World!Of
Numbers

WON plate
219 |

[ 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.

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/world-of-numbers-plate-219/

 BASE 2 Prime byappending 1's [1][1^^1] [4][1^^2] [12][1^^3] [13][1^^4] [42][1^^7] [43][1^^24] [73][1^^2552] [658][1^^800516] [2292][1^^12918431] [23668][1^^ >15000000] In Base 2 the problem is already searched extensively (which is the same as the Riesel problem). Related links: The Riesel Problem: Definition and Status About the Riesel Problem Riesel Problem statistics See alsoA050412, A052333, A040081, A038699, A046069, A057026For prime kA257495

 BASE 3 Prime byappending 1's Prime byappending 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 byappending 1's Prime byappending 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 k-values 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 k-values 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_Liskovets-Gallot.

 BASE 5 Prime byappending 1's Prime byappending 2's Prime byappending 3's Prime byappending 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 k-values == 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 byappending 1's Prime byappending 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 byappending 1's Prime byappending 2's Prime byappending 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 byappending 4's Prime byappending 5's Prime byappending 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 byappending 1's Prime byappending 3's Prime byappending 5's Prime byappending 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 k-values 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 k-values 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 byappending 1's Prime byappending 2's Prime byappending 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 byappending 5's Prime byappending 7's Prime byappending 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 k-values == 5 or 6 mod 10, there are no possible primes, since all [k][1^^n] are divisible by either 2 or 5. For k-values 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 k-values == 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 k-values 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 b-1 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. Where does the difference of 1 comes from in the k-values, I asked Chen? First some examples: Base 6 Appending 5's (1596 versus 1597) with 1597 (6.3M) at http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm Same with Base 5 Appending 4's (4905 versus 4906) with 4906, ... (4.8M) at http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base5-reserve.htm Same with Base 4 Appending 3's (9518 versus 9519) with 9519 (8.38M) at http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.htm Same with Base 3 Appending 2's (3677877 vs 3677878) with 3677878 (5M) at http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base3-reserve.htm Same with Base 2 Appending 1's (23668 versus 23669) with 23669 (15M) at http://www.noprimeleftbehind.net/crus/Riesel-conjecture-base2-reserve.htm Chen replied a while later: This is because k*b^n+(b-1)*(b^n-1)/(b-1) = (k+1)*b^n-1, thus the k for the appending b-1 in base b is equivalent to k+1 in the Riesel problem base b.

 “ 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^n-1)/3 = ((3*k+1)*4^n-1)/3 k-values such that 3*k+1 is a square: Let 3*k+1 = m^2, ((3*k+1)*4^n-1)/3 = (m^2*4^n-1)/3 = (m*2^n-1) * (m*2^n+1) / 3, the only possible prime cases are m*2^n-1 = 3 and m*2^n+1 = 3, but if m*2^n+1 = 3 then m*2^n = 2 then m*2^n-1 = 1 is not prime, thus m*2^n-1 = 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^n-1)/3 = (k+1)*4^n-1 k-values such that k+1 is a square: Let k+1 = m^2, (k+1)*4^n-1 = m^2*4^n-1 = (m*2^n-1) * (m*2^n+1), and since k⩾1, thus m⩾2, and hence m*2^n-1 ⩾ 2*2^1-1 = 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^n-1)/7 = ((7*k+1)*8^n-1)/7 k-values such that 7*k+1 is a cube: Let 7*k+1 = m^3, ((7*k+1)*8^n-1)/7 = (m^3*8^n-1)/7 = (m*2^n-1) * (m^2*4^n+m*2^n+1) / 7, the only possible prime cases are m*2^n-1 = 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^n-1 = 1 is not prime, thus m*2^n-1 = 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^n-1)/7 = (k+1)*8^n-1 k-values such that k+1 is a cube: Let k+1 = m^3, (k+1)*8^n-1 = m^3*8^n-1 = (m*2^n-1) * (m^2*4^n+m*2^n+1), and since k⩾1, thus m⩾2, and hence m*2^n-1 ⩾ 2*2^1-1 = 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^n-1)/8 = ((8*k+1)*9^n-1)/8 k-values such that 8*k+1 is a square: Let 8*k+1 = m^2, ((8*k+1)*9^n-1)/8 = (m^2*9^n-1)/8 = (m*3^n-1) * (m*3^n+1) / 8, and since k⩾1, thus m⩾2, and hence m*3^n-1 ⩾ 2*3^1-1 = 5 > 4, m*3^n+1 ⩾ 2*3^1+1 = 7 > 4, thus the only possible prime cases are m*3^n-1 = 8 and m*3^n+1 = 8, but if m*3^n-1 = 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^n-1 = 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^n-1)/8 = (k+1)*9^n-1 k-values such that k+1 is a square: Let k+1 = m^2, (k+1)*9^n-1 = m^2*9^n-1 = (m*3^n-1) * (m*3^n+1), and since k⩾1, thus m⩾2, and hence m*3^n-1 ⩾ 2*3^1-1 = 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 N-1 primality test, factorization of N-1 and primality certificate of large prime factor of N-1) [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 N-1 primality test, factorization of N-1 and primality certificate of large prime factor of N-1) [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 N-1 primality test, factorization of N-1 and primality certificate of large prime factor of N-1) [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 N-1 primality test, factorization of N-1 and primality certificate of large prime factor of N-1) [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 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: 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 (b-1)'s case to other bases b, see http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm and http://www.noprimeleftbehind.net/crus/Riesel-conjectures-powers2.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 n-digit 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 n-digit 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 n-digit number and a 2*n-digit 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.s3-website.us-west-1.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^n-1 and (5*2^n+1)/3 are both primes, or 5*2^n+1 and (5*2^n-1)/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^n-1 and (7*2^n+1)/3 are both primes, or 7*2^n+1 and (7*2^n-1)/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^n-1 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^n-1 and 9*2^n+1 are both primes. For semiprime [k][3^^n] with k+1 square, this is equivalent to (k+1)*2^n-1 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^n-1 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^n-1 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^n-1 and 4*3^n+1 are twin primes, references: A005537 and A005540. ```

A000219 Prime Curios! Prime Puzzle
Wikipedia 219 Le nombre 219
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