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[ December 6, 2015 ]
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.

Last update : [ December 29, 2016 ]

Let me start with a simple example to illustrate things
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.

[5][11] = [5][1] = 51 = 3 x 17 = composite
[5][12] = [5][11] = 511 = 7 x 73 = composite
[5][13] = [5][111] = 5111 = 19 x 269 = composite
[5][14] = [5][1111] = 51111 = 3 x 3 x 3 x 3 x 631 = composite
[5][15] = [5][11111] = 511111 = prime!

So if this is our next record delayed prime
than [5][15] 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][3n] divisible by 3.
Also for instance [37][1n] and [38][1n] 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.

(a few days later...) Yet with [603] things becomes very hard.
But there are three more columns left for you to fill in.
Maybe that the slope of increase of appended digits to reach
a prime will be less steep than in the first column.

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?


Prime by
appending 1's
Prime by
appending 3's
Prime by
appending 7's
Prime by
appending 9's
[1][11][1][31][1][71][1][91]
[2][12][13][314][2][72][4][92]
[5][15][40][3483][11][73][11][95]
[11][117][410][337398][20][76][31][928]
[12][1136][817][3 > 50000][29][748][88][933]
[45][1772] [73][766][97][990]
[56][118470] [95][72904][449][911958]
[603][1 > 300000] [480][711330][1342][929711]
  [851][728895][1802][945881]
  [1881][747927][1934][951836]
   [4420][9 > 1690000]
    
    
    
Found by
PDG
Found by
Gary Barnes
Upto [480] by
Jeff Heleen
From [851] by
Gary Barnes**
Upto [449] by
Jeff Heleen
From [1342] by
Gary Barnes***
*** Message from Gary Barnes [ December 14, 2016 ]

In the column "Prime by appending 9's" this problem
is the same as CRUS's Riesel base 10 problem as shown at
http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm.
Due to the substantial searching done by project CRUS
many additional terms can be added to the right column.
[4420] is still being searched with no prime yet found.
Because the appending of 9's after a [k]-value can be
reduced to the form (k+1)*10^n-1 all of these are proven primes.

** Message from Gary Barnes [ December 28, 2016 ]

In the column "Prime by appending 7's" I have
found the following PRP's:
1. (7666*10^28895-7)/9
2. (16936*10^47927-7)/9
These convert to:
1. 851*10^28895+(10^28895-1)*7/9
2. 1881*10^47927+(10^47927-1)*7/9
All k's in between these were searched for append 7.
k=891 has a covering set and so is always composite. See below.
I have not searched k > 1881.

Cases that produce only infinite composites
Classification (a.k.a. covering sets)

c1_1 = permutation of factors (11, 3, 11, 13, 3, 7)
c1_2 = permutation of factors (11, 3, 11, 37, 3, 7)
c1_3 = permutation of factors (7, 3, 37, 13, 3, 37)
c1_4 = infinite pattern of semiprimes

c3_1 = permutation of factors (13, 11, 37, 11, 7, 11)

c7_1 = permutation of factors (37, 11, 3, 11, 13, 11)

c9_1 = permutation of factors (7, 11, 37, 11, 13, 11)
[37][1n] = c1_3
[38][1n] = c1_4
[176][1n] = c1_1
[209][1n] = c1_1
[371][1n] = c1_3
[381][1n] = c1_4
[407][1n] = c1_2
[814][1n] = c1_2
[936][1n] = c1_2
[1023][1n] = c1_1
[1222][1n] = c1_1
[1353][1n] = c1_1
[1519][1n] = c1_1
[1750][1n] = c1_2
[1761][1n] = c1_1
[1904][1n] = c1_1
[1937][1n] = c1_1
[3 x n][3n]



[4070][3n] = c3_1
[7 x n][7n]



[891][7n] = c7_1
[3 x n][9n]



[10175][9n] = c9_1
[4070][3n] and [891][7n] likely found by
https://www.rose-hulman.edu/~rickert/Compositeseq/
[10175][9n] found by
http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm
Sequence [k] in order of increasing added digits
Each value < n in [k][1n] must be composite
[1][11]   
[2][12]   
[16][13]   
[36][14]   
[5][15]   
[29][16]   
[73][17]   
[17][18]   
[14][19]   
[100][110]   
[165][111]   
[197][112]   
[396][113]   
[185][114]   
[313][115]   
Continued here   
Narcissistic cases [p][1q] whereby p equals q
[1][11]   
[2][12]   
[5][15]   
Palindromic cases [p][1q] whereby p and q are palindromic
[1][11]   
[2][12]   
[5][15]   
[242][133]   
[595][188]   
Crossreferenced cases [p][1q] whereby p != q
[p][1q] & [q][1p]
None found sofar
   



Here is an elaborated example of how factorizations of
([38][1n]) 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][13], [38][16], [38][112] 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][121] = 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][121]{3 +3} becomes
[2][3m] x [16][3m-1][5][6m-1][7]
So the whole range of numbers is covered and
no primes can arise from [38][1n] !

Here is the extracted list of genuine semiprimes for [38][121]{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_322 ) x ( 16_321_5_621_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.com/nrr/2/23333.htm





A000197 Prime Curios! Prime Puzzle
Wikipedia 197 Le nombre 197














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E-mail address : pdg@worldofnumbers.com