World!Of
Numbers

WON plate
197 |

[ December 6, 2015 ] [ Last update November 24, 2023 ]
Adding 1's, 3's, 7's and 9's to n so that n becomes a record
delayed prime and more derived challenges along the way

Let me coin these numbers FEP's or First Encountered Primes

in order to get familiar with intention and notation.

Suppose the number 5 is our next number to investigate
let me append 1's as long as it stays composite and stop if
the extension of 5 becomes prime.

( '^^' is symbol for concatenation )

[5][1^^1] = [5][1] = 51 = 3 x 17 = composite
[5][1^^2] = [5][11] = 511 = 7 x 73 = composite
[5][1^^3] = [5][111] = 5111 = 19 x 269 = composite
[5][1^^4] = [5][1111] = 51111 = 3 x 3 x 3 x 3 x 631 = composite
[5][1^^5] = [5][11111] = 511111 = prime!

So if this is our next record delayed prime
than [5][1^^5] will be added to the table.
(Note : it is indeed a record number by the way).

There are cases whereby appending any number of 1's, 3's, 7's or 9's
always produce composites ad infinitum. Of course, these cases
are discarded. E.g. [15][3^^n] divisible by 3.
Also for instance [37][1^^n] and [38][1^^n] but are more complicated.

For the sake of this wonplate it is not about enlisting all factors
of the composite. Though it might be useful to detect infinite patterns.
Also when numbers get larger I will accept PRP (PRobable Prime)
as valid entries.

Already number [12] set a milestone as it needed 136 1's
appended before the number got prime! (Source)
So the next record number must have at least 137 1's.

 For starters on PRP'ing here are a few links that will be helpfull. Download the latest version at SourceForge: http://sourceforge.net/projects/openpfgw/files/?source=navbar Introduction into the world of OpenPFGW (or PrimeForm): https://primes.utm.edu/bios/page.php?id=432 How to determine whether a large number is prime What is the fastest deterministic primality test?

Click on the header titles to open the related worksheets.
The summary is at → Appending digits to k.htm

Here is an elaborated example of how the factorization of
[38][1^^n] allows to detect infinite composite patterns.

The first 21 factorizations produce the next list

```
[38][1^^1] = 3 x 127
[38][1^^2] = 37 x 103
[38][1^^3] = 23 x 1657
[38][1^^4] = 3 x 127037
[38][1^^5] = 17 x 37 x 73 x 83
[38][1^^6] = 233 x 163567
[38][1^^7] = 3^2 x 42345679
[38][1^^8] = 37 x 113 x 613 x 1487
[38][1^^9] = 31 x 2333 x 526957
[38][1^^10] = 3 x 2399 x 52954163
[38][1^^11] = 37 x 103003003003
[38][1^^12] = 23333 x 1633356667
[38][1^^13] = 3 x 73 x 1740233384069
[38][1^^14] = 37 x 2287 x 45038479669
[38][1^^15] = 353 x 661 x 163333566667
[38][1^^16] = 3^2 x 131 x 323249458109509
[38][1^^17] = 37 x 114346289 x 900798827
[38][1^^18] = 19 x 227 x 541 x 2857 x 5716953331
[38][1^^19] = 3 x 879449 x 1140233 x 126685261
[38][1^^20] = 37 x 393380951 x 261840342653
[38][1^^21] = 17 x 73 x 1372549 x 22374429543379
```
One sees immediately that the first, the fourth, the seventh, etc.
or {1 +3} for short are all divisible by 3, so these can be put aside.
Idem dito for {2, +3} where all numbers can be divided by 37.

If I shift these to be ignored cases you see what is left over.

```
[38][1^^1] = 3 x 127
[38][1^^2] = 37 x 103
[38][1^^3] = 23 x 1657
[38][1^^4] = 3 x 127037
[38][1^^5] = 17 x 37 x 73 x 83
[38][1^^6] = 233 x 163567
[38][1^^7] = 3^2 x 42345679
[38][1^^8] = 37 x 113 x 613 x 1487
[38][1^^9] = 31 x 2333 x 526957
[38][1^^10] = 3 x 2399 x 52954163
[38][1^^11] = 37 x 103003003003
[38][1^^12] = 23333 x 1633356667
[38][1^^13] = 3 x 73 x 1740233384069
[38][1^^14] = 37 x 2287 x 45038479669
[38][1^^15] = 353 x 661 x 163333566667
[38][1^^16] = 3^2 x 131 x 323249458109509
[38][1^^17] = 37 x 114346289 x 900798827
[38][1^^18] = 19 x 227 x 541 x 2857 x 5716953331
[38][1^^19] = 3 x 879449 x 1140233 x 126685261
[38][1^^20] = 37 x 393380951 x 261840342653
[38][1^^21] = 17 x 73 x 1372549 x 22374429543379
```

Spotting [38][1^^3], [38][1^^6], [38][1^^12] one sees that we are
dealing with semiprimes or numbers with two primefactors.
When more factors are given, these can always be brought back
to two factors [be it composite for one or both].

Take e.g. [38][1^^21] = 17 x 73 x 1372549 x 22374429543379
which is in fact [17 x 1372549] x [73 x 22374429543379]
or [ 23333333 ] x [ 1633333356666667 ]

The general formula for [38][1^^21]{3 +3} becomes
[2][3^^m] x [16][3^^m-1][5][6^^m-1][7]
So the whole range of numbers is covered and
no primes can arise from [38][1^^n] !

Here is the extracted list of genuine semiprimes for [38][1^^21]{3 +3}

```[38][1^^3] ➜ m = 1 ➜ 23 x 1657
[38][1^^6] ➜ m = 2 ➜ 233 x 163567
[38][1^^12] ➜ m = 4 ➜ 23333 x 1633356667
[38][1^^66] ➜ m = 22 ➜ [2][3^^22] x [16][3^^21][5][6^^21][7] or
23333333333333333333333 x
1633333333333333333333356666666666666666666667
```

Can you find more of these semiprimes ?

Note : a preliminary search revealed that the next semiprime
is greater than [38][1^^300000] !
Source http://stdkmd.net/nrr/2/23333.htm

 “ We should show why some families only contain composites ” a quote from Xinyao Chen. ``` Why some families only contain composites Base 10 Appending 1's: k = 37: n == 2 mod 3: factor of 3 n == 0 mod 3: factor of 37 n == 1 mod 6: factor of 7 n == 4 mod 6: factor of 13 k = 38: The form is 38*10^n+(10^n-1)/9 = (343*10^n-1)/9 For n == 0 mod 3, (343*10^n-1)/9 = (7*10^(n/3)-1)/3 * (49*10^(2*n/3)+7*10^(n/3)+1)/3, and (7*10^(n/3)-1)/3 >= (7*10^1-1)/3 = 23 > 1, (49*10^(2*n/3)+7*10^(n/3)+1)/3 >= (49*10^2+7*10^1+1)/3 = 1657 > 1, thus this factorization is nontrivial, and no number of this form is prime. n == 1 mod 3: factor of 3 n == 2 mod 3: factor of 37 k = 176: n == 0 mod 2: factor of 11 n == 1 mod 3: factor of 3 n == 5 mod 6: factor of 7 n == 3 mod 6: factor of 13 k = 209: n == 0 mod 2: factor of 11 n == 1 mod 3: factor of 3 n == 3 mod 6: factor of 7 n == 5 mod 6: factor of 13 k = 371: n == 1 mod 3: factor of 3 n == 2 mod 3: factor of 37 n == 0 mod 6: factor of 7 n == 3 mod 6: factor of 13 k = 381: The form is 381*10^n+(10^n-1)/9 = (343*10^(n+1)-1)/9 For n == 2 mod 3, (343*10^(n+1)-1)/9 = (7*10^((n+1)/3)-1)/3 * (49*10^(2*(n+1)/3)+7*10^((n+1)/3)+1)/3, and (7*10^((n+1)/3)-1)/3 >= (7*10^1-1)/3 = 23 > 1, (49*10^(2*(n+1)/3)+7*10^((n+1)/3)+1)/3 >= (49*10^2+7*10^1+1)/3 = 1657 > 1, thus this factorization is nontrivial, and no number of this form is prime. n == 0 mod 3: factor of 3 n == 1 mod 3: factor of 37 k = 407: n == 0 mod 2: factor of 11 n == 1 mod 3: factor of 3 n == 0 mod 3: factor of 37 n == 5 mod 6: factor of 7 Base 10 Appending 7's: k = 891: n == 0 mod 2: factor of 11 n == 0 mod 3: factor of 3 n == 1 mod 3: factor of 37 n == 5 mod 6: factor of 13 k = 1261: n == 1 mod 2: factor of 11 n == 2 mod 3: factor of 3 n == 1 mod 3: factor of 37 n == 0 mod 6: factor of 13 Primality certificates of the primes > 10^299, for the k less than the smallest k with unknown status: ( '^^' is symbol for concatenation ) [45][1^^772] = [451][1^^771]: http://factordb.com/cert.php?id=1100000000291663725 [56][1^^18470] = [561][1^^18469]: http://factordb.com/cert.php?id=1100000000301454592 [133][1^^2890]: http://factordb.com/cert.php?id=1100000000801396568 [215][1^^653]: http://factordb.com/cert.php?id=1100000000825681624 [232][1^^684]: http://factordb.com/cert.php?id=1100000000825681635 [243][1^^478]: http://factordb.com/cert.php?id=1100000000801396507 [320][1^^864]: http://factordb.com/cert.php?id=1100000000937772989 [453][1^^1943]: http://factordb.com/cert.php?id=1100000000937773059 [482][1^^1704]: http://factordb.com/cert.php?id=1100000000937773081 [529][1^^2778]: http://factordb.com/cert.php?id=1100000000843975000 [40][3^^483] = [403][3^^482]: http://factordb.com/cert.php?id=1100000000291649394 [580][3^^461]: http://factordb.com/cert.php?id=1100000000937220865 [736][3^^995]: http://factordb.com/cert.php?id=1100000001533509778 [95][7^^2904] = [957][7^^2903]: http://factordb.com/cert.php?id=1100000000291761426 [296][7^^434]: http://factordb.com/cert.php?id=1100000003573663500 [337][7^^2184]: http://factordb.com/cert.php?id=1100000000291742096 [436][7^^390]: http://factordb.com/cert.php?id=1100000003573663713 [480][7^^11330]: http://factordb.com/cert.php?id=1100000000843959754 [531][7^^316]: http://factordb.com/cert.php?id=1100000003573664422 [599][7^^2508]: http://factordb.com/cert.php?id=1100000000895929194 [711][7^^1648]: http://factordb.com/cert.php?id=1100000000895929179 [825][7^^391]: (proven prime by N-1 primality test, factorization of N-1 and primality certificate of large prime factor of N-1) [1184][7^^4646]: http://factordb.com/cert.php?id=1100000003573667026 [1258][7^^952]: http://factordb.com/cert.php?id=1100000003573667076 [1299][7^^1355]: http://factordb.com/cert.php?id=1100000003578024741 [1360][7^^1066]: http://factordb.com/cert.php?id=1100000003578024768 [1446][7^^335]: http://factordb.com/cert.php?id=1100000003578025624 [1551][7^^725]: http://factordb.com/cert.php?id=1100000003578025666 [1668][7^^428]: http://factordb.com/cert.php?id=1100000003578025806 [1709][7^^2838]: http://factordb.com/cert.php?id=1100000003578025432 [1745][7^^990]: http://factordb.com/cert.php?id=1100000003578026099 [1756][7^^1308]: http://factordb.com/cert.php?id=1100000003578026171 [1779][7^^428]: http://factordb.com/cert.php?id=1100000003578026485 [1811][7^^6068]: http://factordb.com/cert.php?id=1100000003578027866 [1914][7^^607]: http://factordb.com/cert.php?id=1100000003578028426 [1921][7^^778]: http://factordb.com/cert.php?id=1100000003578028510 [1987][7^^3600]: http://factordb.com/cert.php?id=1100000003573667101 ( [410][3^^37398], [851][7^^28895], [1881][7^^47927], are only PRPs, they are strong PRP to bases 2, 3, 5, 7, 11, 13, 17, 19, 23 and trial factored to 10^11, verified by PFGW ) These certificates do not include "appending digits b-1 in base b" since in this special case the numbers can be easily proven prime with the N+1 primality test, since their N+1 are trivially 100% factored. References: Reference of Base 10 Appending 1's: http://www.users.globalnet.co.uk/~perry/maths/wildeprimes/wildeprimes.htm (archived) Reference of Base 10 Appending 9's: Primes of the form k*10^n–1 ```

Xinyao Chen investigated this topic
also for other bases. Here is the link to that page
WONplate 219

A000197 Prime Curios! Prime Puzzle
Wikipedia 197 Le nombre 197
```

```