Bimp’s Decreasingly Frequent Progress Report
I am still working daily on my mathematics study, but want to get better at dropping a word now and then about my progress.
Below is a text file created (without Perl script help, except in running my “gimme the next prime” wrapper routine) in just the past couple of days, which I am using to examine the Series of Ceiling Squares for each of the odd prime numbers, as a step toward trying to understand how those relate to the odd discrete semiprimes, and possibly coming up with ideas on how to reverse engineer those latter numbers’ expression as (Adjusted) Ceiling Square minus Remainder.
One interesting side trip on which this is taking me has to do with the fact that if an odd positive integer m is congruent to 1 modulo 4, or, equivalently, if (m+1)/2 is odd, then any expression of m as s2-t2 that might give a factorization of m will have t be even, and, further, the Adjusted Ceiling Square, as I am working with it now, will leave an even Remainder. So for constructing the Series of Ceiling Squares (or Roots) using the Adjusted values as I have chosen to do, sometimes I am left with an even Remainder and need to figure out how to express that smaller number’s Series of Ceiling Squares, which will continue the Series for m. Since n+1/2 is not an integer when n is even, this is a challenge that I think I can solve neatly and deterministically if I think it out. But it is also sending me back to the one anomalous prime number behavior in all of this: The Ceiling Square of 2 is 4, and its Remainder is 2. This makes thinking about a corresponding Adjusted Ceiling Square concept for odd numbers at least a little slippery and tricky.
But hey! This is still fun, and I’m still occasionally getting that little thrill of excitement that I could be onto something.
3 = 4 - 1 XX
5 = 9 - 4 XX.
7 = 16 - 9 XX..
11 = 16 - ( 9 - 4 ) XXX.
13 = 25 - ( 16 - 4 ) XX.X
17 = 25 - ( 9 - 1 ) X.X.X
19 = 36 - ( 25 - ( 9 - 1 ) ) XX.X.X
23 = 36 - ( 25 - ( 16 - 4 ) ) XXX.X.
29 = 49 - ( 25 - ( 9 - 4 ) ) X.X.XX.
31 = 36 - ( 9 - 4 ) X..XX.
37 = 49 - ( 16 - 4 ) X..X.X.
41 = 49 - ( 9 - 1 ) X...X.X
43 = 64 - ( 25 - 4 ) X..X..X.
47 = 64 - ( 25 - ( 9 - 1 ) ) X..X.X.X
53 = 81 - ( 36 - ( 9 - 1 ) ) X..X..X.X
59 = 64 - ( 9 - 4 ) X....XX.
61 = 81 - ( 25 - ( 9 - 4 ) ) X...X.XX.
67 = 100 - ( 36 - ( 4 - 1 ) ) X...X...XX
71 = 100 - ( 49 - ( 25 - ( 9 - 4 ) ) ) X..X.X.XX.
73 = 81 - ( 9 - 1 ) X.....X.X
79 = 100 - ( 25 - 4 ) X....X..X.
83 = 100 - ( 25 - ( 9 - 1 ) ) X....X.X.X
89 = 121 - ( 36 - 4 ) X....X...X.
97 = 121 - ( 25 - 1 ) X.....X...X
101 = 121 - ( 25 - ( 9 - 4 ) ) X.....X.XX.
103 = 144 - ( 49 - ( 9 - 1 ) ) X....X...X.X
107 = 144 - ( 49 - ( 16 - 4 ) ) X....X..X.X.
109 = 121 - ( 16 - 4 ) X......X.X.
113 = 121 - ( 9 - 1 ) X.......X.X
127 = 144 - ( 25 - ( 9 - 1 ) ) X......X.X.X
131 = 144 - ( 25 - ( 16 - 4 ) ) X......XX.X.
137 = 169 - ( 36 - 4 ) X......X...X.
139 = 144 - ( 9 - 4 ) X........XX.
149 = 169 - ( 25 - ( 9 - 4 ) ) X.......X.XX.
151 = 196 - ( 49 - 4 ) X......X....X.
157 = 169 - ( 16 - 4 ) X........X.X.
163 = 196 - ( 49 - 16 ) X......X..X...
167 = 196 - ( 49 - ( 25 - ( 9 - 4 ) ) ) X......X.X.XX.
173 = 225 - ( 64 - ( 16 - 4 ) ) X......X...X.X.
179 = 196 - ( 25 - ( 9 - 1 ) ) X........X.X.X
181 = 225 - ( 49 - ( 9 - 4 ) ) X.......X...XX.
191 = 196 - ( 9 - 4 ) X..........XX.
193 = 225 - ( 36 - 4 ) X........X...X.
197 = 225 - ( 36 - ( 9 - 1 ) ) X........X..X.X
199 = 256 - ( 81 - ( 25 - 1 ) ) X......X...X...X
211 = 256 - ( 49 - 4 ) X........X....X.
223 = 256 - ( 49 - 16 ) X........X..X...
227 = 256 - ( 49 - ( 25 - ( 9 - 4 ) ) ) X........X.X.XX.
229 = 289 - ( 64 - 4 ) X........X.....X.
233 = 289 - ( 64 - ( 9 - 1 ) ) X........X....X.X
239 = 256 - ( 25 - ( 9 - 1 ) ) X..........X.X.X
241 = 289 - ( 49 - 1 ) X.........X.....X
251 = 256 - ( 9 - 4 ) X............XX.
257 = 289 - ( 36 - 4 ) X..........X...X.
263 = 324 - ( 64 - ( 4 - 1 ) ) X.........X.....XX
269 = 289 - ( 25 - ( 9 - 4 ) ) X...........X.XX.
271 = 324 - ( 81 - ( 36 - ( 9 - 1 ) ) ) X........X..X..X.X
277 = 289 - ( 16 - 4 ) X............X.X.
281 = 361 - ( 81 - 1 ) X.........X.......X
283 = 324 - ( 49 - ( 9 - 1 ) ) X..........X...X.X
293 = 361 - ( 81 - ( 25 - ( 16 - 4 ) ) ) X.........X...XX.X.
307 = 324 - ( 25 - ( 9 - 1 ) ) X............X.X.X
311 = 324 - ( 16 - ( 4 - 1 ) ) X.............X.XX
313 = 361 - ( 49 - 1 ) X...........X.....X
317 = 361 - ( 49 - ( 9 - 4 ) ) X...........X...XX.
331 = 400 - ( 81 - ( 16 - 4 ) ) X..........X....X.X.
337 = 361 - ( 25 - 1 ) X.............X...X
347 = 400 - ( 81 - ( 36 - ( 9 - 1 ) ) ) X..........X..X..X.X
349 = 361 - ( 16 - 4 ) X..............X.X.
353 = 361 - ( 9 - 1 ) X...............X.X
359 = 400 - ( 49 - ( 9 - 1 ) ) X............X...X.X
367 = 400 - ( 49 - 16 ) X............X..X...
373 = 441 - ( 81 - ( 25 - ( 16 - 4 ) ) ) X...........X...XX.X.
379 = 400 - ( 25 - 4 ) X..............X..X.
383 = 400 - ( 25 - ( 9 - 1 ) ) X..............X.X.X
389 = 441 - ( 64 - ( 16 - 4 ) ) X............X...X.X.
397 = 441 - ( 49 - ( 9 - 4 ) ) X.............X...XX.
401 = 441 - ( 49 - 9 ) X.............X...X..
409 = 441 - ( 36 - 4 ) X..............X...X.
419 = 484 - ( 81 - 16 ) X............X....X...
421 = 441 - ( 25 - ( 9 - 4 ) ) X...............X.XX.
431 = 484 - ( 81 - ( 36 - ( 9 - 1 ) ) ) X............X..X..X.X
433 = 441 - ( 9 - 1 ) X.................X.X
439 = 484 - ( 49 - 4 ) X..............X....X.
443 = 484 - ( 49 - ( 9 - 1 ) ) X..............X...X.X
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