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.
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A000219 
Last update September 26, 2023 ]
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