ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Mon, 21 Jun 2021 20:19:41 +0200Thue-Mahler equationhttps://ask.sagemath.org/question/57577/thue-mahler-equation/Let $F \in \mathbb{Z}[X,Y]$ be an homogenous polynomial. An equation of the form
$$ F(X,Y) = d p_1^{a_1}\cdots p_n^{a_n},$$
with $p_1, \cdots, p_n$ are prime numbers and $d \in \mathbb{Z}$ is called a Thue-Mahler equation. Is there an explicit function in sage that can solve this kind of equations even if is a simpler case such as F(x,y) = p^a ?. I have been looking for this with no result.
Tue, 15 Jun 2021 20:42:36 +0200https://ask.sagemath.org/question/57577/thue-mahler-equation/Comment by rburing for <p>Let $F \in \mathbb{Z}[X,Y]$ be an homogenous polynomial. An equation of the form
$$ F(X,Y) = d p_1^{a_1}\cdots p_n^{a_n},$$
with $p_1, \cdots, p_n$ are prime numbers and $d \in \mathbb{Z}$ is called a Thue-Mahler equation. Is there an explicit function in sage that can solve this kind of equations even if is a simpler case such as F(x,y) = p^a ?. I have been looking for this with no result. </p>
https://ask.sagemath.org/question/57577/thue-mahler-equation/?comment=57578#post-id-57578It seems not. There is Magma code in the master's thesis [Implementation of a Thue-Mahler equation solver](https://open.library.ubc.ca/cIRcle/collections/ubctheses/24/items/1.0072341) by Kyle Hambrook.Tue, 15 Jun 2021 21:29:41 +0200https://ask.sagemath.org/question/57577/thue-mahler-equation/?comment=57578#post-id-57578Answer by Max Alekseyev for <p>Let $F \in \mathbb{Z}[X,Y]$ be an homogenous polynomial. An equation of the form
$$ F(X,Y) = d p_1^{a_1}\cdots p_n^{a_n},$$
with $p_1, \cdots, p_n$ are prime numbers and $d \in \mathbb{Z}$ is called a Thue-Mahler equation. Is there an explicit function in sage that can solve this kind of equations even if is a simpler case such as F(x,y) = p^a ?. I have been looking for this with no result. </p>
https://ask.sagemath.org/question/57577/thue-mahler-equation/?answer=57652#post-id-57652One can rely on the Thue equation solver present in PARI. Here is a sample code for solving the equation $f(x,y)=n$:
def solvethue(f,n):
assert f.is_homogeneous()
t = gp.thueinit( f.subs({f.variables()[1]:1}) )
return gp.thue(t,n)
R.<x,y> = PolynomialRing(QQ)
print( solvethue(x^3+y^3,1) )Mon, 21 Jun 2021 20:19:41 +0200https://ask.sagemath.org/question/57577/thue-mahler-equation/?answer=57652#post-id-57652