[ May 19, 2022 ] [ Last update June 8, 2024 ]
Adding digits to n beyond base 10 so that n becomes
a record delayed prime {or a FEP = First Encountered Prime}
A sequel to WONplate 197 by Xinyao Chen (email)
First of all if this topic is new to you
familiarise yourself with it by studying WONplate 197
which deals only with the base 10 variant.
Xinyao Chen went further and delved into
the other bases as well. This gives cause to a lot of new material to list
and lots of opportunity to extend the tables.
So let us start with BASE 2
Possible digits to add are 0 and 1. But appending zero's will of course
never produce primes so only the digit 1 must be considered.
Note that the first number between the square brackets is
represented with its decimal expansion, for compactness.
So [12][1^^3] is to be interpreted as [1100]_{2}[1^^3] or
[1100][111] or 1100111 or in base 10 our prime 103 !
( '^^' is symbol for concatenation )
Click on the header titles to open the related worksheets from Xinyao Chen.
The main page is at → https://sites.google.com/view/worldofnumbersplate219/
BASE 3 
Prime by appending 1's  Prime by appending 2's 
[1][1^^2]  [1][2^^1] 
[5][1^^22]  [11][2^^2] 
[35][1^^46]  [31][2^^3] 
[48][1^^3131]  [41][2^^10] 
[59][1^^8972]  [281][2^^23] 
[156][1^^24761]  [301][2^^2091] 
[806][1^^ >107000]  [19255][2^^2347] 
 [95415][2^^5933] 
 [201885][2^^39101] 
 [301095][2^^49050] 
 [343371][2^^178255] 
 [3591445][2^^191177] 
 [3677877][2^^ >5000000] 
 
 
 
BASE 4 
Prime by appending 1's  Prime by appending 3's 
[1][1^^1]  [1][3^^1] 
[2][1^^2]  [13][3^^2] 
[19][1^^3]  [38][3^^12] 
[30][1^^17]  [73][3^^1276] 
[35][1^^4553]  [658][3^^400258] 
[250][1^^6615]  [4585][3^^6459215] 
[603][1^^ >70000]  [9518][3^^ >8388608] 
 
 
 
For kvalues such that 3*k+1 is square,
except k = 1, there are no possible primes,
since all [k][1^^n] can be factored as
difference of squares. 
For kvalues such that k+1 is square,
there are no possible primes, since all
[k][3^^n] can be factored as
difference of squares. 
Base 4 Appending 1's, k = 603 a coincide with
Base 10 Appending 1's : the smallest k has no prime or PRP
(nor can prove that such prime does not exist) is also 603. 
Base 4 Appending 1's : k = 120, 306, 371, 481, 495
have a covering set {3, 5, 7, 13} and so is always
composite (although k = 120 and 481 also have that 3*k+1 is square). 
The Base 4 Appending 1's except k = 35 and k = 124 are given by A177330,
other primes are fully searched by Chen, and he found some large primes as well:
[250][1^^6615], [386][1^^5628], [390][1^^2855], [396][1^^3404] 
The test limit of [9518][3^^n] in base 4 is given by
https://www.rieselprime.de/ziki/Riesel_prime_2_9519
http://www.rieselprime.de/Related/LiskovetsGallot.htm and
https://www.rieselprime.de/ziki/CRUS_LiskovetsGallot. 
BASE 5 
Prime by appending 1's  Prime by appending 2's  Prime by appending 3's  Prime by appending 4's 
[1][1^^2]  [1][2^^1]  [1][3^^2]  [1][4^^4] 
[18][1^^3]  [5][2^^2]  [5][3^^4]  [31][4^^8] 
[19][1^^12]  [11][2^^4]  [13][3^^10]  [33][4^^163] 
[23][1^^18]  [15][2^^5]  [53][3^^48]  [85][4^^2058] 
[29][1^^30]  [29][2^^6]  [65][3^^390]  [427][4^^9704] 
[122][1^^43]  [81][2^^7]  [83][3^^1552]  [661][4^^14628] 
[187][1^^54]  [103][2^^8]  [184][3^^26527]  [1395][4^^1146713] 
[222][1^^69]  [141][2^^11]  [796][3^^ >37000]  [3621][4^^7558139] 
[248][1^^565]  [153][2^^19]   [4905][4^^ >4800000] 
[434][1^^25415]  [179][2^^118]   
[1452][1^^ >36000]  [309][2^^279]   
 [467][2^^1560]   
 [639][2^^31357]   
 [1965][2^^ >51000]   
   
   
   
Appending 1's: For kvalues == 3 or 4 mod 6, there are no possible primes,
since all [k][1^^n] are divisible by either 2 or 3. 
Appending 4's: 4905 ( > 4800000, not a record if n ⩽ 7558139)
Related links :
The Sierpinski/Riesel Base 5 Project
Sierpinski/Riesel Base 5 project stats 
BASE 6 
Prime by appending 1's  Prime by appending 5's 
[1][1^^1]  [1][5^^1] 
[4][1^^2]  [12][5^^2] 
[8][1^^4]  [36][5^^4] 
[42][1^^15]  [53][5^^6] 
[50][1^^3008]  [68][5^^10] 
[525][1^^27871]  [91][5^^49] 
[1247][1^^ >86500]  [308][5^^557] 
 [908][5^^780] 
 [1029][5^^1199] 
 [1596][5^^ >6300000] 
 
 
 
Appending 1's:
Only the primes [525][1^^27871], [848][1^^7056] and the test limit of k=1247 are given by A217377. 
BASE 7 
Prime by appending 1's  Prime by appending 2's  Prime by appending 3's 
[1][1^^4]  [1][2^^3]  [1][3^^2] 
[5][1^^18]  [17][2^^4]  [16][3^^3] 
[12][1^^127]  [31][2^^5]  [19][3^^4] 
[13][1^^424]  [37][2^^26]  [20][3^^5] 
[23][1^^468]  [59][2^^102]  [26][3^^7] 
[52][1^^5907]  [263][2^^387]  [29][3^^32] 
[61][1^^15118]  [327][2^^389]  [49][3^^104] 
[113][1^^ >42000]  [761][2^^624]  [79][3^^4896] 
 [1359][2^^1078]  [98][3^^181761] 
 [2047][2^^32613]  [358][3^^ >38000] 
 [4011][2^^ >36000]  
  
  
  
Prime by appending 4's  Prime by appending 5's  Prime by appending 6's 
[1][4^^1]  [1][5^^2]  [1][6^^1] 
[3][4^^2]  [3][5^^4]  [7][6^^4] 
[13][4^^78]  [16][5^^5]  [73][6^^5] 
[57][4^^188]  [17][5^^10]  [107][6^^7] 
[325][4^^226]  [27][5^^14]  [163][6^^21] 
[405][4^^239]  [48][5^^403]  [239][6^^24] 
[507][4^^ >34000]  [59][5^^816]  [515][6^^37] 
 [63][5^^910]  [557][6^^382] 
 [69][5^^12274]  [6413][6^^399] 
 [116][5^^ >35000]  [6751][6^^805] 
  [7733][6^^953] 
  [17243][6^^2703] 
  [48251][6^^3758] 
  [48583][6^^56816] 
  [253717][6^^63295] 
  [315767][6^^ >1000000] 
  
  
  
Appending 1's:
k = 76 has a covering set {2, 3, 5, 13, 19} 
Appending 3's:
358 ( > 38000, not a record if n ⩽ 181761) 
Appending 5's:
k = 43 has a covering set {2, 3, 19, 43} 
BASE 8 
Prime by appending 1's  Prime by appending 3's  Prime by appending 5's  Prime by appending 7's 
[1][1^^2]  [1][3^^1]  [1][5^^1]  [1][7^^2] 
[3][1^^3]  [4][3^^2]  [2][5^^2]  [4][7^^4] 
[4][1^^12]  [11][3^^3]  [9][5^^3]  [10][7^^18] 
[6][1^^21]  [14][3^^8]  [11][5^^4]  [36][7^^851] 
[13][1^^314]  [40][3^^5607]  [17][5^^10]  [73][7^^2632] 
[34][1^^ >60000]  [6865][3^^9949]  [27][5^^11]  [235][7^^5258] 
 [8552][3^^18060]  [32][5^^40]  [246][7^^ >766666] 
 [13892][3^^ >30000]  [54][5^^53]  
  [143][5^^302]  
  [183][5^^519]  
  [188][5^^3512]  
  [252][5^^3657]  
  [486][5^^ >37000]  
   
   
   
Appending 1's: For kvalues such that 7*k+1 is cube, except k=1 and k=9, there are no possible
primes, since all [k][1^^n] can be factored as differences of cubes.
k == 21, 100, 169, 183 (mod 195) have a covering set {3, 5, 13}
and so are always composite. 
Appending 7's: For kvalues such that k+1 is cube, there are no possible primes
since all [k][7^^n] can be factored as differences of cubes.
k == 13, 111, 115, 147 (mod 195) have a covering set {3, 5, 13}
and so are always composite.
k = 657 has a covering set {3, 5, 19, 37, 73} 
The test limit of [246][7^^n] in base 8 is given by
https://www.rieselprime.de/ziki/Riesel_prime_2_247 and
http://www.prothsearch.com/riesel2.html (page is not yet updated)
and http://www.15k.org/riesellist.html (archived),
base 2 converted to base 8. 
BASE 9 
Prime by appending 1's  Prime by appending 2's  Prime by appending 4's 
[2][1^^1]  [1][2^^1]  [1][4^^1] 
[7][1^^2]  [7][2^^2]  [5][4^^11] 
[11][1^^8]  [13][2^^536]  [35][4^^23] 
[17][1^^12]  [67][2^^718]  [59][4^^4486] 
[43][1^^16]  [307][2^^ >33000]  [469][4^^12380] 
[51][1^^36]   [915][4^^ >34000] 
[88][1^^1797]   
[132][1^^2967]   
[598][1^^ >40000]   
  
  
  
Prime by appending 5's  Prime by appending 7's  Prime by appending 8's 
[1][5^^2]  [1][7^^2]  [1][8^^1] 
[8][5^^3]  [9][7^^12]  [13][8^^8] 
[9][5^^4]  [31][7^^14]  [93][8^^12] 
[19][5^^442]  [33][7^^54]  [157][8^^69] 
[123][5^^ >34000]  [73][7^^428]  [301][8^^2849] 
 [104][7^^10171]  [385][8^^ >53000] 
 [514][7^^ >37000]  
  
  
  
Appending 1's:
For kvalues == 5 or 6 mod 10, there are no possible primes,
since all [k][1^^n] are divisible by either 2 or 5.
For kvalues such that 8*k+1 is square, there are no possible primes,
since all [k][1^^n] can be factored as difference of squares. 
Appending 5's:
k = 78 has a covering set {2, 7, 13, 73}. 
Appending 7's:
For kvalues == 2 or 5 mod 10, there are no possible primes,
since all [k][7^^n] are divisible by either 2 or 5. 
Appending 8's:
For kvalues such that k+1 is square, there are no possible primes,
since all [k][8^^n] can be factored as difference of squares.
k = 73 has a covering set {5, 7, 13, 73}. 
The primes [51][1^^36] and [5][4^^11] correspond to the minimal primes
(see WONplate 218) 56111111111111111111111111111111111111
and 544444444444, respectively, in base 9. 
BASES 2 to 10 
Overwiew for Bases 2 to 10 is given here at Composite Sequences.
The list only includes the k < (the smallest k such that the numbers
are always composite), so for example,
for Base 5 Appending digits 1, that page only includes k < 3.
Unfortunately the list date and/or last update is not given.
By John Rickert, Professor of Mathematics. 
The large primes for "appending digits b1 in base b"
(e.g. appending digits 3 in base 4, appending digits 4 in base 5)
are not searched by Xinyao Chen, they are given by the
CRUS Riesel problem page.

“ We should show why some families only contain composites ”
a quote from Xinyao Chen. 
Why some families only contain composites
Base 4 Appending 1's:
The form is k*4^n+(4^n1)/3 = ((3*k+1)*4^n1)/3
kvalues such that 3*k+1 is a square: Let 3*k+1 = m^2,
((3*k+1)*4^n1)/3 = (m^2*4^n1)/3 = (m*2^n1) * (m*2^n+1) / 3,
the only possible prime cases are m*2^n1 = 3 and m*2^n+1 = 3,
but if m*2^n+1 = 3 then m*2^n = 2 then m*2^n1 = 1 is not prime,
thus m*2^n1 = 3, and hence (m,n) = (1,2) or (2,1),
but if m = 1 then k = 0, which is not included in this puzzle,
thus m = 2 (and k = 1) and n = 1 is the only possible prime case.
k = 120:
n == 0 mod 2: factor of 5
n == 0 mod 3: factor of 3
n == 2 mod 3: factor of 7
n == 1 mod 6: factor of 13
k = 306:
n == 1 mod 2: factor of 5
n == 0 mod 3: factor of 3
n == 1 mod 3: factor of 7
n == 2 mod 6: factor of 13
k = 371:
n == 1 mod 2: factor of 5
n == 1 mod 3: factor of 3
n == 0 mod 3: factor of 7
n == 2 mod 6: factor of 13
k = 481:
n == 1 mod 2: factor of 5
n == 2 mod 3: factor of 3
n == 1 mod 3: factor of 7
n == 0 mod 6: factor of 13
k = 495:
n == 0 mod 2: factor of 5
n == 0 mod 3: factor of 3
n == 1 mod 3: factor of 7
n == 5 mod 6: factor of 13
Base 4 Appending 3's:
The form is k*4^n+3*(4^n1)/3 = (k+1)*4^n1
kvalues such that k+1 is a square: Let k+1 = m^2,
(k+1)*4^n1 = m^2*4^n1 = (m*2^n1) * (m*2^n+1), and since k⩾1,
thus m⩾2, and hence m*2^n1 ⩾ 2*2^11 = 3 > 1,
m*2^n+1 ⩾ 2*2^1+1 = 5 > 1,
thus this factorization is nontrivial,
and no number of this form is prime.
Base 5 Appending 1's:
k == 3 mod 6:
n == 1 mod 2: factor of 2
n == 0 mod 2: factor of 3
k == 4 mod 6:
n == 0 mod 2: factor of 2
n == 1 mod 2: factor of 3
Base 7 Appending 1's:
k = 76:
n == 0 mod 2: factor of 2
n == 2 mod 3: factor of 3
n == 0 mod 3: factor of 19
n == 3 mod 4: factor of 5
n == 1 mod 12: factor of 13
k = 215:
n == 1 mod 2: factor of 2
n == 1 mod 3: factor of 3
n == 0 mod 4: factor of 5
n == 0 mod 6: factor of 43
n == 2 mod 12: factor of 13
Base 7 Appending 5's:
k = 43:
n == 1 mod 2: factor of 2
n == 1 mod 3: factor of 3
n == 2 mod 3: factor of 19
n == 0 mod 6: factor of 43
k = 306:
n == 0 mod 2: factor of 2
n == 0 mod 3: factor of 3
n == 1 mod 3: factor of 19
n == 5 mod 6: factor of 43
Base 8 Appending 1's:
The form is k*8^n+(8^n1)/7 = ((7*k+1)*8^n1)/7
kvalues such that 7*k+1 is a cube: Let 7*k+1 = m^3,
((7*k+1)*8^n1)/7 = (m^3*8^n1)/7 = (m*2^n1) * (m^2*4^n+m*2^n+1) / 7,
the only possible prime cases are m*2^n1 = 7 and m^2*4^n+m*2^n+1 = 7,
but if m^2*4^n+m*2^n+1 = 7 then m*2^n = 2 then m*2^n1 = 1 is not prime,
thus m*2^n1 = 7, and hence (m,n) = (1,3) or (2,2) or (4,1),
but if m = 1 then k = 0, which is not included in this puzzle,
thus m = 2 (and k = 1) and n = 2 and m = 4 (and k = 9) and n = 1 are
the only possible prime cases.
k == 21 mod 195:
n == 0 mod 2: factor of 3
n == 3 mod 4: factor of 5
n == 1 mod 4: factor of 13
k == 100 mod 195:
n == 1 mod 2: factor of 3
n == 0 mod 4: factor of 5
n == 2 mod 4: factor of 13
k == 169 mod 195:
n == 1 mod 2: factor of 3
n == 2 mod 4: factor of 5
n == 0 mod 4: factor of 13
k == 183 mod 195:
n == 0 mod 2: factor of 3
n == 1 mod 4: factor of 5
n == 3 mod 4: factor of 13
Base 8 Appending 7's:
The form is k*8^n+7*(8^n1)/7 = (k+1)*8^n1
kvalues such that k+1 is a cube: Let k+1 = m^3,
(k+1)*8^n1 = m^3*8^n1 = (m*2^n1) * (m^2*4^n+m*2^n+1),
and since k⩾1, thus m⩾2, and hence m*2^n1 ⩾ 2*2^11 = 3 > 1,
m^2*4^n+m*2^n+1 ⩾ 2^2*4^1+2*2^1+1 = 21 > 1,
thus this factorization is nontrivial,
and no number of this form is prime.
k == 13 mod 195:
n == 1 mod 2: factor of 3
n == 2 mod 4: factor of 5
n == 0 mod 4: factor of 13
k == 111 mod 195:
n == 0 mod 2: factor of 3
n == 1 mod 4: factor of 5
n == 3 mod 4: factor of 13
k == 115 mod 195:
n == 1 mod 2: factor of 3
n == 0 mod 4: factor of 5
n == 2 mod 4: factor of 13
k == 147 mod 195:
n == 0 mod 2: factor of 3
n == 3 mod 4: factor of 5
n == 1 mod 4: factor of 13
Base 9 Appending 1's:
The form is k*9^n+(9^n1)/8 = ((8*k+1)*9^n1)/8
kvalues such that 8*k+1 is a square: Let 8*k+1 = m^2,
((8*k+1)*9^n1)/8 = (m^2*9^n1)/8 = (m*3^n1) * (m*3^n+1) / 8,
and since k⩾1, thus m⩾2, and hence m*3^n1 ⩾ 2*3^11 = 5 > 4,
m*3^n+1 ⩾ 2*3^1+1 = 7 > 4, thus the only possible prime cases
are m*3^n1 = 8 and m*3^n+1 = 8,
but if m*3^n1 = 8 then m*3^n = 9 then m*3^n+1 = 10 is not prime,
and if m*3^n+1 = 8 then m*3^n = 7 then m*3^n1 = 6 is not prime,
thus this factorization is nontrivial,
and no number of this form is prime.
k == 5 mod 10:
n == 1 mod 2: factor of 2
n == 0 mod 2: factor of 5
k == 6 mod 10:
n == 0 mod 2: factor of 2
n == 1 mod 2: factor of 5
Base 9 Appending 5's:
k = 78:
n == 0 mod 2: factor of 2
n == 1 mod 3: factor of 7
n == 0 mod 3: factor of 13
n == 5 mod 6: factor of 73
Base 9 Appending 7's:
k == 2 mod 10:
n == 0 mod 2: factor of 2
n == 1 mod 2: factor of 5
k == 5 mod 10:
n == 1 mod 2: factor of 2
n == 0 mod 2: factor of 5
Base 9 Appending 8's:
The form is k*9^n+8*(9^n1)/8 = (k+1)*9^n1
kvalues such that k+1 is a square: Let k+1 = m^2,
(k+1)*9^n1 = m^2*9^n1 = (m*3^n1) * (m*3^n+1), and since k⩾1,
thus m⩾2, and hence m*3^n1 ⩾ 2*3^11 = 5 > 1,
m*3^n+1 ⩾ 2*3^1+1 = 7 > 1,
thus this factorization is nontrivial,
and no number of this form is prime.
k = 73:
n == 1 mod 2: factor of 5
n == 1 mod 3: factor of 7
n == 2 mod 3: factor of 13
n == 0 mod 6: factor of 73
Primality certificates of the primes > 10^299,
for the k less than the smallest k with unknown status:
( '^^' is symbol for concatenation )
Base 3:
[48][1^^3131] = [145][1^^3130] = [436][1^^3129]:
http://factordb.com/cert.php?id=1100000000778119927
[59][1^^8972] = [178][1^^8971] = [535][1^^8970]:
http://factordb.com/cert.php?id=1100000000854475920
[96][1^^685] = [289][1^^684]:
http://factordb.com/cert.php?id=1100000000778120663
[156][1^^24761] = [469][1^^24760]:
http://factordb.com/cert.php?id=1100000000854475741
[299][1^^1240]: http://factordb.com/cert.php?id=1100000000778120737
[405][1^^1126]: http://factordb.com/cert.php?id=1100000000914446771
[498][1^^20847]: http://factordb.com/cert.php?id=1100000000854475829
[506][1^^1233]: http://factordb.com/cert.php?id=1100000000914446784
[546][1^^1297]: http://factordb.com/cert.php?id=1100000001545712771
[599][1^^3876]: http://factordb.com/cert.php?id=1100000001545712764
[651][1^^1384]: http://factordb.com/cert.php?id=1100000001545712756
Base 4:
[35][1^^4553] = [141][1^^4552] = [565][1^^4551]:
http://factordb.com/cert.php?id=1100000000350048535
[124][1^^2508] = [497][1^^2507]:
http://factordb.com/cert.php?id=1100000000781123914
[166][1^^648]: (proven prime by N1 primality test, factorization of N1
and primality certificate of large prime factor of N1)
[250][1^^6615]: http://factordb.com/cert.php?id=1100000000891891792
[386][1^^5628]: http://factordb.com/cert.php?id=1100000001551808249
[390][1^^2855]: http://factordb.com/cert.php?id=1100000001533524542
[396][1^^3404]: http://factordb.com/cert.php?id=1100000001551808435
Base 5:
[248][1^^565] = [1241][1^^564]:
http://factordb.com/cert.php?id=1100000000934850627
[434][1^^25415]: http://factordb.com/cert.php?id=1100000003523841628
[583][1^^6238]: http://factordb.com/cert.php?id=1100000003529464162
[704][1^^2489]: http://factordb.com/cert.php?id=1100000003529464635
[713][1^^908]: http://factordb.com/cert.php?id=1100000003569065204
[863][1^^436]: http://factordb.com/cert.php?id=1100000003569065122
[905][1^^2124]: http://factordb.com/cert.php?id=1100000003529464944
[954][1^^3823]: http://factordb.com/cert.php?id=1100000003529465492
[1032][1^^755]: http://factordb.com/cert.php?id=1100000003569065391
[1085][1^^14720]: http://factordb.com/cert.php?id=1100000003529466005
[1147][1^^16892]: http://factordb.com/cert.php?id=1100000003529466486
[1175][1^^524]: http://factordb.com/cert.php?id=1100000003569065537
[1184][1^^4905]: http://factordb.com/cert.php?id=1100000003529466992
[1189][1^^1152]: http://factordb.com/cert.php?id=1100000003569065585
[467][2^^1560]: http://factordb.com/cert.php?id=1100000000934850656
[1091][2^^2128]: http://factordb.com/cert.php?id=1100000000907434346
[1211][2^^2302]: http://factordb.com/cert.php?id=1100000003573645684
[1271][2^^610]: http://factordb.com/cert.php?id=1100000003573645893
[1313][2^^3282]: http://factordb.com/cert.php?id=1100000003573645941
[1479][2^^835]: http://factordb.com/cert.php?id=1100000003573646375
[1485][2^^12899]: http://factordb.com/cert.php?id=1100000000907434332
[1859][2^^526]: (proven prime by N1 primality test, factorization of N1
and primality certificate of large prime factor of N1)
[83][3^^1552] = [418][3^^1551]:
http://factordb.com/cert.php?id=1100000003529469440
[187][3^^988]: http://factordb.com/cert.php?id=1100000003573647710
[349][3^^804]: http://factordb.com/cert.php?id=1100000003573648403
[392][3^^479]: http://factordb.com/cert.php?id=1100000003573648424
[545][3^^2370]: http://factordb.com/cert.php?id=1100000002183244201
[626][3^^1711]: http://factordb.com/cert.php?id=1100000003573648892
[639][2^^31357] is only PRP,
it is strong PRP to bases 2, 3, 5, 7, 11, 13, 17, 19, 23
and trial factored to 10^11, verified by PFGW )
Base 6:
[50][1^^3008] = [301][1^^3007]:
http://factordb.com/cert.php?id=1100000000775146416
[358][1^^414]: http://factordb.com/cert.php?id=1100000000775146470
[848][1^^7056]: http://factordb.com/cert.php?id=1100000000854476279
[904][1^^1392]: http://factordb.com/cert.php?id=1100000000775146499
[1148][1^^1189]: http://factordb.com/cert.php?id=1100000001083192073
( [525][1^^27871] is only PRP, it is strong PRP to bases 2, 3, 5, 7, 11,
13, 17, 19, 23 and trial factored to 10^11, verified by PFGW )
Base 7:
[13][1^^424] = [92][1^^423]:
http://factordb.com/cert.php?id=1100000000854476434
[23][1^^468]: (proven prime by N1 primality test, factorization of N1
and primality certificate of large prime factor of N1)
[52][1^^5907]: http://factordb.com/cert.php?id=1100000000887911292
[61][1^^15118]: http://factordb.com/cert.php?id=1100000000887911299
[65][1^^938]: http://factordb.com/cert.php?id=1100000000887902040
[75][1^^398]: http://factordb.com/cert.php?id=1100000000900877290
[80][1^^735]: http://factordb.com/cert.php?id=1100000001526113753
[263][2^^387] = [1843][2^^386]:
http://factordb.com/cert.php?id=1100000001526114027
[327][2^^389] = [2291][2^^388]:
(proven prime by N1 primality test, factorization of N1
and primality certificate of large prime factor of N1)
[761][2^^624]: http://factordb.com/cert.php?id=1100000003569054026
[1359][2^^1078]: http://factordb.com/cert.php?id=1100000003569054008
[1501][2^^416]: http://factordb.com/cert.php?id=1100000003569061159
[1775][2^^394]: http://factordb.com/cert.php?id=1100000003569061101
[1935][2^^467]: http://factordb.com/cert.php?id=1100000003569061057
[2771][2^^780]: http://factordb.com/cert.php?id=1100000003658037125
[2811][2^^619]: (proven prime by N+1 primality test, factorization of N+1
and primality certificate of large prime factor of N+1)
[3099][2^^1315]: http://factordb.com/cert.php?id=1100000003658039367
[3403][2^^510]: http://factordb.com/cert.php?id=1100000003658040991
[3451][2^^519]: http://factordb.com/cert.php?id=1100000003658042038
[3631][2^^1956]: http://factordb.com/cert.php?id=1100000003658044253
[79][3^^4896]: http://factordb.com/cert.php?id=1100000000887911277
[149][3^^600]: http://factordb.com/cert.php?id=1100000000900877143
[209][3^^1052]: http://factordb.com/cert.php?id=1100000000887911460
[214][3^^3815]: http://factordb.com/cert.php?id=1100000000887911327
[218][3^^385]: http://factordb.com/cert.php?id=1100000000900877277
[247][3^^716]: http://factordb.com/cert.php?id=1100000001526113848
[326][3^^1051]: http://factordb.com/cert.php?id=1100000001526113949
(appending digit 4 has no prime number > 10^299 for k < 507)
[48][5^^403]: http://factordb.com/cert.php?id=1100000003569060790
[59][5^^816]: http://factordb.com/cert.php?id=1100000003569060734
[63][5^^910]: http://factordb.com/cert.php?id=1100000003569060669
[69][5^^12274]: http://factordb.com/cert.php?id=1100000003575248452
( [2261][2^^13096], [3601][2^^24699] and [98][3^^181761] are only PRP,
they are strong PRP to bases 2, 3, 5, 7, 11, 13, 17, 19, 23
and trial factored to 10^11, verified by PFGW )
Base 8:
(appending digit 1 has no prime number > 10^299 for k < 34)
[40][3^^5607] = [323][3^^5606] = [2587][3^^5605]:
http://factordb.com/cert.php?id=1100000000891670908
[554][3^^4467] = [4435][3^^4466]:
http://factordb.com/cert.php?id=1100000003578218898
[1025][3^^1498] = [8203][3^^1497]:
http://factordb.com/cert.php?id=1100000003578218992
[1394][3^^2118] = [11155][3^^2117]:
http://factordb.com/cert.php?id=1100000003578219707
[2264][3^^336]: http://factordb.com/cert.php?id=1100000003578219914
[2282][3^^1109]: http://factordb.com/cert.php?id=1100000003578220269
[3023][3^^785]: http://factordb.com/cert.php?id=1100000003578221448
[3367][3^^485]: http://factordb.com/cert.php?id=1100000003578221501
[3385][3^^930]: http://factordb.com/cert.php?id=1100000003578221574
[4250][3^^795]: http://factordb.com/cert.php?id=1100000003578221595
[4289][3^^562]: http://factordb.com/cert.php?id=1100000003578221834
[5210][3^^530]: http://factordb.com/cert.php?id=1100000003578222238
[5359][3^^1312]: http://factordb.com/cert.php?id=1100000003578222249
[5434][3^^335]: http://factordb.com/cert.php?id=1100000003578222268
[5575][3^^402]: http://factordb.com/cert.php?id=1100000003578222280
[5692][3^^455]: http://factordb.com/cert.php?id=1100000003578222289
[6865][3^^9949]: http://factordb.com/cert.php?id=1100000003569053927
[6994][3^^574]: http://factordb.com/cert.php?id=1100000003578222819
[7124][3^^582]: http://factordb.com/cert.php?id=1100000003578222943
[7213][3^^964]: http://factordb.com/cert.php?id=1100000003578223237
[7382][3^^808]: http://factordb.com/cert.php?id=1100000003578223598
[7484][3^^2107]: http://factordb.com/cert.php?id=1100000003578223890
[7588][3^^789]: http://factordb.com/cert.php?id=1100000003578224061
[7657][3^^3133]: http://factordb.com/cert.php?id=1100000003578224702
[8047][3^^959]: http://factordb.com/cert.php?id=1100000003578225115
[8359][3^^538]: http://factordb.com/cert.php?id=1100000003578225349
[8617][3^^2011]: http://factordb.com/cert.php?id=1100000003578226322
[9985][3^^5855]: http://factordb.com/cert.php?id=1100000003578227236
[10325][3^^341]: http://factordb.com/cert.php?id=1100000003578227649
[10442][3^^1021]: http://factordb.com/cert.php?id=1100000003578227753
[10804][3^^367]: http://factordb.com/cert.php?id=1100000003578228158
[10988][3^^436]: http://factordb.com/cert.php?id=1100000003578228390
[11143][3^^557]: http://factordb.com/cert.php?id=1100000003578228531
[11167][3^^2067]: http://factordb.com/cert.php?id=1100000003578230050
[11594][3^^639]: http://factordb.com/cert.php?id=1100000003578230512
[12383][3^^377]: http://factordb.com/cert.php?id=1100000003578230528
[12388][3^^1210]: http://factordb.com/cert.php?id=1100000003578230847
[12773][3^^658]: http://factordb.com/cert.php?id=1100000003578231155
[12842][3^^819]: http://factordb.com/cert.php?id=1100000003578231310
[13388][3^^472]: http://factordb.com/cert.php?id=1100000003578231515
[183][5^^519]: http://factordb.com/cert.php?id=1100000003569053824
[188][5^^3512]: http://factordb.com/cert.php?id=1100000003569053769
[252][5^^3657]: http://factordb.com/cert.php?id=1100000003569053799
[296][5^^1502]: http://factordb.com/cert.php?id=1100000003569064669
[461][5^^428]: http://factordb.com/cert.php?id=1100000003569064623
( [8552][3^^18060] is only PRP,
it is strong PRP to bases 2, 3, 5, 7, 11, 13, 17, 19, 23
and trial factored to 10^11, verified by PFGW )
Base 9:
[88][1^^1797]: http://factordb.com/cert.php?id=1100000000934850943
[127][1^^708]: http://factordb.com/cert.php?id=1100000000934850856
[132][1^^2967]: http://factordb.com/cert.php?id=1100000003569089441
[204][1^^1167]: http://factordb.com/cert.php?id=1100000003569089475
[297][1^^564]: http://factordb.com/cert.php?id=1100000003569090317
[322][1^^815]: http://factordb.com/cert.php?id=1100000003569090350
[452][1^^593]: http://factordb.com/cert.php?id=1100000003569090604
[13][2^^536] = [119][2^^535]:
http://factordb.com/cert.php?id=1100000000934847239
[67][2^^718]: http://factordb.com/cert.php?id=1100000000934851029
[59][4^^4486] = [535][4^^4485]:
http://factordb.com/cert.php?id=1100000000854475920
[145][4^^1565]: http://factordb.com/cert.php?id=1100000000778119927
[289][4^^342]: http://factordb.com/cert.php?id=1100000000778120663
[299][4^^620]: http://factordb.com/cert.php?id=1100000000778120737
[405][4^^563]: http://factordb.com/cert.php?id=1100000000914446771
[469][4^^12380]: http://factordb.com/cert.php?id=1100000000854475741
[599][4^^1938]: http://factordb.com/cert.php?id=1100000001545712764
[651][4^^692]: http://factordb.com/cert.php?id=1100000001545712756
[19][5^^442]: http://factordb.com/cert.php?id=1100000000890678886
[73][7^^428]: http://factordb.com/cert.php?id=1100000003573090435
[104][7^^10171]: http://factordb.com/cert.php?id=1100000003584935099
[129][7^^1128]: http://factordb.com/cert.php?id=1100000003584955284
[143][7^^2258]: http://factordb.com/cert.php?id=1100000003584966842
[163][7^^480]: http://factordb.com/cert.php?id=1100000003584979201
[174][7^^377]: http://factordb.com/cert.php?id=1100000003584988062
[321][7^^444]: http://factordb.com/cert.php?id=1100000003585010661
These certificates do not include "appending digits b1 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:
For the repunit (i.e. Appending 1's, and k itself is a repunit in base b)
case to other bases b, see A084740,
http://www.fermatquotient.com/PrimSerien/GenRepu.txt,
http://www.users.globalnet.co.uk/~aads/primes.html (archived),
http://www.primenumbers.net/Henri/us/MersFermus.htm,
Generalized Repunit Primes (Harvey Dubner)
For the Appending (b1)'s case to other bases b,
see http://www.noprimeleftbehind.net/crus/Rieselconjectures.htm
and http://www.noprimeleftbehind.net/crus/Rieselconjecturespowers2.htm
(only include the k < (the smallest k such that the numbers are
always composite)), and there is a condensed table.
Semiprimes:
Many [k][d^^n] in base b can be factored as difference of squares or
difference of cubes, they are also infinite pattern of semiprimes
like [38][1^^n] in base 10 (see WONplate 197).
Heuristically, the number of semiprimes in all these families (including
[38][1^^n] with n divisible by 3 in base 10, in WONplate 197) is finite,
unlike the number of primes in the families [k][d^^n] in this WONplate
and WONplate 197 and the families x(d^^n)y in WONplate 218, which is
heuristically to be infinite, unless they can be proven to only contain
composites or only contain finitely many primes, by covering congruence,
algebraic factorization, or a combination of them.
Since by the prime number theorem, the probability that a random ndigit
number in base b is prime is about 1/(n*ln(b)), thus for the families
[k][d^^n] in this WONplate and WONplate 197 and the families x(d^^n)y
in WONplate 218, if we conjecture that the numbers in such families behave
similarly we would expect (1/1 + 1/2 + 1/3 + 1/4 + …)/ln(b) = ∞ primes
in these families (of course, this does not always happen, since some
families can be proven to only contain composites or only contain finitely
many primes (by covering congruence, algebraic factorization, or a combination
of them), and every family has its own Nash weight (or difficulty), families
which can be proven to only contain composites or only contain finitely many
primes have Nash weight (or difficulty) 0, but it is at least a reasonable
conjecture in the absence of evidence to the contrary). However, in this case
of semiprimes, we require two numbers to be prime simultaneously, thus in the
case of difference of squares (i.e. the cases of base 4 and base 9), we
require two ndigit numbers to be prime simultaneously, thus we would expect
(1/1 + 1/4 + 1/9 + 1/16 + …)/ln(b) = (pi^2)/6/ln(b) semiprimes in
these families, and in the case of difference of cubes (i.e. the cases of
base 8 and base 10 family [38][1^^n] with n divisible by 3), we require an
ndigit number and a 2*ndigit number to be prime simultaneously, thus we
would expect (1/2 + 1/8 + 1/18 + 1/32 + …)/ln(b) = (pi^2)/12/ln(b)
semiprimes in these families (for the case of base 10 family [38][1^^n] with
n divisible by 3, since the n must be divisible by 3, the number should be
multiplied by 1/3, thus we would expect (pi^2)/36/ln(10) semiprimes in the
family [38][1^^n] in base 10 with n divisible by 3).
Base 4:
[5][1^^n] is semiprime for n = 1, 2, 3, 5, 11, 15, 17, 29, 59, 125
and the next semiprime (if exists) must be greater than [5][1^^65008905],
[5][1^^n] is semiprime (except the case n=2) is equivalent to the Mersenne
number 2^(n+2)1 and the Wagstaff number (2^(n+2)+1)/3 are both primes
(see A107360 for the status, also see A000043 and A000978, currently
all Mersenne exponents ⩽ 65008907 have been checked,
see https://www.mersenne.org),
and this is already studied in the New Mersenne Conjecture, references:
http://www.hoegge.dk/mersenne/NMC.html,
https://primes.utm.edu/mersenne/NewMersenneConjecture.html,
https://primes.utm.edu/glossary/xpage/NewMersenneConjecture.html,
http://www.primenumbers.net/rl/nmc/ and
http://mprime.s3website.uswest1.amazonaws.com/new_mersenne_conjecture.html (archived).^{ }
[8][1^^n] is semiprime for n = 1, 2, 3, 12, 72 and the next semiprime^{ }
(if exists) must be greater than [8][1^^9000000], see A001770 and A002254,
[8][1^^n] is semiprime is equivalent to 5*2^n1 and (5*2^n+1)/3 are both
primes, or 5*2^n+1 and (5*2^n1)/3 are both primes.
[16][1^^n] is semiprime for n = 1, 4, 6 and the next semiprime (if exists)
must be greater than [16][1^^5000000], see A001771 and A032353,
[16][1^^n] is semiprime is equivalent to 7*2^n1 and (7*2^n+1)/3 are
both primes, or 7*2^n+1 and (7*2^n1)/3 are both primes.
The only semiprime [120][1^^n] is 13201 (481 in decimal), since all
numbers [120][1^^n] are divisible by at least one of {3, 5, 7, 13}.
[8][3^^n] is semiprime for n = 1, 2, 6, 18 and the next semiprime
(if exists) must be greater than [8][3^^17150000], see A002235 and A002253,
[8][3^^n] is semiprime is equivalent to 3*2^n1 and 3*2^n+1 are both
primes.
[80][3^^n] is semiprime for n = 1, 3, 7, 43, 63, 211 and the next semiprime
(if exists) must be greater than [80][3^^6500000], see A002236 and A002256,
[80][3^^n] is semiprime is equivalent to 9*2^n1 and 9*2^n+1 are both
primes.
For semiprime [k][3^^n] with k+1 square, this is equivalent to
(k+1)*2^n1 and (k+1)*2^n+1 are twin primes, such semiprime does not
exist for k = 56168, since all numbers [56168][3^^n] are divisible
by at least one of {5, 7, 13, 17, 241}, and for all such k < 56168,
there is known such semiprime except k = 12320, 15128, 23408, 25280,
29240, 33488, 35720, 47960, 49283, 50624, since k must be == 2 mod 3
(k cannot be == 0 mod 3 since these k are excluded from [k][3^^n],
and k cannot be == 1 mod 3 since square cannot be == 2 mod 3),
k+1 must be divisible by 3, and they are indeed studied in
https://www.rieselprime.de/Related/RieselTwinSG.htm,
http://www.noprimeleftbehind.net/gary/twins100K.htm,
http://www.noprimeleftbehind.net/gary/twins1M.htm and
https://www.primepuzzles.net/problems/prob_049.htm.
Base 8:
[73][1^^n] is semiprime for n = 2, 6, 28 and the next semiprime (if exists)
must be at least [73][1^^13466914], [73][1^^n] is semiprime (except the
case n=6) is equivalent to 2^(n+3)1 and (4^(n+3)+2^(n+3)+1)/7 are
both primes.
[26][7^^n] is semiprime for n = 1, 2, 3, 6, 18, 38 and the next semiprime
(if exists) must be greater than [26][7^^18000000], [26][7^^n] is semiprime
is equivalent to 3*2^n1 and 9*4^n+3*2^n+1 are both primes,
reference: A002235.
Base 9:
[1][1^^n] is semiprime for n = 1, 2, 6, 12 and the next semiprime (if exists)
must be greater than [1][1^^5000000], [1][1^^n] is semiprime (except the case
n=1) is equivalent to (3^(n+1)1)/2 and (3^(n+1)+1)/4 are both primes,
references: A028491 and A007658.
[3][1^^n] is semiprime for n = 2, 6, 48 and the next semiprime (if exists)
must be greater than [3][1^^100000].
The only semiprime [6][1^^n] is 61 (55 in decimal), since all numbers
[6][1^^n] are divisible by at least one of {2, 5}.
[3][8^^n] is semiprime for n = 1, 2 and the next semiprime (if exists) must
be greater than [3][8^^1360104], [3][8^^n] is semiprime is equivalent
to 2*3^n1 and 2*3^n+1 are twin primes, references:
A003306 and A003307 (this problem is also related to
“three consecutive numbers with exactly different four prime factors”
(see three consecutive numbers with exactly different four prime factors
and A325204) and “three consecutive numbers with primitive roots”
(see A305237).
[15][8^^n] is semiprime for n = 1, 3, 15 and the next semiprime (if exists)
must be greater than [15][8^^423253], [15][8^^n] is semiprime is equivalent
to 4*3^n1 and 4*3^n+1 are twin primes, references: A005537 and A005540.

