Adjusted Scatter Plot for Integer Factorization Study


The plot below is similar to the one I put in my previous post. However, the data points are, I think, a better set for exposing properties, since I am using the concept of Adjusted Ceiling Square as follows: Let m be an odd positive integer. Define the Adjusted Ceiling Square of m as the smallest perfect square greater than m that is congruent to (m+1)/2 modulo 2; i.e. it has the same oddness/evenness as the larger square of the difference-of-squares expression of m’s trivial factorization as 1*m: m = ((m+1)/2)^2 – ((m-1)/2)^2. So if the Ceiling Square as I have defined it in the past is of opposite oddness/evenness as the larger square thus obtained, and thus of the larger square of any difference of squares corresponding to any other factorization of m (see earlier posts and links to papers for why this must be so), then the Adjusted Ceiling Square is one greater. Otherwise, they are equal. At this point, we can then calculate adjusted remainder and Ascent values, as also discussed earlier.

Okay, enough explanation, and now to share the plot.

Adjusted Remainder vs. Adjusted Ascent

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