ...
0
votes
1answer
36 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 ...
4
votes
1answer
174 views

Solve in integers $ (y^3+xy-1)(x^2+x-y)=(x^3-xy+1)(y^2+x-y)$

Solve in integers: $$ (y^3+xy-1)(x^2+x-y)=(x^3-xy+1)(y^2+x-y)$$ My idea: $$\Longleftrightarrow (y^3+xy-1)(x^2+x-y)-(x^3-xy+1)(y^2+x-y)=0$$ $$\Longleftrightarrow ...