d;n

Abstract

Abstract

I investigate the use of the notation \(d;n\), meaning repeat the digit \(d\), \(n\) times. The convergence of rational expressions and the primality of larger sequences are explored. A way of ruling some non-trivial numbers out in \(\mathrm{log}_{10}(digits)\) is found for prime searches. The relationship of numbers written in such notation and thier prime decompositions is investigated.

Investigation

Let \(d;n\) mean repeat the digit \(d\), \(n\) times. Let \(|\) be a digitwise concatenation, such that \(2|3;4|6=233336\) This allows constants to be created for example \[C=\lim_{n\to\infty} \frac{1|3;n|8}{2|8;n|7}\]

For a given sequence it is not entirely clear if convergence will be achieved. We may take a partial series. For \(n=24\) we have \(C_{24}=0.461538461538461538461538653254437869822485207100604251251706...\) which appears to repeat at first with a \(6\) digit sequence \(.461538\), repeating \(4\) times. FOr \(n=40\) we have \(C_{40}=0.461538461538461538461538461538461538461557633136094674556213...\) which repeats 6.5 times. If we assume convergence we could then argue that \[C_\infty = 0.\overline{461538} =\frac{6}{13}\]

This doesn’t sound, too ridiculous when one considers that \(0.|9;\infty=1\) However one can notice that \[\lim_{n\to\infty} \frac{1|3;n|8}{2|8;n|7}=\lim_{n\to\infty} \frac{1|3;n}{2|8;n}\]

In fact the end digits wil be very small, so as we go to infinity, this information will probably be lost.

\[\lim_{n\to\infty} \frac{1|3;n}{0|8;n} = 1.5\overline{0} =\frac{12}{8} \\ \lim_{n\to\infty} \frac{1|3;n}{1|8;n} = 0.\overline{7058823529411764} =\frac{12}{17} \\ \lim_{n\to\infty} \frac{1|3;n}{2|8;n} = 0.\overline{461538} =\frac{12}{26} \\ \lim_{n\to\infty} \frac{1|3;n}{3|8;n} = 0.3\overline{428571} =\frac{12}{35} \\ \lim_{n\to\infty} \frac{1|3;n}{4|8;n} = 0.\overline{27} =\frac{12}{44} \\\]

Which would at least imply \[\lim_{n\to\infty} \frac{1|3;n}{d|8;n}=\frac{12}{(d|8) - d}\]

Then we have \[\lim_{n\to\infty} \frac{1|2;n}{0|8;n} = 1.375\overline{0} =\frac{11}{8} \\ \lim_{n\to\infty} \frac{1|2;n}{1|8;n} =0.\overline{6470588235294117} = \frac{11}{17} \\\]

Which would at then imply \[\lim_{n\to\infty} \frac{1|d_1;n}{d_2|8;n}=\frac{(1|d_1) - 1}{(d_2|8) - d_2}\]

\[\lim_{n\to\infty} \frac{0|2;n}{1|8;n} =0.\overline{1176470588235294} = \frac{2}{17} \\ \lim_{n\to\infty} \frac{1|2;n}{1|8;n} =0.\overline{6470588235294117} = \frac{11}{17} \\ \lim_{n\to\infty} \frac{2|2;n}{1|8;n} =1.\overline{1764705882352941} = \frac{20}{17} \\ \lim_{n\to\infty} \frac{3|2;n}{1|8;n} =1.\overline{7058823529411764} = \frac{29}{17} \\\]

Which would at then imply \[\lim_{n\to\infty} \frac{d_1|d_2;n}{d_3|8;n}=\frac{(d_1|d_2) - d_1}{(d_3|8) - d_3}\]

And then perhaps \[\lim_{n\to\infty} \frac{d_1|d_2;n}{d_3|d_4;n}=\frac{(d_1|d_2) - d_1}{(d_3|d_4) - d_3}\]

Hence if the beginning digits are made to be \(0\). We have \[\frac{a;\infty}{b;\infty}=\frac{a}{b}\]

Which can only be true, if the digits are single digits, i.e \(d\in[0,9]\) Thus for longer digit phrases for example \[10;5 \ne 1010101010 \\ 10;5 = 111110\]

However such a phrase allows for insights such as \[10;5=(1;5)|0 \\ 100;5=(1;5)|00\\\]

Or more generally \[(d|0;n);m =(d;m)|(0;n), \;\; d\in[0,9]\]

Likewise we have \[(0;n|d);m=(0;n)|(d;m), \;\; d\in[0,9]\]

For the pipe and semi-comma operations we have identity \[0|a = a \\ a;1 = a \\ a;0 = 1 or 1???\]

Interesting expressions such as \[10101;1=10101 \\ 10101;2=111111 \\ 10101;3=1121211 \\ 10101;4=11222211 \\ 10101;5=112232211 \\\]

Drive a cellular automaton? Pascals triangle generator?

Thing

\[\lim_{n\to\infty}\frac{9|8;n}{7|6;n}=\frac{89}{69}\]

Try 339/108, for \(\pi\) like test... Have \[\frac{(d_1|d_2)-d_1}{(d_3|d_4)-d_3}=\frac{339}{108} \to \frac{(4|03) - 4}{(1|09)-1}\]

Consider \[03;5 = 033333 \\ 09;5 = 099999 \\\]

attempt to see error convergence with added digits\[\frac{403}{109}-\pi= 0.555655052832225110161209827729671427729436105212050142327807...\\ \frac{4033}{1099}-\pi= 0.528107073434774550436355706802389745466160900897869611236156...\\ \frac{40333}{10999}-\pi=0.525377070930617707987033860106259457833924336419057284761940...\\\]

Pascals

Try to generate pascals triangle or similar. \[010 \\ 0110 \\ 01210 \\ 013310 \\ 0146410 \\\]

The first line is \(010;1\).
The second line is \(010;2\).
The third line is \((010;2);2\).
The n^th line is \(010;2_1;2_2;2_3...;2_{n-1}\)
We can attempt similar triangles with other starting strings.

1 1;3= 111 111;3=

111 111 111 = 12321

12321;3=

12321 12321 12321 = 1367631

1367631;3

1:3:6:7:6:3:1 :1:3:6:7:6:3:1 :1:3:6:7:6:3:1

1:4:10:16:19:16:10:4:1

\[1 \\ 1:1:1 \\ 1:2:3:2:1 \\ 1:3:6:7:7:6:3:1 \\ 1:4:10:16:19:16:10:4:1\]

12

12 12 =132

132 132 ==1452

1452 1452 = 15972

\[12 \\ 132 \\ 1452 \\ 15972 \\\]

1001

1001 1001 = 11011

11011 11011 = 121121

121121 121121 = 1332331

1332331 1332331 = 14655641

\[1001 \\ 11011 \\ 121121 \\ 1332331 \\ 14655641\]

Primality

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 & 11×909090909090909091×1111111