These charts are from my Numbers file for a recent study of Ceiling Squares, Remainders, and Ascents for the odd integers 1 through 999, with definitions of the quantities indicated as follows:

If m is a positive non-prime integer:

Let c be the integer ceiling of the square root of m. Adjust the value by adding 1 if necessary to make it odd or even depending on whether (m+1)/2 is odd or even. The value of c thus adjusted is the **Ceiling Root** of m, and c^{2} is the least perfect square greater than or equal to m, its **Ceiling Square**.

Any expression of m as the difference of perfect squares m=s^{2}-t^{2} for odd m will have s odd or even, and t even or odd respectively, according to whether m is congruent to 1 or 3 modulo 4 respectively; or, equivalently, whether (m+1)/2 is odd or even, respectively. It will also match the oddness/evenness of the Ceiling Square and Root. s will be greater than or equal to c, the Ceiling Square, and the difference s-c will be an even number. Call ths number the **Ascent** of m.

I believe that is sufficient information to grasp the meaning of the following charts, for all positive odd integers less than 1000.

**CORRECTION**: In the chart screenshots below, when it says Ceiling Square, replace it with Ceiling Root, as the value for the x axis is c, not c^{2}.

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