0
votes
1answer
36 views

Using existential quantifiers to turn equalities into inequalities

I have a formula of the form $f(x)^2 = 0$, where $x \in \mathbb{R}^n$ and $f$ is a Diophantine polynomial. I am wondering if there is a general way to produce a formula of the form $\exists y \in ...
3
votes
2answers
76 views

Solve diophantine equation $x^2 - 2y^2 = x - 2y$

Thanks to internet, I found and understand how to solve diophantine $x^2 - Dy^2 = 1$. Now I would like to solve the following diophantine equation : $$x^2 - 2y^2 = x - 2y$$ but I don't know how to do ...
0
votes
0answers
17 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) ...
0
votes
0answers
32 views

Solve the following symmetric equations:

Solve the following equations: \begin{align}\left\{ \begin{array}{c} x_1+x_2+x_3+x_4=6 \\ x_1{}^2+x_2{}^2+x_3{}^2+x_4{}^2=10 \\ x_1{}^3+x_2{}^3+x_3{}^3+x_4{}^3=18 \\ ...
3
votes
1answer
124 views

On $x^3+y^3+z^3 = 1$ and a Pell equation

Given, $$(1-ac+bc)^3 + (a+c^2-ac^3)^3 + (ac^3-b-c^2)^3 = 1\tag{1}$$ where, $$a,b,c,r = 12qrt,\;\; 3(q-r)(3q+r)t,\;\; 3s^2t^2,\;\; p-18qs^3t^3$$ then $(1)$ holds true if $p,q,s,t$ satisfies, ...
3
votes
1answer
117 views

Testing polynomial equivalence

Suppose I have two polynomials, P(x) and Q(x), of the same degree and with the same leading coefficient. How can I test if the two are equivalent in the sense that there exists some $k$ with ...
1
vote
1answer
22 views

Show how the Diophantine sets are closed under concatenation.

It follows easily from Matiyasevich's Theorem that the Diophantine sets are closed under concatenation. I am trying to figure out the mechanism by which they are closed under concatenation. In other ...
0
votes
0answers
70 views

How to find the nearest power product?

We call power products the integers of the form $x^m*y^n$ for $m$, $n$, $x$, $y \in \mathbb{N}$. Given a number $u \in \mathbb{N}$, find the closest power product. How does one solve this ...
5
votes
3answers
199 views

Can anyone show me, how to solve these system of Equations:

$$\begin{align*} x+y+z &= 2 \\ (x+y)(y+z)+(y+z)(z+x)+(z+x)(x+y) &= 1 \\ x^2(y+z)+y^2(z+x)+z^2(x+y) &= -6 \end{align*}$$ Can anyone explain me the solution. I asked it in mathoverflow but ...
1
vote
3answers
190 views

Way to show $x^n + y^n = z^n$ factorises as $(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n$

For odd $n$ the Fermat equation $x^n + y^n = z^n$ factorises as $$(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n,$$ where $\zeta = e^{2 \pi i/n}$. I tried seeing this was true by multiplying ...
0
votes
1answer
99 views

find all positive integers for a given diophantine equation involving 4 or 7 variables

Given equation: Ap + Bq + Cr + Ds + Et + Gu + Vg = K; (Eq. in 7 variables); suppose we have A, B, C, D initialize with = 1,2,5,10,20,50 and 100 respectively; and K = 50000; How do we solve it? ...
2
votes
1answer
34 views

Which diophantine polynomials generate these diophantine sets?

Via Matiyasevich's Theorem, it is easy to prove that the following sets are diophantine: $\{k\}$ $\{0, 1, \dots, k-1, k+1, k+2, \dots \}$ $\{0, 1, \dots, k\}$ $\{k+1, k+2, \dots\}$ Number 1 is ...
3
votes
1answer
110 views

Diophantine equations and Groebner bases

I'm trying to teach myself the basics of algebraic geometry and have run into something that I don't understand. I know that the problem of deciding whether a Diophantine equation $P(\vec{x}) = 0$ ...
0
votes
2answers
169 views

Quadratic equations

Does anyone know how to find integer solutions of the quadratic equation $$y^2+y+z=f$$ where $z$ is a fixed odd prime or $1$ and $f$ is a fixed odd prime greater than $3$? This problem arose from ...
34
votes
1answer
1k views

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 ...
0
votes
1answer
83 views

Minimal bivariate diophantine equation solution space

I am facing the following type of diophantine equations: $$ axy + bx + cy + d = 0 $$ Where $a$, $b$, $c$, $d$ are integers and solutions for $x$, $y$ in the integers are seeked. If $a=0$ one can ...
13
votes
1answer
347 views

Roots with equal fractional parts

Question. ¿Does there exist an integer $n>1$ such that there exist positive integers $a,b$ such that $\{\sqrt[n]{a}\}=\{\sqrt[n]{b}\},a\neq b$ and $a$ and $b$ aren't perfect n-th powers? ( $\{x\}$ ...
1
vote
2answers
319 views

Solution of Diophantine equation for polynomials

I'm trying to solve this polynomial Diophantine equation for $R$ and $S$: $ AR + BS = G $ where $A(x) = a_0x^{n_a} + a_1x^{n_a-1} + a_2x^{n_a-2} + ... + a_{n_a}$ $B(x) = b_0x^{n_b} + b_1x^{n_b-1} + ...
4
votes
1answer
146 views

A simple-looking diophantine equation

Consider the diophantine equation $Q(x,y,z)=0$, where $x$, $y$ and $z$ are nonnegative integer unknowns and $$ Q(x,y,z)=x^3 + (-2y + 2)x^2 + ((z - 6)y + (2z + 1))x + ((2z - 4)y + 3z) $$ Since the ...
2
votes
1answer
116 views

Is there no univariate integer polynomial that takes on the same positive values as the multivariate polynomial $x^2+y^2$?

Is there no univariate integer polynomial that takes on the same positive values as the multivariate polynomial $x^2+y^2$? The values are numbers such that each prime factor of the form $4k+3$ occurs ...
1
vote
1answer
409 views

Discriminant of derivative of cubic equation being a perfect square

Is it possible for the discriminant of the first derivative of a cubic polynomial (x+a)(x+b)(x+c), where a, b and c are distinct non-zero integers (i.e. Discriminant[d((x+a)(x+b)(x+c))/dx] in ...
12
votes
2answers
672 views

What is the simplest ellipse that goes through exactly 13 lattice points?

The ellipse $-30 x + 3 x^2 - 10 y - 3 x y + 4 y^2$ goes through exactly 11 lattice points. Another such ellipse is $4 - 30 x + 2 x^2 - 5 y - x y + 3 y^2$. What is the simplest ellipse that goes ...
3
votes
1answer
226 views

Factoring a trivariate polynomial

I would appreciate some help with factoring a trivariate polynomial. The polynomial in question is $$p(x,y,z)=a_1 x^7+a_2 x^5y+a_3 x^3y^2+a_4 xy^3+a_5 x^4z+a_6 x^2yz+a_7 y^2z+a_8 xz^2,$$ where the ...
4
votes
10answers
771 views

Polynomial satisfying $p(x)=3^{x}$ for $ x \in \mathbb{N}$

Let $p(x)$ be a polynomial with integer coefficients, which is not constant. Then is this condition possible: $$p(x)=3^{x}$$ whenever $x \in \mathbb{N}$. Motivation: ...