31 is nothing fancy, but a bit complex. The goal is to discern how/where/why paired twin primes can be found. Here is a simple example using decade two. (The first decade is: 0 1 2 3 4 5 6 7 8 9.) (In a decade pattern scheme, decade 10 is actually a 1 because digit addition is used to determine decade patterns.) 10 |

111213141516171819Note: Only an integer whose final digit is "5" can be at the center of a paired twin primes set because an integer whose final digit is "5" will be encountered if the center is at an integer whose final digit is "1" or "3" or "7" or "9" (11 + 4 = 15 13 + 2 = 15 17 - 2 = 15 19 - 4 = 15). Step 1. That integer whose final digit is "5" must be evenly divisible by "3". Step 2. In the decade being investigated/ the integers whose final digits are "2" and "8" must be evenly divisible by "6". Step 3. Also, one of the integers noted in Step 2 must be evenly divisible by "4". In the example/ that integer is 12. Interestingly, in every even decade "4" divides evenly into the final-digit-2 integer; and in every odd decade "4" divides evenly into the final-digit-8 integer.[Formula for finding Decade #: (decade's final-digit-5 integer + 5) divided by 10](285 + 5)/10 Note 1 --- This is where complexity arrives. --- Example: 280 281 282 283 284 285 286 287 288 289 Odd-integer squares follow this final-digit pattern: 1 9 5 9 1 as in 1 x 1 = 1, 3 x 3 = 9, 5 x 5 = 25, 7 x 7 = 49, 9 x 9 = 81. Three ways to find them are: 1. (a natural number summation sequence term times 8) + 1 2. (an integer times an adjacent integer times 4) + 1 note: the sum of the two adjacent integers chosen equals the square root 3. (an even integer times an adjacent even integer) + 1 note: the odd integer between the adjacent even integers is the square root I'm going to use the latter here. 288/4 = 72 72/2 = 36 36/2 = 18 288/18 = 16 18 - 1 or 16 + 1 17 Also:18/2 = 9 72/9 = 8 8 x 9 = 72 8 + 9 = 17 17 is 289's square root. 17 x 17 = 289. [ sub-note: 5 35 65 95 (x 3) These are 90-apart locations, but 30-apart locations also

sometimes containpaired twin primes. Both the 30-apart and the 90-apart decades begin at decade 2. 5 (15/3) is in decade 2 and 95 (285/3) is in decade 29. 5 x 1 = 5 5 x 7 = 35 5 x 13 = 65 5 x 19 = 95 90-apart 5 x 1 = 5 5 x 3 = 15 5 x 5 = 25 5 x 7 = 35 30-apart To get the actual integers/ multiply the products in these intentionally-collapsed sequences by "3". ] Note 2 -- 0 4 8 and 2 6 are the two available division-by-4 patterns in each decade Note 3 -- Regarding the odd squares: Since the odd squares pattern is 1 9 5 9 1 | 1 9 5 9 1, beginning at the fifth odd square (81), adjacent occurrences of squares having "1" as a final digit/ happen every 5 squares after 81 and after 121. Squares having "1" as a final digit/ also, starting at 1/ occur every ten odd squares (1 x 1 and 19 x 19). They interlock, however, as 19 x 19 is adjacent to 21 x 21. Integers: 1, 81, 121, 361, 441, 841, 961, 1521, 1681 Decades: 1 9 13 37 45 85 97 153 169 The final-digit-9 squares run: Integers: 9, 49, 169, 289, 529, 729, 1089, 1369, 1849 Decades: 1 5 17 29 53 73 109 137 185 Note 4 -- final-digit-5 integers evenly divisible by "3" 15 45 75 105 135 165 195 225 255 285 - 315 345 375 405 435 465 495 525 555 585 - 615 645 675 705 735 765 795 825 855 885 [ sub-note: The decade pattern for those decades in which "3" divides evenly into a final-digit-5 integer is: 2 5 8. Therefore, only those decades can contain, but may not contain, paired twin primes. One of these decades, 29 (2 + 9 = 11 = 2), has 17 x 17 (289) in it. ] For the above three rows of integers, the decades are: 2 5 8 11 14 17 20 23 26 29 - 32 35 38 41 44 47 50 53 56 59 - 62 65 68 71 74 77 80 83 86 89 Note 5 -- Whenever "3" divides evenly into adecade, it also divides evenly into the final-digit-1 integer in that decade. Some f-d-1 pown squares are in these decades. One is in decade 153: 39 x 39 (1521). 153 digit sums to 9, and so is not a decade in which "3" divides evenly into its f-d-5 integer. Note 6 -- More information on decades and "3": In decade 1, "3" divides evenly into 3 and 6 and 9; and does the same every third decade from thereon: that is, it divides evenly into an integer whose final digit is a 3 or 6 or 9. - In decade 3, "3" divides evenly into 21 and 24 and 27, integers whose final digits are 1, 4 and 7 respectively; and does the same every third decade from thereon. - In decade 2, "3" divides evenly into 12 and 15 and 18, integers whose final digits are 2, 5 and 8 respectively; and does the same every third decade from thereon. - From this it should be obvious why paired twin primes cannot be found in decades 1, 4, 7, 10, 13, 16, etc., or in decades 3, 6, 9, 12, 15, 18, etc. Note 7 -- Digit addition patterns for integers in decades > 1: final digit 9: 1 2 3 4 5 6 7 8 9 f-d-1: 2 3 4 5 6 7 8 9 1 f-d-2: 3 4 5 6 7 8 9 1 2 f-d-3: 4 5 6 7 8 9 1 2 3 f-d-4: 5 6 7 8 9 1 2 3 4 f-d-5: 6 7 8 9 1 2 3 4 5 f-d-6: 7 8 9 1 2 3 4 5 6 f-d-7: 8 9 1 2 3 4 5 6 7 f-d-8: 9 1 2 3 4 5 6 7 8 f-d-0: 1 2 3 4 5 6 7 8 9 - for even squares (if there is a pattern): 4 is 4 16 is 7 36 is 9 64 is 1 100 is 1 144 is 9 196 is 7 256 is 4 324 is 9 | 400 is 4 484 is 7 576 is 9 676 is 1 784 is 1 900 is 9 1024 is 7 1156 is 4 1296 is 9 4 7 9 1 1 9 7 4 9 - for odd squares (if there is a pattern): 9 is 9 25 is 7 49 is 4 81 is 9 121 is 4 169 is 7 225 is 9 289 is 1 361 is 1 9 7 4 9 4 7 9 1 1 - Curious related facts: 17 is 8; 17 x 17 = 289, a 1; 289 is in decade 29, a 2. 39 is 3; 39 x 39 = 1521, a 9; 1521 is in decade 153, a 9. 6:17 PM Addendum 1: Began this in Note 3; but moments ago, because I wanted to know the changes cycle for odd squares having "1" as a final digit, this: 1 x 1 = 1 9 x 9 = 81 (9) 11 x 11 = 121 (4) 19 x 19 = 361 (1) 21 x 21 = 441 (9) 29 x 29 = 841 (4) 31 x 31 = 961 (7) 39 x 39 = 1521 (9) 41 x 41 = 1681 (7) 49 x 49 = 2401 (7) 51 x 51 = 2601 (9) 59 x 59 = 3481 (7) 61 x 61 = 3721 (4) 69 x 69 = 4761 (9) 71 x 71 = 5041 (1) 79 x 79 = 6241 (4) 81 x 81 = 6561 (9) 89 x 89 = 7921 (1) 18-term cycle for which the decades are: 1 9 13 37 45 85 | 97 153 169 241 261 349 | 373 477 505 625 657 793 Because the final digit for each of these squares is a "1"/ each square/ digit sums to same number its decade does. This being so, f-d-1 squares do not occur in the decades where "3" divides evenly into an f-d-5 integer. Those decades digit sum to 2, 5 and 8 only. These decades digit sum to 1, 4, 7 and 9 only. Addendum 2: The f-d-9 squares occur in decades which digit sum to 1, 2, 5and 8 only, which is why the 289 square in decade 29 prevents that decade from containing paired twin primes. 8 PM - This is the 18-term changes cycle for f-d-9 squares: 3 x 3 = 9 (9) 7 x 7 = 49 (4) 13 x 13 = 169 (7) 1 5 8 17 x 17 = 289 (1) 23 x 23 = 529 (7) 27 x 27 = 729 (9) 2 8 1 33 x 33 = 1089 (9) 37 x 37 = 1369 (1) 43 x 43 = 1849 (4) 1 2 5 47 x 47 = 2209 (4) 53 x 53 = 2809 (1) 57 x 57 = 3249 (9) 5 2 1 63 x 63 = 3969 (9) 67 x 67 = 4489 (7) 73 x 73 = 5329 (1) 1 8 2 77 x 77 = 5929 (7) 83 x 83 = 6889 (4) 87 x 87 = 7569 (9) 8 5 1 See also: sw00878math/odd-primes-and-digit-addition sw00879math/digit-addition-families sw00881math/pown-and-pewn-squares-notes sw00882math/descension-by-squares-groups sw00883math/my-paired-twin-primes-conjecture sw00884math/more-views-of-my-conjecture #Brian A. J. Salchert

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

## About Me

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- Rhodingeedaddee is my node blog. See
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[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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## Wednesday, May 14, 2008

### sw00887math--my-paired-twin-primes-proof

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