In an effort to elicit potentially useful patterns from my mathematical study, and particularly from the Zone Grid idea I discovered, it would seem that I have come up with another way to help me visualize what is going on with the Remainder Series associated with the Ceiling Square and its Ascents. This one is geometric.

When dealing with an odd positive composite number m, with more factorizations than the trivial m=1*m, I have noticed that the zone lines and the Ascents produce on the Zone Grid a trapezoid. In fact, for a value of m with more than four divisors, there will be more than one trapezoid. But there will be a factorization of composite m with least ascent. The integer point on the zone boundary of the Zone Grid corresponding to this factorization, when known, together with the point on the zone boundary that corresponds to the Trivial Factorization, both located on the Remainder Line, are the endpoints of a line segment that is the diagonal of a trapezoid made by the two parallel lines corresponding to the two x values of the factorizations’ Ascents, crossed by the two zone boundaries. I can then take the area of this trapezoid based on Cartesian Plane units. Since there may or may not be other integer zone boundary intersections, and thus smaller trapezoids that one can form, I call this one, made by the least Ascent and the Trivial Factorization’s Ascent, the Grand Trapezoid for the Remainder Line of the particular odd composite m on the Zone Grid.

More meaningful numbers for me to examine, and more research follows. What fun as my Factorization Study approaches its year mark.

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