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.
One 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()
return gp.thue(f.subs({f.variables()[1]:1}), n).sage()
R.<x,y> = PolynomialRing(QQ)
print( solvethue(x^3+y^3,1) )
Asked: 2021-06-15 20:42:36 +0100
Seen: 509 times
Last updated: Apr 16
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It seems not. There is Magma code in the master's thesis Implementation of a Thue-Mahler equation solver by Kyle Hambrook.