I’ve systematized, somewhat, the recursive expression of an arbitrary positive integer as a recursive sequence of differences of systematically adjusted ceiling squares with their remainders.

I am hoping this will allow me to continue on a path to greater revelations about integer factorization.

As outlined below, slightly surprisingly to me this morning, I’m not quite done teasing out all of the details of this systematization that need further study. However, I am happy to share what I have developed so far.

**POSTSCRIPT**: It occurs to me to correct the inelegant-seeming Type a method as described below, simply by letting the Adjusted Ceiling Square I use be the least even perfect square greater than or equal to the integer, but only if the least odd perfect square greater than or equal to the integer fails to provide an expression for the integer as the difference of two perfect squares. This will leave a remainder of Type a, and will be what I consider to be pleasingly symmetric to how Adjusted Ceiling Squares come out for the other Types. I did not, however, employ this in the list below. My own list here at home, I will update.

```
Classification of Positive Integers by Residue Modulo 4
0: Type a.
1: Type b.
2: Type c.
3: Type d.
Type a always has an expression as (n+2)^2 - n^2 for some non-negative
integer n. There may, however, be a difference of perfect squares closer
to the integer's Ceiling Square. This may benefit from further study and
systematization. There may be another derivation that fits in more
harmoniously with the other Types' derivations.
Type b has an odd Adjusted Ceiling Square, and a Remainder of Type a.
Type c has no expression as s^2 - t^2. Therefore choose an Adjusted Ceiling
Square for c to be the least odd perfect square greater than c. The
Remainder will then be Type d, which always has an expression as the
difference of perfect squares as below.
Type d has an even Adjusted Ceiling Square, and a Remainder of Type b.
The only semiprime of Type a is 4. Other even semiprimes are 2p for some
prime p.
Odd perfect squares are all of Type b, and factor readily.
Attention to the semiprimes thus focuses on the odd discrete semiprimes,
hereafter abbreviated as ODSs. These can be of Type b or d.
To express an ODS of Type b as s^2 - t^2, s must be odd and t must be even.
To express an ODS of Type d as s^2 - t^2, s must be even and t must be odd.
PROCEDURE: Determine, given an ODS m, a Telescoping Sequence for m by first
finding m's Adjusted Ceiling Square that is even or odd as (m+1)/2 is even
or odd, respectively, and then recursively finding the Adjusted Ceiling
Square of the Remainder, and that Remainder's Remainder, and so on.
```

Here are the first fifty positive integers telescoped by this systematic procedure:

**1 = 1^2-0^2.**

**2 = 3^2-(4^2-3^2).**

**3 = 2^2-1^2.**

**4 = 2^2-0^2.**

**5 = 3^2-2^2.**

**6 = 3^2-(2^2-1^2).**

**7 = 4^2-3^2.**

**8 = 3^2-1^2.**

**9 = 3^2-0^2.**

**10 = 5^2-(4^2-1^2).**

**11 = 4^2-(3^2-2^2).**

**12 = 4^2-2^2.**

**13 = 5^2-(4^2-2^2).**

**14 = 5^2-(4^2-(3^2-2^2)).**

**15 = 4^2-1^2.**

**16 = 4^2-0^2.**

**17 = 5^2-(3^2-1^2).**

**18 = 5^2-(4^2-3^2).**

**19 = 6^2-(5^2-(3^2-1^2)).**

**20 = 6^2-4^2.**

**21 = 5^2-2^2.**

**22 = 5^2-(2^2-1^2).**

**23 = 6^2-(5^2-(4^2-2^2)).**

**24 = 5^2-1^2.**

**25 = 5^2-0^2.**

**26 = 7^2-(5^2-(4^2-2^2)).**

**27 = 6^2-3^2.**

**28 = 8^2-6^2.**

**29 = 7^2-(6^2-4^2).**

**30 = 7^2-(6^2-(5^2-(3^2-1^2)))**

**31 = 6^2-(3^2-2^2).**

**32 = 6^2-2^2.**

**33 = 7^2-4^2.**

**34 = 7^2-(4^2-1^2).**

**35 = 6^2-1^2.**

**36 = 6^2-0^2.**

**37 = 7^2-(4^2-2^2).**

**38 = 7^2-(4^2-(3^2-2^2)).**

**39 = 8^2-5^2.**

**40 = 7^2-3^2.**

**41 = 7^2-(3^2-1^2).**

**42 = 7^2-(4^2-3^2).**

**43 = 8^2-(5^2-2^2).**

**44 = 12^-10^2. (This illustrates the need for a Type-a derivation with greater harmony.)**

**45 = 7^2-2^2.**

**46 = 7^2-(2^2-1^2).**

**47 = 8^2-(5^2-(3^2-1^2)).**

**48 = 7^2-1^2.**

**49 = 7^2-0^2.**

**50 = 9^2-(6^2-(3^2-2^2)).**

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