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[ December 6, 2015 ] [  Last update October 26, 2024 ]
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
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.
( '^^' 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.
Click on the header titles to open the related worksheets.
The summary is at → Appending digits to k.htm
| BASE 10 |
Prime by
appending 1's | Prime by
appending 3's | Prime by
appending 7's | Prime by
appending 9's |
| [1][1^^1] | [1][3^^1] | [1][7^^1] | [1][9^^1] |
| [2][1^^2] | [13][3^^14] | [2][7^^2] | [4][9^^2] |
| [5][1^^5] | [40][3^^483] | [11][7^^3] | [11][9^^5] |
| [11][1^^17] | [410][3^^37398] | [20][7^^6] | [31][9^^28] |
| [12][1^^136] | [817][3^^ >554789] | [29][7^^48] | [88][9^^33] |
| [45][1^^772] | | [73][7^^66] | [97][9^^90] |
| [56][1^^18470] | | [95][7^^2904] | [449][9^^11958] |
| [603][1^^ >300000] | | [480][7^^11330] | [1342][9^^29711] |
| | | [851][7^^28895] | [1802][9^^45881] |
| | | [1881][7^^47927] | [1934][9^^51836] |
| | | [2038][7^^76206] | [4420][9^^ >5000000] |
| | | [2174][7^^94146] | |
| | | [4444][7^^ >100000] | |
| | | | |
| | | | |
| | | | |
Found by
PDG |
Found by
Gary Barnes |
Up to [480] by
Jeff Heleen
[851] & [1881]
by Gary Barnes**
[2038] to [4444]
by PDG |
Up to [449] by
Jeff Heleen
From [1342] by
Gary Barnes***
[4420] update by
Xinyao Chen |
See also
A069568 &
A083747
For prime k
A257459 |
See also
A090584
For prime k
A232210 |
See also
A090464 &
A363922
For prime k
A257460 |
See also
A090465
For prime k
A257461 |
Base 10 Appending 3’s:
k = 817 has been searched to 554789 digits with no prime or PRP found.
See https://www.rose-hulman.edu/~rickert/Compositeseq/#b10d3.
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*** 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.
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** 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, 11, 3, 7)
c1_2 = permutation of factors (11, 3, 11, 37, 11, 7)
c1_3 = permutation of factors (7, 3, 37, 13, 3, 37)
c1_4 = infinite pattern of semiprimes
c1_5 = permutation of factors (11, 3, 11, 13, 11, 37)
c1_6 = permutation of factors (11, 7, 11, 13, 11, 37)
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][1^^n] = c1_3
[38][1^^n] = c1_4
[176][1^^n] = c1_1
[209][1^^n] = c1_1
[371][1^^n] = c1_3
[381][1^^n] = c1_4
[407][1^^n] = c1_2
[814][1^^n] = c1_2
[936][1^^n] = c1_2 or c1_6
[1023][1^^n] = c1_1
[1222][1^^n] = c1_1 or c1_6
[1353][1^^n] = c1_1
[1519][1^^n] = c1_1
[1750][1^^n] = c1_2
[1761][1^^n] = c1_1
[1904][1^^n] = c1_1
[1937][1^^n] = c1_1
[2091][1^^n] = c1_1
[2146][1^^n] = c1_2
[2596][1^^n] = c1_1
[2739][1^^n] = c1_2 or c1_5
[2893][1^^n] = c1_1 |
[3 x n][3^^n]
[4070][3^^n] = c3_1 |
[7 x n][7^^n]
[891][7^^n] = c7_1
[1261][7^^n] = c7_1
[2889][7^^n] = c7_1
[3263][7^^n] = c7_1
[3300][7^^n] = c7_1
|
[3 x n][9^^n]
[10175][9^^n] = c9_1 |
[4070][3^^n] and [891][7^^n] likely found by
https://www.rose-hulman.edu/~rickert/Compositeseq/
[10175][9^^n] found by
http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm
k ⩾ 2091 appending 1's by Xinyao Chen |
Sequence [k] in order of increasing added digits
Each value < n in [k][d^^n] must be composite |
| By PDG | By Xinyao Chen |
| [1][1^^1] | [1][3^^1] | [1][7^^1] | [1][9^^1] |
| [2][1^^2] | [14][3^^2] | [2][7^^2] | [4][9^^2] |
| [16][1^^3] | [20][3^^3] | [11][7^^3] | [25][9^^3] |
| [36][1^^4] | [32][3^^4] | [81][7^^4] | [28][9^^4] |
| [5][1^^5] | [62][3^^5] | [32][7^^5] | [11][9^^5] |
| [29][1^^6] | [58][3^^6] | [20][7^^6] | [167][9^^6] |
| [73][1^^7] | [253][3^^7] | [52][7^^7] | [190][9^^7] |
| [17][1^^8] | [25][3^^8] | [51][7^^8] | [109][9^^8] |
| [14][1^^9] | [292][3^^9] | [59][7^^9] | [121][9^^9] |
| [100][1^^10] | [179][3^^10] | [37][7^^10] | [95][9^^10] |
| [165][1^^11] | [418][3^^11] | [302][7^^11] | [176][9^^11] |
| [197][1^^12] | [509][3^^12] | [40][7^^12] | [64][9^^12] |
| [396][1^^13] | [133][3^^13] | [940][7^^13] | [385][9^^13] |
| [185][1^^14] | [13][3^^14] | [408][7^^14] | [403][9^^14] |
| [299][1^^15] | [484][3^^15] | [275][7^^15] | [58][9^^15] |
| Continued here at seq197.htm |
| Narcissistic cases [p][1^^q] whereby p equals q |
| [1][1^^1] | [1][3^^1] | [1][7^^1] | [1][9^^1] |
| [2][1^^2] | | [2][7^^2] | |
| [5][1^^5] | | | |
Palindromic cases [p][1^^q] whereby p and q are palindromic
Only the cases of palindromic k followed by smallest 10^e digits n |
| [1][1^^1] | [1][3^^1] | [1][7^^1] | [1][9^^1] |
| [191][1^^11] | [434][3^^11] | [2112][7^^11] | [9449][9^^11] |
| [42324][1^^131] | [73637][3^^232] | [6996][7^^191] | [2882][9^^121] |
| [k^^pal][1^^????] | [k^^pal][3^^????] | [k^^pal][7^^????] | [k^^pal][9^^????] |
| [k^^pal][1^^?????] | [k^^pal][3^^?????] | [k^^pal][7^^?????] | [k^^pal][9^^?????] |
| Crossreferenced cases [p][1^^q] whereby p != q |
[p][1^^q]
& [q][1^^p] | [p][3^^q]
& [q][3^^p] | [p][7^^q]
& [q][7^^p] | [p][9^^q]
& [q][9^^p] |
Xinyao Chen couldn't find a single "crossreferenced" case
for [p][d^^q] & [q][d^^p], for any of d = 1, 3, 7, 9 |
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.localnet.co.uk/~perry/maths/wildeprimes/wildeprimes.htm (archived)
Reference of Base 10 Appending 9's:
Primes of the form k*10^n–1
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Xinyao Chen investigated this topic
also for other bases. Here is the link to that page
WONplate 219
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A000197 
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