Are Results Found Of An Elliptic Curve By SageMathCell Proven (does There Exists No More Solutions)?
Answer : The documentation of integral_points cites the algorithm found in Henri Cohen, Number Theory, Vol. I: Tools and Diophantine Equations. GTM 239, Springer, 2007. You can read the implementation e.g. by entering E.integral_points?? into a SageMath session. The algorithm finds all the integral points, but it depends crucially on having computed the Mordell-Weil group. On this topic, the documentation reads: mw_base - list of EllipticCurvePoint generating the Mordell-Weil group of E (default: ‘auto’ - calls self.gens() ) Note: The complexity increases exponentially in the rank of curve E. The computation time (but not the output!) depends on the Mordell-Weil basis. If mw_base is given but is not a basis for the Mordell-Weil group (modulo torsion), integral points which are not in the subgroup generated by the given points will almost certainly not be listed. In turn, the documentation of gens() , about its optional argument proof , says: p...