On the Diophantine equation \(x^2+2^a\cdot 3^b \cdot 11^c = y^n\)
There are many results in the literature about the equation \(x^2+C=y^n\), where \(C>0\) is either fixed, or only its prime factors are in a fixed finite set. The paper under review deals with the case when the prime factors of \(C\) belong to the set \{2, 3, 11\}. Namely, the authors consider the equation NEWLINE\[NEWLINE x^2+2^a\cdot 3^b\cdot 11^c=y^n NEWLINE\]NEWLINE in nonnegative integer unknowns \((x,y,n,a,b,c)\) and solve it completely under the assumptions \(n \geq 3\) and \(\gcd (x,y)=1\).NEWLINENEWLINEIn the proof the authors distinguish two cases according to \(n \in \{3,4\}\) and \(n \geq 5\) a prime. In the first case the problem is reduced to finding \(S\)-integral points on certain elliptic curves where \(S=\{2,3,11\}\). Then the program package MAGMA is used to compute explicitly the \(S\)-integral points on the above elliptic curves. The case \(n \geq 5\) a prime is handled using the Primitive Divisor Theorem concerning Lucas-sequences.
scientific article; zbMATH DE number 6201655
scientific article