We can try an establish a check for primality in this more tractable notation for large numbers, take \(1;n\), we can check the primality for varying \(n\).
\[\begin{array}{|c|c|c|} \hline & prime & rep \\ \hline 1;1=1 & 0 & 1 \\ 1;2=11 & 1 & 11 \\ 1;3=111 & 0 & 3×37 \\ 1;4 & 0 & 11×101 \\ 1;5 & 0 & 41×271 \\ 1;6 & 0 & 3×7×11×13×37 \\ 1;7 & 0 & 239×4649 \\ 1;8 & 0 & 11×73×101×137 \\ 1;9 & 0 & 3^2×37×333667 \\ 1;10 & 0 & 11×41×271×9091 \\ 1;11 & 0 & 21649×513239 \\ 1;12 & 0 & 3×7×11×13×37×101×9901 \\ 1;13 & 0 & 53×79×265371653 \\ 1;14 & 0 & 11×239×4649×909091 \\ 1;15 & 0 & 3×31×37×41×271×2906161 \\ 1;16 & 0 & 11×17×73×101×137×5882353 \\ 1;17 & 0 & 2071723×5363222357 \\ 1;18 & 0 & 3^2×7×11×13×19×37×52579×333667 \\ 1;19 & 1 & 1;19 \\ 1;20 & 0 & 11×41×101×271×3541×9091×27961 \\ 1;21 & 0 & 3×37×43×239×1933×4649×10838689 \\ 1;22 & 0 & 11^2×23×4093×8779×21649×513239 \\ 1;23 & 1 & 1;23 \\ 1;24 & 0 & 3×7×11×13×37×73×101×137×9901×99990001 \\ 1;25 & 0 & 41×271×21401×25601×182521213001 \\ 1;26 & 0 & 11×53×79×859×265371653×1058313049 \\ 1;27 & 0 & 3^3×37×757×333667×440334654777631 \\ 1;28 & 0 & 11×29×101×239×281×4649×909091×121499449 \\ 1;29 & 0 & 3191×16763×43037×62003×77843839397 \\ 1;30 & 0 & 3×7×11×13×31×37×41×211×241×271×2161×9091×2906161 \\ 1;31 & 0 & 2791×6943319×57336415063790604359 \\ 1;32 & 0 & 11×17×73×101×137×353×449×641×1409×69857×5882353 \\ 1;33 & 0 & 3×37×67×21649×513239×1344628210313298373 \\ 1;34 & 0 & 11×103×4013×2071723×5363222357×21993833369 \\ 1;35 & 0 & 41×71×239×271×4649×123551×102598800232111471 \\ 1;36 & 0 & 3^2×7×11×13×19×37×101×9901×52579×333667×999999000001 \\ 1;37 & 0 & 2028119×247629013×2212394296770203368013 \\ 1;38 & 0 & 11×909090909090909091×1111111111111111111 \\ 1;39 & 0 & 3×37×53×79×265371653×900900900900990990990991 \\ 1;40 & 0 & 11×41×73×101×137×271×3541×9091×27961×1676321×5964848081 \\ 1;41 & 0 & 83×1231×538987×201763709900322803748657942361 \\ 1;42 & 0 & 3×7^2×11×13×37×43×127×239×1933×2689×4649×459691×909091×10838689 \\ 1;43 & 0 & 173×1527791×1963506722254397×2140992015395526641\\ 1;44 & 0 & 11^2×23×89×101×4093×8779×21649×513239×1052788969×1056689261 \\ 1;45 & 0 & 3^2×31×37×41×271×238681×333667×2906161×4185502830133110721 \\ 1;46 & 0 & 11×47×139×2531×549797184491917×11111111111111111111111 \\ 1;47 & 0 & 35121409×316362908763458525001406154038726382279 \\ 1;48 & 0 & 3×7×11×13×17×37×73×101×137×9901×5882353×99990001×9999999900000001 \\ 1;49 & 0 & 239×4649×505885997×1976730144598190963568023014679333 \\ 1;50 & 0 & 11×41×251×271×5051×9091×21401×25601×182521213001×78875943472201 \\ \end{array}\]
Finding sequences of \(901\) components \[\begin{array}{|c|c|c|} \hline & prime & rep \\ \hline 1;1 & 0 & 1 \\ 1;2 & 1 & 11 \\ 1;3 & 0 & 3×37 \\ 1;4 & 0 & 11×101 \\ 1;5 & 0 & 41×271 \\ 1;6 & 91 & 3×7×11×13×37 \\ 1;7 & 0 & 239×4649 \\ 1;8 & 0 & 11×73×101×137 \\ 1;9 & 0 & 3^2×37×333667 \\ 1;10 & 9091 & 11×41×271×9091 \\ 1;11 & 0 & 21649×513239 \\ 1;12 & 91,9901 & 3×7×11×13×37×101×9901 \\ 1;13 & 0 & 53×79×265371653 \\ 1;14 & 909091 & 11×239×4649×909091 \\ 1;15 & 0 & 3×31×37×41×271×2906161 \\ 1;16 & 0 & 11×17×73×101×137×5882353 \\ 1;17 & 0 & 2071723×5363222357 \\ 1;18 & 91,999001 & 3^2×7×11×13×19×37×52579×333667 \\ 1;19 & 1 & 1;19 \\ 1;20 & 9091,99009901 & 11×41×101×271×3541×9091×27961 \\ 1;21 & 0 & 3×37×43×239×1933×4649×10838689 \\ 1;22 & 0 & 11^2×23×4093×8779×21649×513239 \\ 1;23 & 1 & 1;23 \\ 1;24 & 91,9901,99990001 & 3×7×11×13×37×73×101×137×9901×99990001 \\ 1;25 & 0 & 41×271×21401×25601×182521213001 \\ 1;26 & 0 & 11×53×79×859×265371653×1058313049 \\ 1;27 & 0 & 3^3×37×757×333667×440334654777631 \\ 1;28 & 909091 & 11×29×101×239×281×4649×909091×121499449 \\ 1;29 & 0 & 3191×16763×43037×62003×77843839397 \\ 1;30 & 91,9091,9999900001,999000999001 & 3×7×11×13×31×37×41×211×241×271×2161×9091×2906161 \\ 1;31 & 0 & 2791×6943319×57336415063790604359 \\ 1;32 & 0 & 11×17×73×101×137×353×449×641×1409×69857×5882353 \\ 1;33 & 0 & 3×37×67×21649×513239×1344628210313298373 \\ 1;34 & 0 & 11×103×4013×2071723×5363222357×21993833369 \\ 1;35 & 0 & 41×71×239×271×4649×123551×102598800232111471 \\ 1;36 & 91,9901,999001,999999000001 & 3^2×7×11×13×19×37×101×9901×52579×333667×999999000001 \\ 1;37 & 0 & 2028119×247629013×2212394296770203368013 \\ 1;38 & 909090909090909091 & 11×909090909090909091×1111111111111111111 \\ 1;39 & 900900900900990990990991 & 3×37×53×79×265371653×900900900900990990990991 \\ 1;40 & 9091,99009901 & 11×41×73×101×137×271×3541×9091×27961×1676321×5964848081 \\ 1;41 & 0 & 83×1231×538987×201763709900322803748657942361 \\ 1;42 & 91,909091 & 3×7^2×11×13×37×43×127×239×1933×2689×4649×459691×909091×10838689 \\ 1;43 & 0 & 173×1527791×1963506722254397×2140992015395526641\\ 1;44 & 0 & 11^2×23×89×101×4093×8779×21649×513239×1052788969×1056689261 \\ 1;45 & 0 & 3^2×31×37×41×271×238681×333667×2906161×4185502830133110721 \\ 1;46 & 0 & 11×47×139×2531×549797184491917×11111111111111111111111 \\ 1;47 & 0 & 35121409×316362908763458525001406154038726382279 \\ 1;48 & 91,9901,99990001,9999999900000001 & 3×7×11×13×17×37×73×101×137×9901×5882353×99990001×9999999900000001 \\ 1;49 & 0 & 239×4649×505885997×1976730144598190963568023014679333 \\ 1;50 & 9091 & 11×41×251×271×5051×9091×21401×25601×182521213001×78875943472201 \\ \end{array}\]
The second column above is the possibility to form the sequence. Some of the prime factors needed will overlap. But this gives a divisibility by check.
Consider, the “squared” [not under multiplication some other operation] of a prime... leading to another, some kind of sequence combination/manipulation/convolution.
Can condider an operation of the form \[O_1(9090)+1 \to 9091 \\ O_2(9090)+1 \to 99009901 \\ O_3(9090)+1 \to 999000999001 \\\]
This kind of sequence extention is curious. In the above precedings is there a sense in which \((9090)+1\otimes(9090)+1=99009901\)?
Divisors of \(n\) and prime divisors of \(1;n\) comparison table.
\[\begin{array}{|c|c|c|} \hline n & \sigma_0(n) & p_d(1;n) \\ \hline 1 & 1 & 0 \\ 2 & 2 & 1 \\ 3 & 2 & 2 \\ 4 & 3 & 2 \\ 5 & 2 & 2 \\ 6 & 4 & 5 \\ 7 & 2 & 2 \\ 8 & 4 & 4 \\ 9 & 3 & 3^* \\ 10 & 4 & 4 \\ 11 & 2 & 2 \\ 12 & 6 & 7 \\ 13 & 2 & 3 \\ 14 & 4 & 4 \\ 15 & 4 & 6 \\ 16 & 5 & 6 \\ 17 & 2 & 2 \\ 18 & 6 & 8^* \\ 19 & 2 & 1 \\ 20 & 6 & 7 \\ \end{array}\]
^* \(3^2\) was counted as one.
We can observe a potential pattern that evey other \(1;2n\in\mathbb{Z}\) string of ones is divisible by \(11\), also \(1;3n\in\mathbb{Z}\) is divisible by \(37\)... We also note that if this trend continues there should be only be distinct primes of the form \(1;p\), where \(p\) must be prime, but not any prime will do. [2,19,23,...] special primes. Due to the nature of the occurence of divisors, the number of prime factors on \(1;n\), should be directly proportional to the number of divisors of \(n\)!
Let us tabulate \[\begin{array} \hline 1;mn & \mathrm{div} \; \mathrm{by} \\ \hline 1;n\in\mathbb{Z} & 1 & 1\\ 1;2n\in\mathbb{Z} & 11 & 1;2\\ 1;3n\in\mathbb{Z} & 3×37 & 1;3\\ 1;4n\in\mathbb{Z} & 101 & 101\\ 1;5n\in\mathbb{Z} & 41×271 & 1;5\\ 1;6n\in\mathbb{Z} & 13 & 1;6/8547 \\ 1;7n\in\mathbb{Z} & 239×4649 & 1;7 \\ 1;8n\in\mathbb{Z} & 73 \\ 1;9n\in\mathbb{Z} & 3^2×333667 & 3003003\\ 1;10n\in\mathbb{Z} & 9091 & 9091\\ 1;11n\in\mathbb{Z} & 21649×513239 & 1;11 \\ 1;12n\in\mathbb{Z} & 9901 & 9901\\ 1;13n\in\mathbb{Z} & 53×79×265371653 & 1;13\\ 1;14n\in\mathbb{Z} & 909091 & 909091\\ 1;15n\in\mathbb{Z} & 2906161 \\ 1;16n\in\mathbb{Z} & 5882353 \\ 1;17n\in\mathbb{Z} & 2071723×5363222357 & 1;17\\ 1;18n\in\mathbb{Z} & 52579?? \\ 1;19n\in\mathbb{Z} & 1;19 & 1;19\\ 1;20n\in\mathbb{Z} & 27961 \\ 1;21n\in\mathbb{Z} & 10838689 \\ 1;22n\in\mathbb{Z} & 513239 \\ 1;23n\in\mathbb{Z} & 1;23 & 1;23\\ 1;24n\in\mathbb{Z} & 99990001 & 99990001\\ 1;25n\in\mathbb{Z} & 21401×25601×182521213001 & 100001000010000100001\\ 1;50n\in\mathbb{Z} & 251×5051×78875943472201 & 99999000009999900001\\ \hline \end{array}\]
This is curious as the prime is not in ascending order. There are multiples, which together would make some other factor. We have for \(3\) and \(9\), [thus \(27\) ...] an interesting relationship where the number is as divisible by \(3\) as the integer prefactor \(m\) is...
We note that \(n\in\mathbb{P}\) that are not \(2\) appear to have two, larger primes introduced as divisible factors. (sometimes more? \(13\). [semiprime?].
\(\forall \; p\in\mathbb{P}, n\in\mathbb{Z}\;:\;1;pn\) is divisible by \(1;p\).
Automatically true for \(n=1\).
No proof for other \(n\), only based on the evidence above. Is a conjecture for now.
\(\forall \; p_i\in\mathbb{P}, m\in\mathbb{Z}\;:\;1;p_1p_2\cdots p_N n\) is divisible by \(1;p_1,1;p_2,...,1;p_N\).
We may single out any of the \(p_i\) and make \(1;p_in\), if \(Theorem \; 1\) holds, then it is divisible by \(1;p_i\), i1,,N.
\(10\),\(12\),\(14\), may or may not have some curious relationship as thier numbers are \(9091,9901,909091\)...\(99990001\), Think [False concatenation of \(90\) -> \((90;;3)+1\)], [think added permutations], does this apply to \(333667\) as \((3;3|6;3)+1\). This sounds ridiculous which is why it’s exciting.
Addition to this idea: Try "Any number \(n\) whose \(1;n\) spare representation is of the form (9,0,+1), multiplied to another \(m\) whose \(1;m\) spare representation is of the form \(1;a\), gives that \(1;nm\) has spare represntation the form (9,0,+1).
Groups of equal numbers of \(9,0\), with \(1\) added, sorted into all permutations. [One is prime??]
To find more factors, one could use the previously found factors to quickly divide the number. For example if we wanted to find the prime factors of \(1;50\), [given the previously foudn information of up to \(1;25\)], we may look at the divisors of \(50\), \(1,2,5,10,25,50\), and identify that the prime factorisation of \(1;50\) will be \(11×41×271×9091×21401×25601×182521213001×F\), where \(F\) are the associated divisors with \(n=50\). We know \(1;50\) is not prime, as \(50\) is not prime. We learn frommthis calculation that \(F=99999000009999900001\) which is not prime but is of our \(9,0,+1\) form... This gives \(251×5051×78875943472201\) for \(1;50n\in\mathbb{Z}\).
Task: construct a sieve of Sieve of Eratosthenes style algorithm which uses this concept as a test for primality?
Information required: The appending patterns of the form \(d_1d_2...d_N;n\). This is a big task.
The special primes, are apprently \(OEIS:A004023\), and this has already been investigated from the references there for \(1;n\) although not in this notation. See, repunits.