Goal: Given odd positive integer m , its ceiling square c, and the remainder r=c^{2}-m, determine the column x and row y in the previously posted chart at which m is located. The range of integers between (c-1)^{2} and c^{2} falling on the upper half of one of the diagonals holding consecutive values and the lower half of the next one, this should be an easy task. Now that I am typing about it, I actually think I might have that formula tonight. Mind you, how helpful it will be for the factorization of an arbitrary m, I don’t yet know. Let’s just see what the x and y turn out to be… and what they do with row polynomials expressed only in terms of x, and column polynomials only in terms of y … I think.

Stay tuned!

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