I belive I now have all the information cobbled together sufficiently to give a concise characterization of all positive integers’ “Trivial” characterizations as differences of squares.

For any positive integer m, let the Ceiling Square of m be the least perfect square greater than m. For odd m, let the Adjusted Ceiling Square of m be the least perfect square greater than m that is even if (m+1)/2 is even, and odd otherwise. For even m, let the Adjusted Ceiling Square of m be the least odd perfect square greater than m.

Odd m will always (“trivially”) have an expression as the difference of two perfect squares of consecutive integers: m = [(m+1)/2]^{2}-[(m-1)/2]^{2}.

An even value of m congruent to 0 modulo 4 – i.e., a multiple of 4 – can appear as the difference of two perfect squares of integers differing by 2: When m=4n for positive integer n, m=(n+1)^{2}-(n-1)^{2}.

An even value of m congruent to 2 modulo 4 has no expression as the difference of two perfect squares, but does have a “trivial” expression in the form a^{2}-(b^{2}-c^{2}) where a is the Adjusted Ceiling Square of even m as described above, and b and c are the consecutive positive integers that produce the odd remainder a^{2}-m.

None of this is earth-shattering – as far as I know (I have not yet researched thoroughly beyond doing my own work with these ideas), Pierre de Fermat could well have characterized all of this in his own work on factorization. But my study will now take me further into whether these “trivial” forms interplay with other ways of expressing odd positive integers, and their Adjusted Ceiling Squares and remainders, in ways that give insight to that most intriguing, elusive, and central quantity to my thinking: the amount one must add to the Adjusted Ceiling Square to get a value that differs from m by a perfect square amount, and thus gives a non-trivial factorization of any odd composite m.

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