Integer Factorization: A Possibly Not So Big Idea, but Who Knows?

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The reason I took that recent side-trip into looking at the “Fermat factorization” of multiples of 4, further using the fact that 4n = (n+1)2-(n-1)2 to decide upon the even/odd “polarity” to determine an appropriate Ceiling Square and its Root, was that I had gotten a bit excited about the ability to use quadratic functions to look at the remainders when subtracting the odd positive integer m from its Ceiling Square c2 and the two similarly even or odd perfect squares above it, (c+2)2 and (c+4)2. The differences between those three remainders may still lead to a clue as to a method for determining the desired Ascent of the Ceiling Root formulaically; however, the actual prospect still needs more investigation to determine if it’s real or a pipe dream.

The essential equivalent problem is as explained before: Given m=c2-r, for m, c, and r positive integers, what is the smallest positive value of x for which the polynomial x2+2cx+r gives a perfect square value?

[EDITED TO ADD] Or, in a different form of the problem obtained by completing the square of x2+2cx+r and remembering that m=c2-r, what is the least perfect square whose sum with m results in another perfect square?

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