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  • Growing Concatenate Primes

    Introduction

    I begin to investigate the concept of a growing concatenate prime. Let us define the notation \[a\dagger b = a|a+1|...|b-1|b,\]

    for \(b > a \), where, as in the last few articles \(|\) indicates a digitwise concatenation, therefore to give a few examples

    \[5 \dagger 7 = 567 \\ 3 \dagger 8 = 345678 \\ 13 \dagger 15 = 131415 \\ 1 \dagger 1 = 1\]

    The reason for trying this is primarily out of curiosity. However, by fixing the ’form’ of numbers (for example in the d;n notation), we have already shown it is possible to extract information for very large numbers in previous work.

    Now, it is clear that the numeric interpretation of such digits strings will grow in magnitude extremely fast, a number such as \(100 \dagger 120\) will be \(60\) digits long. It is also clear that the numbers made in this fashion are base dependent.

    A good idea at first may be to tabulate conceptual intervals such as \(n \dagger n+1\) \[\begin{array}{|r|r r|} \hline n & n \dagger n+1 & \Delta\\ \hline -3 & -32 & - \\ -2 & -21 & 11 \\ -1 & -10 & 11 \\ 0 & 1 & 11 \\ 1 & 12 & 11 \\ 2 & 23 & 11 \\ ... & ... & 11 \\ 7 & 78 & 11 \\ 8 & 89 & 11 \\ 9 & 910 & 821 \\ 10 & 1011 & 101 \\ 11 & 1112 & 101 \\ ... & ... & 101 \\ 18 & 1819 & 101 \\ 19 & 1920 & 101 \\ 20 & 2021 & 101 \\ ... & ... & 101 \\ 98 & 9899 & 101 \\ 99 & 99100 & 89201 \\ 100 & 100101 & 1001 \\ \hline \end{array}\]

    We see here the special numbers \(11,101,1001\) forming in the differences. These numbers and their concatenates, \(10101, 1001001, ...\) form the repdigit basis. But there are the other numbers \(821,89201, 8992001\), which are made by \(910-89 , 99100-9899, 9991000-998999, ...\).

    These two sets of numbers can be described compactly then by \[1|0;k|1 , \;\;\; k \in \mathbb{N}^0 \\ 8|9;k|2|0;k|1, \;\;\; k \in \mathbb{N}^0\]

    By arranging this family of numbers in a table of divisors we can look for patterns, for example:
    \[\begin{array}{| c | l |} \hline k & Factors \;\; of \;\; 8|9;k|2|0;k|1 \in \mathbb{P} \\ \hline 0 & 821 \\ 1 & 7 \times 12743 \\ 2 & 29×149×2081 \\ 3 & 899920001 \\ 4 & 139×2239×289181 \\ 5 & 12041×747445561 \\ 6 & 31×29032255483871 \\ 7 & 7×12857142742857143 \\ 8 & 5939×259681×5835646939 \\ 9 & 31×120167×241599258215113 \\ 10 & 19×4736842105221052631579 \\ 11 & 2078217726601×4330633833401 \\ 12 & 899999999999920000000000001 \\ 13 & 7×13151×977655148440631347969193 \\ 14 & 19×131×709×2539×329641069×6093507681211 \\ 15 & 647×8762041×158757023470472699268463 \\ 16 & 5386471×3336989011711×5007067047060521 \\ 17 & 5386471×3336989011711×5007067047060521 \\ 18 & 199×1699×1119415751×93500544191×25432584853061 \\ 19 & 7×47×273556231003039513675379939209726443769 \\ 20 & 59×152542372881355932203254237288135593220339 \\ 21 & 31×21391×131117953566049×10351125295707468458809169 \\ 22 & 421×22621×2701121747698887529×3498681867987224789609 \\ 23 & 199×977×5661367×8176615206505410031316643554451717361 \\ 24 & 31×2408264179×12055263005477825957505994169206911982949 \\ 25 & ...\\ 40 & 89999999999999999999999999999999999999999200000000000000000000000000000000000000001 \end{array}\]

    Primes could occur on the following sequences... A034956 , A052482, A120304, A007993, A053043, A032093, Check 3, 12, 40, 70...

    Interesting that we have the sequences \[\begin{array}{|c|c|} \hline k & \\ \hline 1 & 7 \times 12743 & 1|274|3\\ 7 & 7 \times 12857142742857143 & 1|285714|274|285714|3 \\ 13 & 7 \times 12857142857142742857142857143 & 1|285714|285714|274|285714|285714|3 \\ \hline \end{array}\]

    We can state that the integer \(7\) provides a mapping \[8|9;k|2|0;k|1 \;\; [\to_7] \;\; 1|285714;n|274|285714;n|3, \\ \forall k=1+6n, \;\; n \in \mathbb{N}^0\]

    We note that \(6\) is the period of the repdigit in the right hand concatenate, but also that \[\frac{274}{7} = 39.\overline{142857},\] with period \(6\), and the order of the sequence is jumbled somewhat. Also note that \(12743\) and \(821\) are prime, (that is the \(k=0\) and \(n=0\) cases). It may be a coincidence that \(13\) the outermost pairing on the right hand side is \(3\) times \(39\). \[\begin{array}{|c|c|} \hline k & \\ \hline 3 & \mathrm{prime} \\ 6 & 31 \times 2903225|5483871 \\ 9 & 31 \times 2903225|806193|5483871 \\ 12 & \mathrm{prime} \\ 15 & - \\ 18 & - \\ 21 & 31×2903225|806|451612903225|548387096774|193|5483871 \\ 24 & 31×2903225|806|451612903225|806|193|548387096774|193|5483871\\ 27 & - \\ 30 & - \\ 33 & - \\ 36 & 31×2903225|806|451612903225|806|451612903225|548387096774|193|548387096774|193|5483871 \\ 39 & 31×2903225|806|451612903225|806|451612903225|806|193|548387096774|193|548387096774|193|5483871 \in \mathbb{P} \\ \hline \end{array}\] \[\frac{8|9;6+15n|2|0;6+15n|1}{31} = 2903225|806451612903225;n|548387096774193;n|5483871\\ \frac{8|9;9+15n|2|0;6+15n|1}{31} = 2903225|806451612903225;n|806193|548387096774193;n|5483871\\\]

    Now a strange/fantastic result is that in the number \(2903225|806451612903225548387096774193|5483871\) each digit \(0-9\) appears exactly three times in the 30 digit central partition! The numbers \(806\) and \(193\) are ’9 conjugate’, their sum is \(999\). The number is likely to split between the double \(5\) in the remaining 24 digits in the centre, as it did on the \(k=6\).

    N+2

    \[\begin{array}{|r | r r |} \hline n & n\dagger n+2 & \Delta \\ \hline 0 & 12 & - \\ 1 & 123 & 111 \\ 2 & 234 & 111 \\ 3 & 345 & 111 \\ 4 & 456 & 111 \\ 7 & 789 & - \\ 8 & 8910 & 8121 \\ 9 & 91011 & 82101 \\ 10 & 101112 & 10101 \\ 11 & 111213 & 10101 \\ 97 & 979899 & - \\ 98 & 9899100 & 8919201 \\ 99 & 99100101 & 89201001 \\ 100 & 100101102 & 1001001 \\ ... & ... & - \\ 997 & 997998999 & - \\ 998 & 9989991000 & 8991992001 \\ 999 & 99910001001 & 89920010001 \\ 1000 & - & 100010001 \\ \hline \end{array}\]

    This leads to three families of numbers associated with the differences, \[[1|0;k|1|0;k|1], \;\;\; k \in \mathbb{N}^0 \\ [8|9;k|1|9;k|2|0;k|1], \;\;\; k \in \mathbb{N}^0 \\ [8|9;k|2|0;k|1|0;k|01], \;\;\; k \in \mathbb{N}^0\]

    Then we have \[\begin{array}{|l|l|} k & [8|9;k|1|9;k|2|0;k|1] \\ \hline 0 & 3×2707 \\ 1 & 3×367×8101 \\ 2 & 3×7331×408857 \\ 3 & 3×61×49175955847 \\ 4 & 3×461×6507534345047 \\ 5 & 3×29×13068901×7915599323 \\ \end{array}\]

    It appears there is a trend of always being divisible by three forming for this particular family. In this way we can state a new family relationship to the more fundamental set \[[2|9;k|7|3;k|0|6;k|7]=\frac{1}{3}[8|9;k|1|9;k|2|0;k|1], \;\;\; k \in \mathbb{N}^0,\]

    by analysing the form of the ’other’ divisors, that is those not equal to \(3\) in the table above. Then we have \[\begin{array}{|l|l|} ndigs & k & [2|9;k|7|3;k|0|6;k|7] \\ \hline &0 & 2707 \\ &1 & 367×8101 \\ &2 & 7331×408857 \\ &3 & 61×49175955847 \\ &4 & 461×6507534345047 \\ &5 & 29×13068901×7915599323 \\ &6 & 132157×22700271142151431 \\ 25 &7 & 2999999973333333066666667 \\ &8 & 31×134119195669×721553637267553 \\ &9 & 971×3089598351939581187751458977 \\ &10 & 41617×5285509×13638407658308215754239 \\ &11 & 47×2427587751724753×26293503577218173237 \\ &12 & 29×67×191×8083791190283642268886655295467059 \\ &13 & 6823×4922853072028765637×89315947440316415417 \\ &14 & 59×1471×460409×38225179×111669466423×17588499752611651 \\ 49 &15 & 2999999999999999733333333333333306666666666666667 \\ &16 & 97×17378243×1902810619×38565693547×44999108501×538943244589 \\ &17 & 23×1123×854999×208683067618997483×650969803556729644268849819 \\ 58 &18 & 2999999999999999999733333333333333333306666666666666666667 \\ &19 & 43×103×3019×1083463×7470322471885033×27720370278030258074577706955323 \\ &20 & 16253×10657403×52655479033×1327848729329×247710354243298686538325081309 \\ &21 & \ne p \\ &22 & \ne p \\ &23 & \ne p \\ &24 & 29×30643×134967101×25012900528022619660042187099801110590209954791717634851850561 \\ &25 & \ne p \\ &26 & 233×57074713×404869643×557193937725682497848127100766089368113977211817871577285417761 \\ &27 & 59×1523×33386380582481053229019441999324853192665383145813162394767983202940969169532331 \\ &28 & \ne p \\ &29 & \ne p \\ &30 & 643×4665629860031104199066874027993364437532400207361327112493519917055469155002592016588906169 \\ &31 & 509×144203×18071502899×1739144075649997×1300467021504033122697284599677770897073180065778957265077177107 \\ & 32 & \ne p \\ & 33 & \ne p \\ & 34 & \ne p \\ & 35 & \ne p \\ & 36 & \ne p \\ & 37 & \ne p \\ & 38 & \ne p \\ & 39 & \ne p \\ & 40 & 29×43×1447949×5221373×318212193202873202203915859020970384316533247825314047632529269407411811226313804190758990607680705805260093\\ & 41 & \ne p \\ & 42 & 4793×45815352361×13661638669780679124036863023001124578173550024707290545709626024023811674456619238451475241882350377541514421080779\\ & 43 & \ne p \\ & 44 & \ne p \\ & 45 & \ne p \\ & 46 & \ne p \\ %37139×80777619214303023775545922076523331269016397855982480231921520055287792706678514050818097776813233169085507597583851656389958444402559753\\ & 47 & \ne p \\ & 48 & \ne p \\ & 49 & \ne p \\ & 50 & \ne p \\ & 51 & \ne p \\ & 52 & \ne p \\ & 53 & \ne p \\ & 54 & \ne p \\ & 55 & \ne p \\ & 56 & \ne p \\ & 57 & \ne p \\ & 58 & 293×587×17442773168363460878766912222151159072277037751975394061317006897647745133950807503493399848441682025997484364491940468202793557027208788056739403030778742298531124690633037\\ & 59 & \ne p \\ & 60 & 3911×22279601×34429128521134020280530097452556863771517130835644028733975877853802187716818619167678555270320838855615064140652396383384508385807432690210619643741836537573021123186893197\\ & 61 & \ne p \\ \end{array}\]

    \[\begin{array}{|l|l|} k & [2|9;k|7|3;k|0|6;k|7] \\ \hline 5 & 29×103448|183907|954023 \\ 12 & 29×103448|2758620|597|7011494252|7816091|954023 \\ 24 & 29×103448|2758620|689655172413|7011494252|873563218390804505747126436|7816091|954023 \\ \end{array}\]

    Noting that \(183907\), \(7816091\) are prime. There seems to be no mention of this phrase in the next \(27\) digits section. It is very strange that large non-repeating numbers are occurring.

    \[\begin{array}{|l|l|} k & [2|9;k|7|3;k|0|6;k|7] \\ \hline 14 & 59×50847457627118|59887005|6497175|096045197740113 \\ 27 & 59×50847457627118|6440677966101|6497175|1412429378531073446327231638418079|096045197740113 \\ \end{array}\]