Tagged Questions
1
vote
3answers
64 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
40 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?
...
0
votes
0answers
11 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 ...
0
votes
0answers
102 views
prove cubic equation has no positive integer root
prove $q_1t^3+(k_2-1)t^2-k_2((q_1^2-1)k_1+1)^2=0$ has no Positive integer root, t is variable , $q_1$ is constant and $k_1,k_2$ are parameter
$q_1>0, k_1>0, k_2>0$, and all characters ...
3
votes
1answer
74 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
143 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 ...
32
votes
1answer
985 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
56 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
288 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
172 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
113 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
101 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
234 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 ...
11
votes
2answers
502 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
188 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
763 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: ...
