My current approach to integer factorization, elements of which I have described in the last several days on this blog, involves a process I discovered last night that surprised me with its beauty and simplicity: It is a two-dimensional method, straightforwardly described and probably easily programmed, for visualizing quadratically-distributed numbers, namely consecutive perfect squares as well as the remainders r_{0}, r_{1}, r_{2}, … that result from finding the Ceiling Square c^{2} of a positive odd integer m and successively forming the equalities m=c^{2}-r_{0}, m=(c+1)^{2}-r_{1}, m=(c+2)^{2}-r_{2} and so on. The method is such that the quadratically-distributed values of both the perfect squares and the remainder sequence each fall evenly spaced along straight lines. In the case of the perfect squares, they plot in two dimensions redundantly to form a family of intersecting straight lines, while the remainder series I select to intersect along and between the spaces of redundant two-dimensional distribution of perfect squares in its own uniform, straight line.

The approach that I intend to take, and that I hope will give good insight in the days to come, is to characterize the family of lines using analytic geometry techniques, along with the rise and run of the linear remainder sequence that accompanies it for an arbitrary m that gives a particular c, and r_{0}, and thus to derive formulaically when the remainder line intersects one of the family of perfect square lines at an integer point.

An illustration is in order:

Yes, as with many visualizations and conceptualizations I have had before, this might work out to have no value in simplifying integer factorization. But I do not yet know this to be the case, and am pressing on in hope.

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