Prime Representation Transform

Abstract

Abstract

Investigate a recreational maths transform. We turn a number into its prime representation, then turn that into a digit string, and repeat. There may exist numbers other than primes that are invarient under such a transformation, others will grow tending to infinity.

Introduction

Introduce the transform \(T\) on a number \(N\), in base \(10\). Say, \(N=3024\), then the unique prime representation is \(2^4×3^3×7\), and we can write a new number \(T(3024)=22223337\). Repeating, we have \(3×61×121439\) and \(T^2(3024)=361121439\), again \(3×7×43×399913\), giving \(T^3(3024)=3743399913\). etc.

\[\begin{array}{|c|c|c|c|c|c|C|} \hline m & N & T(N) & T^2(N) & T^3(N) & T^4(N) & T^5(N) & T^6(N) & T^7(N) & T^8(N) & T^9(N) & T^{10}(N) & T^{11}(N) & T^{12}(N) & T^{13}(N) & T^{14}(N) & T^{15}(N) & T^{16}(N) & T^{17}(N)\\ \hline \infty & 1 & 1 \\ 0 & 2 & 2 \\ 0 & 3 & 3 \\ 2 & 4 & 2|2 & 2|11 & 211\\ 0 & 5 & 5 \\ 1 & 6 & 2|3 & 23\\ 0 & 7 & 7 \\ 13 & 8 & 2|2|2 & 2|3|3|7 & 3|19|41 & 3|3|3|7|13|13 & 3|11123771 & 7|149|317|941 & 229|31219729 & 11|2084656339 & 3|347|911|118189 & 11|613|496501723 & 97|130517|917327 & 53|1832651281459 & 3|3|3|11|139|653|3863|5107 & 3331113965338635107\\ 2 & 9 & 3|3 & 3|11 & 311 \\ 4 & 10 & 2|5 & 5|5 & 5|11 & 7|73 & 773 \\ 0 & 11 & 11 \\ 1 & 12 & 2|2|3 & 223 \\ 0 & 13 & 13 \\ 5 & 14 & 2|7 & 3|3|3 & 3|3|37 & 47|71 & 13|367 & 13367 \\ 4 & 15 & 3|5 & 5|7 & 3|19 & 11|29 & 1129 \\ 4 & 16 & 2|2|2|2 & 2|11|101 & 3|11|6397 & 3|163|6373 & 31636373 \\ 0 & 17 & 17 \\ 1 & 18 & 2|3|3 & 233 \\ 0 & 19 & 19 \\ 15 & 20 & 2|2|5 & 3|3|5|5 & 5|11|61 & 11|4651 & 3|3|12739 & 17|194867 & 19|41|22073 & 709|273797 & 3|97|137|17791 & 11|3610337981 & 7|3391|4786213 & 3|3|3|3|7|23|31|1815403 & 13|17|23|655857429041 & 7|7|2688237874641409 & 3|31|8308475676071413 & 3318308475676071413\\ 1 & 21 & 3|7 & 37 \\ 1 & 22 & 2|11 & 211 \\ 0 & 23 & 23 \\ 2 & 24 & 2|2|2|3 & 3|3|13|19 & 331319 \\ 3 & 25 & 5|5 & 5|11 & 7|73 & 773 \\ 4 & 26 & 2|13 & 3|71 & 7|53 & 3|251 & 3251 \\ 4 & 27 & 3|3|3 & 3|3|37 & 47|71 & 13|367 & 13367 \\ 1 & 28 & 2|2|7 & 227 \\ 0 & 29 & 29 \\ 2 & 30 & 2|3|5 & 5|47 & 547 \\ 0 & 31 & 31 \\ 2 & 32 & 2|2|2|2|2 & 2|41|271 & 241271 \\ 1 & 33 & 3|11 & 311 \\ 16 & 34 & 2|17 & 7|31 & 17|43 & 3|7|83 & 3|13|97 & 3|29|131 & 11|29921 & 13|23|3779 & 13|433|2351 & 17|23|343561 & 37|46576853 & 3|3|416286317 & 17|1965663901 & 3|3|3|6369098663 & 2897|1151663479 & 3|397|743|2339|13997 & 3397743233913997 \\ 3 & 35 & 5|7 & 3|19 & 11|29 & 1129 \\ 2 & 36 & 2|2|3|3 & 7|11|29 & 71129 \\ 0 & 37 & 37 \\ 2 & 38 & 2|19 & 3|73 & 373 \\ 1 & 39 & 3|13 & 313 \\ 9 & 40 & 2|2|2|5 & 5|5|89 & 3|3|3|3|3|23 & 7|7|59|1153 & 29|2675557 & 3|31|3147049 & 809|1019|4019 & 3|53639|502807 & 3|31|41|92745739 & 3314192745739 \\ 0 & 41 & 41 \\ 2 & 42 & 2|3|7 & 3|79 & 379 \\ 0 & 43 & 43 \\ 9 & 44 & 2|2|11 & 3|11|67 & 3|3|3463 & 13|113|227 & 173|229|331 & 11|15748121 & 541|2062381 & 11|607|810553 & 2281|5088913 & 22815088913 \\ 6 & 45 & 3|3|5 & 5|67 & 3|3|3|3|7 & 17|37|53 & 239|727 & 3|41|1949 & 3411949 \\ 1 & 46 & 2|23 & 223 \\ 0 & 47 & 47 \\ 15 & 48 & 2|2|2|2|3 & 71|313 & 3|11|2161 & 3|13|199|401 & 19|43|109|3517 & 11|17|109|877|1087 & 23|1481|7039|46591 & 3|3|7|53|67|1034726207 & 3|11251223678242069 & 23|4583|2952795526741 & 359|5782291|1130063089 & 835996339|43011938251 & 31|49123|54898161457127 & 467|79367|8496358995643 & 61|61|79|1591356884791277 & 6161791591356884791277 \\ - & 49 & 7|7 & 7|11 & 3|3|79 & 31|109 & 13|2393 & 3|44131 & 17|31|653 & 7|11|43|523 & 11|11|5771019 & 311|35742029 & 7|17|261644891 & 11|19|3431873899 & 11|613|4799|345907 & 3|204751|189066719 & 3|1068250396355573 & 621611|49980213343 & 3|3|6906794442245927 & 73|4615161567701999 & 3|13|18836286194043641 & 3|3|3|43|14369|161461|11627309 & 3|32057|1618455677|2142207827 & 3|1367|2221|5573|475297|1376323127 & 7|3391|51263|25777821480557336017 & 47|67|347|431|120361987|12947236602187 & 3|7|7|17|12809|57470909|57713323|4490256751 & 3096049809383|121823389214993262890297 & 73796236325118712936424989555929478399 & 13|1181|145261411|33089538087518197265265053 & 3|19|521|441731977174163487542111577539726749 & 59|5415617656474189392601483764603009147911 & 13|8423|1466957|3706744784027901056001426046777 & 3|12919|2501509379|96709539317201|1476342474406759 & 3|2039|2713|3121|399320591|151296378525102203388346189 & 13|3119|651853|9121952491|13288820301002347322382772769\\ 2 & 50 & 2|5|5 & 3|5|17 & 3517 \\ 1 & 51 & 3|17 & 317 \\ 1 & 52 & 2|2|13 & 2213 \\ 0 & 53 & 53 \\ 1 & 54 & 2|3|3|3 & 2333 \\ 2 & 55 & 5|11 & 7|73 & 773 \\ 3 & 56 & 2|2|2|7 & 17|131 & 37|463 & 37463 \\ 2 & 57 & 3|19 & 11|29 & 1129 \\ 1 & 58 & 2|29 & 229 \\ 0 & 59 & 59 \\ 2 & 60 & 2|2|3|5 & 3|5|149 & 35149 \\ \hline \end{array}\]

Where \(m\) is the number of transforms needed for the new number to be prime.

It can be seen that some numbers lead to previous numbers and therefore arrive at the same prime number to terminate the sequence. The steps till the resulting number is prime has the sequence \(A(n)\)=0,0,0,2,0,1,0,13,2,4,0,1,0,5,4,4,0,1,0,15,1,1,0,2,3,4,4,1,0,2,0,2,1,16,3,2,0,2,1,9,0,2,0,9,6,1,0,0,-,2,1,1,0,1,2,3,2,1,0,2,... as the prime numbers require \(0\) transforms to be prime.

We could call this measure \(A(n)\) the anti-primality of \(n\), although, at the moment just because all prime numbers express anti-primality of \(0\). But does the index even mean anything like this...

Comparing the divisors sequence \(\sigma_0(n)\) and the \(A(n)\) sequence, we observe that:

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8,
0, 0, 0, 2, 0, 1, 0, 13,2, 4, 0, 1, 0, 5, 4, 4, 0, 1, 0, 15,1 ,1 ,0 ,2 ,3 ,4 ,4 ,1 ,0, 2,
2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12,
0, 2, 1, 16,3, 2, 0, 2, 1, 9, 0, 2, 0, 9, 6, 1, 0, 15, -, 2, 1, 1, 0, 1, 2, 3, 2, 1, 0, 2,

In the first \(40\) terms [not that many, this is speculative]
Every zero in \(A\) after the first is matched with a \(2\) in the divisor sequence, this is the primality of a number and is built in.
Every \(1\) in \(A\) has a corresponding divisor term of either \(4\) or \(6\). [true for first \(8\) \(1\)’s]
Every \(2\) in \(A\) has a corresponding divisor term of either \(3\), \(4\), \(6\), \(8\) or \(9\).
Every \(3\) has \(3\) or \(4\).
Every \(4\) has \(4\) or \(5\).
\(6\) has \(4\)
.. \(9\) has \(8\)
..


Find more terms...

Require a counter proof of the statement, “any non-prime whose ordered concatenation is prime has either \(4\) or \(6\) divisors”.

Counterproof: 531832651281459 has 128 divisors. But it’s prime factors are 3*3*3*11*139*653*3863*5107, which have an ordered concatenation of 3331113965338635107 which is prime.

Better counter proof, 54, has 8 divisors and has concatenation 2333 which is prime.

I like the region \(53\) to \(59\), as we have transforms till prime \(0,1,2,3,2,1,0\).

Aside

Consider that if a number is \[N = \prod_i p_i^{q_i}\] with primes \(p\) and some integer powers \(q\). Then \[log(N)=\sum_j q_jlog(p_j)\]

Now if all the \(p_j\) are orthogonal in some sense, and the infinite number of primes form a Hilbert space, we have a loose anology as the log of a number being a wavefunction as \[\Psi(x,t) = \sum_n a_n\psi_n(x,t)\]

In the reverse we have \[e^{\Psi(x,t)} = \prod_n e^{a_n\psi_n(x,t)}\]

If \(\Psi\) and \(\psi\) are complex valued, we then have a seperation of parts \[\Psi(x,t) = R(x,t) +i\Upsilon(x,t) \\ \psi(x,t) = r(x,t) +iu(x,t) \\ e^{R(x,t)}(\mathrm{cos}(\Upsilon(x,t))+i\mathrm{sin}(\Upsilon(x,t)) = \prod_n e^{a_nr_n(x,t)} (\mathrm{cos}(a_nu_n(x,t))+i\mathrm{sin}(a_nu_n(x,t))\]

Any Numbers

We can do this strange concatenation without focus on the prime factors. One can define left and right concatenation and create a table.

\[\begin{array}{|c|cc|} \hline &0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline 0 & [0] & [1] & 2 & 3 & [4]& 5 & [6]& 7 & [8]& [9] \\ 1 & [10] & 11 & [12] & 13 & [14] & [15] & [16] & 17 & [18] & 19 \\ 2 & [20] & [21] & [22] & 23 & [24] & [25] & [26] & [27] & [28] & 29 \\ 3 & [30] & 31 & [32] & [33] & [34] & [35] & [36] & 37 & [38] & [39] \\ 4 & [40] & 41 & [42] & 43 & [44] & [45] & [46] & 47 & [48] & [49] \\ 5 & [50] & [51] & [52] & 53 & [54] & [55] & [56] & [57] & [58] & 59 \\ 6 & [60] & 61 & [62] & [63] & [64] & [65] & [66] & 67 & [68] & [69] \\ 7 &