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 † 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 † n + 1 {|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 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 ^0 \\ 8|9;k|2|0;k|1, \;\;\; k \in ^0 By arranging this family of numbers in a table of divisors we can look for patterns, for example: {| c | l |} \hline k & Factors \;\; of \;\; 8|9;k|2|0;k|1 \in \\ \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 Primes could occur on the following sequences... A034956 , A052482, A120304, A007993, A053043, A032093, Check 3, 12, 40, 70... Interesting that we have the sequences {|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 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 ^0 We note that 6 is the period of the repdigit in the right hand concatenate, but also that {7} = 39., 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. {|c|c|} \hline k & \\ \hline 3 & \\ 6 & 31 \times 2903225|5483871 \\ 9 & 31 \times 2903225|806193|5483871 \\ 12 & \\ 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 \\ \hline {31} = 2903225|806451612903225;n|548387096774193;n|5483871\\ {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.