28 pewn = positive even whole number pown = positive odd whole number sqrt = square root resultant = remainder G = Group sq = square In a descension-by-squares/ pown squares are subtracted from a pewn square in a lesser to greater order, or pewn squares are subtracted from a pown sq in a lesser to greater order. Each subtraction produces a resultant integer which is always a pown. That pown is also the product of two integers which can be discovered by inspecting the sqrt values in a subtraction. If the values are "18" and "11", simply subtract the 11 from the 18 to get the multiplier and add the 11 to the 18 to get the multiplicand. As I will show, the construction of a d-b-s can quickly become cumbersome, tending to make such constructions less and less useful. The main use, however--finding the final resultant, is easily done. Here are some relevant formulas: (2 x G#) - 2 = pewn sqrt for that group G# = (pewn sqrt + 2)/2 (2 x G#) - 1 = pown sqrt for that group G# = (pown sqrt + 1)/2 (final resultant # + 1)/2 = sqrt (sqrt x 2) - 1 = final resultant # Example using the above: G28 pewn sqrt is 54 final resultant is 107 pown sqrt is 55 final resutant is 109 [ Note: Because they are at level one (the x 1 level), some final resultants are prime numbers. ] First ten groups: - G1 has one object: 1² - 0² = 1² = 1 - G2, wherein 2 + 1 = 3 and 3 + 2 = 5: 2 x 2 = 4 _ _ _ _ _ _ _ _ _ _ 3 x 3 = 9 2² - 1² = 3 _ _ _ _ _ _ _ _ _ 3² - 2² = 5 - G3 4 x 4 = 16 _ _ _ _ _ _ _ _ _ 5 x 5 = 25 4² - 1² = 15 _ _ _ _ _ _ _ _ 5² - 2² = 21 4² - 3² = 7 _ _ _ _ _ _ _ _ _5² - 4² = 9 - G4 6 x 6 = 36 _ _ _ _ _ _ _ _ _ 7 x 7 = 49 6² - 1² = 35 _ _ _ _ _ _ _ _7² - 2² = 45 6² - 3² = 27 _ _ _ _ _ _ _ _7² - 4² = 33 6² - 5² = 11 _ _ _ _ _ _ _ _7² - 6² = 13 - G5 8 x 8 = 64 _ _ _ _ _ _ _ _ 9 x 9 = 81 8² - 1² = 63 _ _ _ _ _ _ _ 9² - 2² = 77 8² - 3² = 55 _ _ _ _ _ _ _ 9² - 4² = 65 8² - 5² = 39 _ _ _ _ _ _ _ 9² - 6² = 45 8² - 7² = 15 _ _ _ _ _ _ _ 9² - 8² = 17 - G6 10 x 10 = 100 _ _ _ _ _ _ 11 x 11 = 121 10² - 1² = 99 _ _ _ _ _ _ 11² - 2² = 117 10² - 3² = 91 _ _ _ _ _ _ 11² - 4² = 105 10² - 5² = 75 _ _ _ _ _ _ 11² - 6² = 85 10² - 7² = 51 _ _ _ _ _ _ 11² - 8² =57 10² - 9² = 19 _ _ _ _ _ _ 11² - 10² = 21 - G7 12 x 12 = 144 _ _ _ _ _ _13 x 13 = 169 12² - 1² = 143 _ _ _ _ _ 13² - 2² = 165 12² - 3² = 135 _ _ _ _ _ 13² - 4² = 149 12² - 5² = 119 _ _ _ _ _ 13² - 6² = 129 12² - 7² = 95 _ _ _ _ __13² - 8² = 105 12² - 9² = 63 _ _ _ _ _ _13² - 10² = 69 12² - 11² = 23 _ _ _ _ _ 13² - 12² = 25 - G8 14 x 14 = 196 _ _ _ _ _ _ 15 x 15 = 225 14² - 1² = 195 _ _ _ _ _ _15² - 2² = 221 14² - 3² = 187 _ _ _ _ _ _15² - 4² = 209 14² - 5² = 171 _ _ _ _ _ _15² - 6² = 189 14² - 7² = 147 _ _ _ _ _ _15² - 8² = 161 14² - 9² = 115 _ _ _ _ _ _15² - 10² = 125 14² - 11² = 75 _ _ _ _ _ _15² - 12² = 81 14² - 13² = 27 _ _ _ _ _ _15² - 14² = 29 - G9 16 x 16 = 256 _ _ _ _ _ _ 17 x 17 = 289 16² - 1² = 255 _ _ _ _ _ _17² - 2² = 285 16² - 3² = 247 _ _ _ _ _ _17² - 4² = 273 16² - 5² = 231 _ _ _ _ _ _17² - 6² = 253 16² - 7² = 207 _ _ _ _ _ _17² - 8² = 225 16² - 9² = 175 _ _ _ _ _ _17² - 10² = 189 16² - 11² = 135 _ _ _ _ _ 17² - 12² = 145 16² - 13² = 87 _ _ _ _ _ _17² - 14² = 93 16² - 15² = 31 _ _ _ _ _ _17² - 16² = 33 - G10 18 x 18 = 324 _ _ _ _ _ _ 19 x 19 = 361 18² - 1² = 323 _ _ _ _ _ _19² - 2² = 357 18² - 3² = 315 _ _ _ _ _ _19² - 4² = 345 18² - 5² = 299 _ _ _ _ _ _19² - 6² = 325 18² - 7² = 275 _ _ _ _ _ _19² - 8² = 297 18² - 9² = 243 _ _ _ _ _ _19² - 10² = 261 18² -11² = 203 _ _ _ _ _ _19² - 12² = 217 18² - 13² = 155 _ _ _ _ _ 19² - 14² = 165 18² - 15² = 99 _ _ _ _ _ _19² - 16² = 105 18² - 17² = 35 _ _ _ _ _ _19² - 18² = 37 [ Note: I was surprised when 4 and 28 had prime pairs which do not match when digit addition is applied to them, but have since learned that there are congruities in two other ways. 28 - 4 = 24, which is divisible by both 3 and 8; 107 - 11 = 96, which is divisible by both 3 and 8. ] Guess what. Unless I've made an error someplace, I've just been thrown another dropping curve ball I cannot hit. 191, 193; 197, 199 are primes. G49 pewn sqrt is 96 final resultant is 191 pown sqrt is 97 final resultant is 193 Plainly put, if the even integer between two primes is not divisible by "4", those primes will not be in the same group. [ Note: What I have been calling prime pairs are usually called twin primes; so I guess one could call 11 13 and 17 19 paired twin primes. In every such pairing, only the even integer between one of the twins is divisible by "4". Not sure, but the integer between the twin primes which is not divisible by "4" may always be divisible by "6". ] 9:27 PM - With help from a table on page 211 of the paperback copy of

The Mathematical ExperienceI own, just discovered that 9431 9433 and 9437 9439 are paired twin primes, and that 9432 is divisible by "4" and 9438 is divisible by "6". Conjecture: Regarding paired twin primes, the even integer between one of the twin primes is always divisible by "4" and the even integer between the other of the twin primes is always divisible by "6". #Brian A. J. Salchert

is a tiny wandering imaginary dinosaur which migrated from AOL in October of 2008.

## About Me

- brian (baj) salchert
- Rhodingeedaddee is my node blog. See
*my other blogs*and*recent posts*.

## Guide

[6-16-2009 Update Insert: Most of what is in this space is now moot. I found out what I was doing wrong and have reinstated Archives and Labels searches. They do work. However, in certain cases you may prefer Labels to Archives. Example: 1976 Today begins in November of 2006 and concludes in December of 2006, but there are other related posts in other months. Note: Labels only shows 20 posts at a time. There are 21 hubs, making 21 (which is for 1976 Today) an older hub.] ********************************* to my online poems and song lyrics using Archives. Use hubs for finding archival locations but do not link through them. Originally an AOL Journal, where the archive system was nothing like the system here, this blog was migrated from there to here in October of 2008. Today (Memorial/Veteran's Day, May 25, 2009) I discovered a glitch when trying to use a Blogger archive. Now, it may be template-related, but I am unable to return to S M or to the dashboard once I am in the Archives. Therefore, I've decided on this approach: a month-by-month post guide. The sw you see in the codes here stood for Salchert's Weblog when I began it in November of 2006. It later became Sprintedon Hollow. AOL provided what were called entry numbers, but they weren't consistent, and they didn't begin at the first cardinal number. That is why the numbers after "sw" came to be part of a post's code. ************** Here then is the month-by-month post guide: *2006* November: 00001 through 00046 - December: 00047 through 00056 -- *2007* January: 00057 through 00137 - February: 00138 through 00241 - March: 00242 through 00295 - April: 00296 through 00356 - May: 00357 through 00437 - June: 00438 through 00527 - July: 00528 though 00550 - August: 00551 through 00610 - September: 00611 through 00625 - October: 00626 through 00657 - November: 00658 through 00729 - December: 00730 through 00762 -- *2008* January: 00763 through 00791 - February: 00792 through 00826 - March: 00827 through 00849 - April: 00850 through 00872 - May: 00873 through 00907 - June: 00908 through 00931 - July: 00932 through 00955 - August: 00956 through 00993 - September 00994 through 01005 - October: 01006 through 01007 - November: 01008 through 01011 - December: 01012 through 01014 -- *2009* January: 01015 through 01021 - February: 01022 through 01028 - March: 01029 through 01033 - April: 01034 through 01036 - May: 01037 through 01044 - ******************************************************* 1976 Today: 2006/11 and 2006/12 -- Rooted Sky 2007: 2007/01/00063rsc -- Postures 2007: 2007/01/sw00137pc -- Sets: 2007/02/sw00215sgc -- Venturings: 2007/03/00216vc -- The Undulant Trees: 2007/03/00266utc -- This Day's Poem: 2007/03/00267tdpc -- Autobio: 2007/04/sw00316ac -- Fond du Lac: 2007/04/00339fdl -- Justan Tamarind: 2007/05/sw00366jtc -- Prayers in December: 2007/05/sw00393pindc -- June 2007: 2007/06/sw00440junec -- Seminary: 2007/07/sw00533semc -- Scatterings: 2008/08/00958sc ** Song Lyrics: 2008/02/sw00797slc ********** 2009-06-02: Have set S M to show 200 posts per page. Unfortunately, you will need to scroll to nearly the bottom of a page to get to the next older/newer page.

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## Friday, May 9, 2008

### sw00882math--descension-by-squares-groups

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