Author – Andrew J Frost 26/09/2021 REV TWO PnonP p17
OBSERVATION 17 – A Comparison of Ending Numbers
The Ending Number Pattern (Today 2019-07-06)
Initially, as is well known and I confirmed to myself, that the ending digit of primes greater or equal to 7 (and including 1), is always 1, 3, 7 or 9. Then I discovered that in the series of Set Saf that these ending digits follow a completely rigid pattern, which equally applies to the odd-composite numbers in the series and are, by the results of my spreadsheets, infinite, as in ‘without end’; as OBSERVATION 5 also see back to OBSERVATION 7.
And these are the starting numbers: –
| Uplift | Ending | Gap |
|---|---|---|
| 1 | 1 | 1 |
| 7 | 7 | 6 |
| 11 | 1 | 4 |
| 1 | 3 | 2 |
| 5 | 7 | 4 |
| 7 | 9 | 2 |
| 11 | 3 | 4 |
| 5 | 9 | 6 |
Look at spreadsheet: – https://www.mediafire.com/file/t4jb6ews48qe995/20190813-list-ending-nos_2nd-try.ods/file
We now count the sequences of the Ending numbers from the first row, numbers 01, 07, 11, 13, 17, 19, 23, 29…..
The Table Compares; Uplift number, Ending number of the Primes and Non-Primes and the Gap between these numbers.
This gives a repeating sequence 1, 7, 1, 3, 7, 9, 3, 9; which if the initial integer sequence is considered, repeats 4 times against the integer ending pattern of 32 sequence units long (not shown in this table).
It seems logical to pursue these patterns further: –
Last Two Ending Digits of the Prime & Odd-Composite Series
The information is arranged starting at the (prime number list) number 1 after these block illustrations, so that the series starts 001, 007, 011 and so on, by adding leading zeros.
The “two ending numbers” sequence is composed of 80 digits being the last 2 ending numbers of any number occurring in the list. The sequence is composed of three lesser sub-sequences of 26 numbers, 28 numbers and another 26 numbers, (refer to post 5, Observation 5).
Three Ending Numbers
“Three ending numbers” require that these 80 numbers are repeated through the blocks ten times, comprising therefore 800 numbers. These sequences per block start with ending numbers 01,07,11. Then the second block starts with 01,03,07 and the last block starts with 03,09,11.
This produces the list of the last three digits in the numbers of the series Set Saf, that repeats regularly through the block (0)001 to (2)999 (block 1 is 800 numbers), then 3001 to 5999 (800 and so on), then 6001 to 8999, 9001 to 11999, 12001 to 14999, 15001 to 17999, 18001 to 20999, 21001 to 23999, 24001 to 26999, 27001 to 29999 (block 10 is 8000 numbers), 30001 to 32999, 33001 to 35999, and so on….
By list number 29999 (non-prime), the sequence of 80 numbers has been repeated 100 times and the sequence of 800 numbers has been repeated 10 times. This block 10 is where the first series of 8000 numbers ends and the next series of 8000 numbers starts. This pattern of 8000 four-ending numbers continues into in five digit numbers.
Looking then at the two ending numbers; this sequence of three sub-sequences totalling 80 numbers occurs once in a block ranging from ###1 to #300. Below, the sub-sequence numbers are separated by | and each repeat of 80 numbers is separated by || The pattern then repeats three times to carry through to number #899. Then an additional sequence (or two sequences) is/are required to bring the pattern in a block of numbers ranging within the parameter limit of 1 to 1000. These numbers comprise of 266 twice and 268 numbers once. So the three blocks below have a layout of 266, 268, 266 numbers, which total 800 numbers.
As far as I can determine the ‘last two digits and last three digits’ sequences are infinitely repeating. The last two digits repeating every 80 numbers in 80 lines and the last three digits repeating every 80 x 10 numbers in 800 lines; that is, there are 10 complete patterns of the sequence in the block 0001 to 2999 and so on.
The first sequence runs 80 numbers three times in this block from 0001 to 0899 (240 numbers), the second from 0901 to 1799 (another 240 numbers), and the third from 1801 to 2699 (another 240). Within the final third of the third block from 2701 to 2999 (the final 80 numbers). A total of 800 numbers ranging through numbers below 1000, then through 1001 to below 2000, then 2001 to below 3000.
Taking the start of the repeating series as integer 1, (0)001), then the whole sequence of three blocks ends with the block-ending-number 2999.
The numbers are as follows (black are primes; red are non-prime-odd-composite ‘prime-factor’ numbers) : –
Illustration of Primes and Non-Primes as the Three-Ending Numbers in Four-Digit Numbers
| (0)001 007 011 013 017 019 023 029 031 037 041 043 047 049 053 059 061 067 071 073 077 079 083 089 091 097 | 101 103 107 109 113 119 121 127 131 133 137 139 143 149 151 157 161 163 167 169 173 179 181 187 191 193 197 199 | 203 209 211 217 221 223 227 229 233 239 241 247 251 253 257 259 263 269 271 277 281 283 287 289 293 299 || 301 307 311 313 317 319 323 329 331 337 341 343 347 349 353 359 361 367 371 373 377 379 383 389 391 397 | 401 403 407 409 413 419 421 427 431 433 437 439 443 449 451 457 461 463 467 469 473 479 481 487 491 493 497 499 | 503 509 511 517 521 523 527 529 533 539 541 547 551 553 557 559 563 569 571 577 581 583 587 589 593 599 || 601 607 611 613 617 619 623 629 631 637 641 643 647 649 653 659 661 667 671 673 677 679 683 689 691 697 | 701 703 707 709 713 719 721 727 731 733 737 739 743 749 751 757 761 763 767 769 773 779 781 787 791 793 797 799 | 803 809 811 817 821 823 827 829 833 839 841 847 851 853 857 859 863 869 871 877 881 883 887 889 893 899 ** || 901 907 911 913 917 919 923 929 931 937 941 943 947 949 953 959 961 967 971 973 977 979 983 989 991 (0)997
| (1)001 003 007 009 013 019 021 027 031 033 037 039 043 049 051 057 061 063 067 069 073 079 081 087 091 093 097 099 | 103 109 111 117 121 123 127 129 133 139 141 147 151 153 157 159 163 169 171 177 181 183 187 189 193 199 || 201 207 211 213 217 219 223 229 231 237 241 243 247 249 253 259 261 267 271 273 277 279 283 289 291 297 | 301 303 307 309 313 319 321 327 331 333 337 339 343 349 351 357 361 363 367 369 373 379 381 387 391 393 397 399 | 403 409 411 417 421 423 427 429 433 439 441 447 451 453 457 459 463 469 471 477 481 483 487 489 493 499 || 501 507 511 513 517 519 523 529 531 537 541 543 547 549 553 559 561 567 571 573 577 579 583 589 591 597 | 601 603 607 609 613 619 621 627 631 633 637 639 643 649 651 657 661 663 667 669 673 679 681 687 691 693 697 699 | 703 709 711 717 721 723 727 729 733 739 741 747 751 753 757 759 763 769 771 777 781 783 787 789 793 799 ** || 801 807 811 813 817 819 823 829 831 837 841 843 847 849 853 859 861 867 871 873 877 879 883 889 891 897 | 901 903 907 909 913 919 921 927 931 933 937 939 943 949 951 957 961 963 967 969 973 979 981 987 991 993 997 (1)999
| (2)003 009 011 017 021 023 027 029 033 039 041 047 051 053 057 059 063 069 071 077 081 083 087 089 093 099 || 101 107 111 113 117 119 123 129 131 137 141 143 147 149 153 159 161 167 171 173 177 179 183 189 191 197 | 201 203 207 209 213 219 221 227 231 233 237 239 243 249 251 257 261 263 267 269 273 279 281 287 291 293 297 299 | 303 309 311 317 321 323 327 329 333 339 341 347 351 353 357 359 363 369 371 377 381 383 387 389 393 399 || 401 407 411 413 417 419 423 429 431 437 441 443 447 449 453 459 461 467 471 473 477 479 483 489 491 497 | 501 503 507 509 513 519 521 527 531 533 537 539 543 549 551 557 561 563 567 569 573 579 581 587 591 593 597 599 | 603 609 611 617 621 623 627 629 633 639 641 647 651 653 657 659 663 669 671 677 681 683 687 689 693 699 ** |||| 701 707 711 713 717 719 723 729 731 737 741 743 747 749 753 759 761 767 771 773 777 779 783 789 791 797 | 801 803 807 809 813 819 821 827 831 833 837 839 843 849 851 857 861 863 867 869 873 879 881 887 891 893 897 899 | 903 909 911 917 921 923 927 929 933 939 941 947 951 953 957 959 963 969 971 977 981 983 987 989 993 (2)999
Ending Four Digits of the Prime & Odd-Composite Series in Five Digit Numbers
It would be logical therefore to project this pattern to the next series of numbers ending in four digits occurring in five-digit numbers, which would be 8000 numbers in 8000 lines. The blocks of numbers would run from 0001 to 9997 (within 0001 to 10000), then 10001 to 19999 (within 10001 to 20000), then 20003 to 29999 (within 20001 to 30000), 30001…..
My conjecture would be that the logical projection of each series is infinite and that the logical increase of ending numbers of 4 digits would be in a five digit number; but containing the previous 3, 2, and 1 number ending patterns. The prime and non-prime-odd-composite 3,2,1 ending patterns will maintain their value of being a prime or odd-composite-non-prime number through out their series pattern. This principal will follow for ending patterns of 5 in a six digit number (1,000,000 to 9,999,999), 6 in a seven digit number (10,000,000 to 99,999,999) and so on, to produce patterns through numbers n104, n105, n106, and so on up through increasing powers; all infinite series where n= Set Saf digits, followed by an exponent of 10.
These are rigid patterns because they are part of Set Saf.
Five Digits of the Prime & Odd-Composite Series
Five digit numbers ending in 4 digits; for example the square of prime 179, producing the odd-composite number 32041.
The ending 4 digits, 2041 (also odd-composite, can be seen occurring in the previous table, as (2)041. The ending 3 digits of this number, i.e. 041 actually occur twice as (0)041 and (2)041 and of course 41 is prime. The ending 2 digits, i.e. 41 occurs 20 times. The ending 1 digit occurs 200 times.
Six Digits of the Prime & Odd-Composite Series
A six digit number ending in 5 digits; for example 132041 (odd-composite), a unique number which is the product of primes 1451, 13, and 7.
The ending 5 digits of this number, i.e. 32041 occur twice as (0)32041 and (1)32041. The ending 4 digits occur 20 times. The ending 3 digits occur 200 times. The ending 2 digits, i.e. 41, occurs 2000 times. The ending 1 digit occurs 20,000 times.
Seven Digits of the Prime & Odd-Composite Series
A seven digit number ending in 6 digits; an example 1332041 (consider (13)32041), a unique number which is the product of primes 2111 and 631; or 1832041 (consider (18)32041), a unique number which is the product of primes 3923 and 467. Therefore the occurrence of the first two 6 digit ending numbers is indicative of the same pattern as previous patterns.
Within Seven Digits
The ending 6 digits occur 2 times. The ending 5 digits occur 20 times. The ending 4 digits occur 200 times. The ending 3 digits occur 2000 times. The ending 2 digits occurs 20000 times. The ending 1 digit occurs 200,000 times.
While I believe that these series are typically expanding and expandable over the pattern of numbers in the spreadsheet; i.e. in greater and greater series up through 8, 9, 10 digit numbers and so on, and that this pattern has regularity; I cannot prove it. The spreadsheet needs to be substantially larger for a visual check, which is not possible with this computer’s capabilities or my time and poor little brain.
The Principal of Ending Numbers
The principal is the important issue. This principal is the fact that these series through Set Saf are completely regular, logical, and repetitive. So the numbers are predictable. The position of the primes within each field of numbers; as an example across the spreadsheet, – say the ‘Ending Seven Digits of the Prime & Odd-Composite Series’; are only predictable because they have already been defined within the fields of numbers up to the end of the previous sequence – ‘Ending Six Digits of the Prime & Odd-Composite Series’. But on analysis of course will not include as predictable any of the primes newly occurring within the ‘Ending Seven Digits of the Prime & Odd-Composite Series’.
Repeating Sequences
80=27+26+27=3×26+2=10×8 > line 89
240=3×80=30×8 > line 249
720=3×240 > line 729
2160 =3×720 > line 2169
6480=3×2160 > line 6489
19440=3×6480 > line 19449
58320=3×19440 > line 58329
Version of 2019-07-06 updated and republished 2021-10-06 af
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Ending Numbers in Eq2
Whilst looking at ending number patterns, another sequence manifests itself in the application of Eq2 (Equation 2 of Sandor Kristyans Paper). I have uploaded a revised copy of spreadsheet – 20211007-PnP-IntegerEnding Numbers.ods – into my MediaFire account. Please use the link below.
Click or paste – https://www.mediafire.com/file/qmdzh03zsblb3nm/20211007-PnP-IntegerEndingNos-2.ods/file
This spreadsheet is a doctored copy of the 7th spreadsheet in the Ratio series of spreadsheets. The number of rows is only 5000 and you need to select Sheet S2.
In this Sheet (S2), I have tested the application of the Integer Series for determining the results of Eq2; n = 2(2ab+a+b)+1 where a>b; which in spreadsheet speak, the equation is written: =IF(introw>=intcol,SUM(2*(2*((introw)*(intcol))+introw+intcol)+1),””) where ‘intcol’ = a and ‘introw’ = b. (See Observation 11).
To construct the Integer Series which constructs the visual application of Eq2, it is necessary to use the following simple formula on the entire list of primes and non-prime odd-composites which occur in Set Saf , which have been formed by the application of ‘My Logical Process’ formula, explained at the very beginning of these blogs for the paper “Primes and Non Prime Patterns”.
Simple Formula
Take one starting value from the list, choosing anywhere to start. Subtract 1 and divide by 2. This results in the value for the integer required for the values of ‘a’ and ‘b’ in the equation, and called ‘introw’ and ‘intcol’ in the spreadsheet formula.
As I said earlier, I am using one of the Ratio spreadsheets cut down to just 5000 rows to do this. Prime and odd-composite numbers start at 7874933 which is the 2,099,992nd number in the spreadsheet series.
On examining these numbers I noticed yet another repeating sequence of ending numbers which is 16 digits long. These are the ending numbers of the integers themselves. The series runs as 3,5,6,8,9,1,4,5, and then 8,0,1,3,4,6,9,0, and repeats thereafter without fail. (16 number sequence).
In Observation 11, I listed the Integer Sequence 3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 24, 26, 29, 30, 33, ….. and illustrated a table of values that relate back to the Gap sequence 6,2,6,4,2,4,2,4, but divided by 2 thus; 3,1,3,2,1,2,1,2.
This developed the table which follows in this Observation 11. Please refer back to the post number 11 in this series. This pattern of integers then produces a spreadsheet table with only the non-primes odd-composite numbers that are complementary to the prime numbers.
When comparing the ending numbers with the actual numbers from the series, it is clear that 16 ending number sequence repeats endlessly providing a rigid format for the infinite series. The difference between eight numbers and the ninth number of the next next eight numbers in the sequence is always 15.
| 3937473 | 3937475 | 3937476 | 3937478 | 3937479 | 3937481 | 3937484 | 3937485 | 3937488 | 3937490 | 3937491 | 3937493 | 3937494 | 3937496 | 3937499 | 3937500 |
One can conclude, as with all these regular patterns in this series of numbers that the linking patterns are the “mesenteric” links through the maths associated with primes and non-prime complementarity. Or they are maps through a landscape, however they are without a reference enabling overall navigation at the moment.
End
