Let l and m be consecutive integers, where l represents the floor of the square root of a whole number N that is not a perfect square. Are there any elementary curve structures that might allow l or m to be pinpointed so as to facilitate a non-trivial factorisation of N, without having to divide by values in a factor base?
If so, is it possible to qualify the characteristics of a class of N, for which this may be feasible?
Edit 18/11/21
The idea was to try and keep this question as simple as possible in order to gather responses. Perhaps a little reformulation of the question may keep it alive.
Is there any basic factoring methodology that is characterised by its ability to locate (potentially) either the floor or the ceiling of a whole number as an output of a distinct polynomial? In so doing, does it enable the index point of the discovery to be employed so as to offer up a non-trivial two factor decomposition of the whole number in a routine manner?