Minimal set of the strings for primes with at least two digits

[sweety439]

* Case (5,1):

** 51 is prime, and thus the only minimal prime in this family.

* Case (5,3):

** 53 is prime, and thus the only minimal prime in this family.

* Case (5,5):

** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes)

*** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes)

**** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime.

**** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes)

***** 500525 is not prime.

***** The smallest prime of the form 5{5}25 is 555555555555525

***** The smallest prime of the form 5{5}025 is 55555025

***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025 (not minimal prime, since 55555025 and 555555555555525 are primes)

***** The smallest prime of the form 5{5}0525 is 5550525

***** The smallest prime of the form 5{5}00525 is 5500525

***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes)

* Case (5,7):

** 57 is prime, and thus the only minimal prime in this family.

[sweety439]

* Case (6,1):

** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes)

*** Since 111, 141, 401, 471, 701, 711, 6101, 6441 are primes, we only need to consider the families 6{0}0{0,1,4,7}1, 6{0,4}1{7}1, 6{0,7}4{1}1, 6{0,1,7}7{4,7}1 (since any digits combo 11, 14, 40, 47, 70, 71, 10, 44 between them will produce smaller primes)

**** For the 6{0}0{0,1,4,7}1 family, since 6007 is prime, we only need to consider the families 6{0}0{0,1,4}1 and 60{1,4,7}7{0,1,4,7}1 (since any digits combo 1007 between (6,1) will produce smaller primes)

***** For the 6{0}0{0,1,4}1 family, since 111, 141, 401, 6101, 6441, 60411 are primes, we only need to consider the families 6{0}1, 6{0}11, 6{0}41 (since any digits combo 10, 11, 14, 40, 41, 44 between (6{0}0,1) will produce smaller primes)

****** All numbers of the form 6{0}1 are divisible by 7, thus cannot be prime.

****** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime.

****** All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime.

***** For the 60{1,4,7}7{0,1,4,7}1 family, since 701, 711, 60741 are primes, we only need to consider the family 60{1,4,7}7{7}1 (since any digits 0, 1, 4 between (60{1,4,7}7,1) will produce smaller primes)

***** Since 471, 60171 is prime, we only need to consider the family 60{7}1 (since any digits 1, 4 between (60,7{7}1) will produce smaller primes)

****** All numbers of the form 60{7}1 are divisible by 7, thus cannot be prime.

**** For the 6{0,4}1{7}1 family, since 417, 471 are primes, we only need to consider the families 6{0}1{7}1 and 6{0,4}11

***** For the 6{0}1{7}1 family, since 60171 is prime, and thus the only minimal prime in the family 6{0}1{7}1.

***** For the 6{0,4}11 family, since 401, 6441, 60411 are primes, we only need to consider the number 6411 and the family 6{0}11

****** 6411 is not prime.

****** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime.

**** For the 6{0,7}4{1}1 family, since 60411 is prime, we only need to consider the families 6{7}4{1}1 and 6{0,7}41

***** For the 6{7}4{1}1 family, since 111, 6777 are primes, we only need to consider the numbers 641, 6411, 6741, 67411, 67741, 677411

****** None of 641, 6411, 6741, 67411, 67741, 677411 are primes.

***** For the 6{0,7}41 family, since 701, 6777, 60741 are primes, we only need to consider the families 6{0}41 and the numbers 6741, 67741 (since any digits combo 07, 70, 777 between (6,41) will produce smaller primes)

****** All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime.

****** Neither of 6741, 67741 are primes.

***** For the 6{0,1,7}7{4,7}1 family, since 747 is prime, we only need to consider the families 6{0,1,7}7{4}1, 6{0,1,7}7{7}1, 6{0,1,7}7{7}{4}1 (since any digits combo 47 between (6{0,1,7}7,1) will produce smaller primes)

****** For the 6{0,1,7}7{4}1 family, since 6441 is prime, we only need to consider the families 6{0,1,7}71 and 6{0,1,7}741 (since any digits combo 44 between (6{0,1,7}7,1) will produce smaller primes)

******* For the 6{0,1,7}71 family, since all numbers of the form 6{0,7}71 are divisible by 7 and cannot be prime, and 111 is prime (thus, any digits combo 11 between (6,71) will produce smaller primes), we only need to consider the family 6{0,7}1{0,7}71

******** Since 717 and 60171 are primes, we only need to consider the family 61{0,7}71 (since any digit combo 0, 7 between (6,1{0,7}71) will produce smaller primes)

********* Since 177 and 6101 are primes, we only need to consider the number 6171 (since any digit combo 0, 7 between (61,71) will produce smaller primes)

********** 6171 is not prime.

****** All numbers in the 6{0,1,7}7{7}1 or 6{0,1,7}7{7}{4}1 families are also in the 6{0,1,7}7{4}1 family, thus these two families cannot have more minimal primes.

[sweety439]

Upload past file, the set is not complete for bases >=7, I want to complete them.

I know some primes in the set which is not listed:

* base 7: 33333333333333331

* base 8: 77774444441, 7777777777771, 555555555555525, 4₂₂₀7

* base 10: 555555555551

* base 11: A0000000001, A0014444444, 40000000A0041, A000000014444, A044444444441, A144444444411, 40000000000401, A0000044444441, A00000000444441, 11111111111111111, 14444444444441111, 44444444444444111, A1444444444444444, A9999999999999996, 1444444444444444444, 4000000000000000A041, A999999999999999999999, A44444444444444444444444441, 40000000000000000000000000041, 440000000000000000000000000001, 999999999999999999999999999999991, 444444444444444444444444444444444444444444441 (they are in the searching where single-digit primes are included, but this puzzle does not include single-digit primes)

* base 13: 80₃₂₀₁₇111

* base 14: 4D₁₉₆₉₈

* base 15: DE₁₄

* base 17: 74₄₉₀₄

(this problem is much harder than the original minimal prime (where single-digit primes are included), see post https://mersenneforum.org/showpost.p...5&postcount=57)

[sweety439]

Found a minimal prime (start with 2 digits) in base 13: 7₁₅₀₄1, which equals (7*13^1505-79)/12

[sweety439]

Consider the "simplest" families x{y} and {x}y, where x,y are base b digits

Necessary conditions are gcd(x,y) = 1, gcd(y,b) = 1

b, x, y, smallest prime
2, {1}, 1: 3
2, 1, {1}: 3
3, {1}, 1: 13
3, 1, {1}: 13
3, {1}, 2: 5
3, 1, {2}: 5
3, {2}, 1: 7
3, 2, {1}: 7
4, {1}, 1: 5
4, 1, {1}: 5
4, {1}, 3: 7
4, 1, {3}: 7
4, {2}, 1: 41
4, 2, {1}: 37
4, {2}, 3: 11
4, 2, {3}: 11
4, {3}, 1: 13
4, 3, {1}: 13
5, {1}, 1: 31
5, 1, {1}: 31
5, {1}, 2: 7
5, 1, {2}: 7
5, {1}, 3: 0
5, 1, {3}: 43
5, {1}, 4: 0
5, 1, {4}: 1249
5, {2}, 1: 11
5, 2, {1}: 11
5, {2}, 3: 13
5, 2, {3}: 13
5, {3}, 1: 2341
5, 3, {1}: 0
5, {3}, 2: 17
5, 3, {2}: 17
5, {3}, 4: 19
5, 3, {4}: 19
5, {4}, 1: 3121
5, 4, {1}: 0
5, {4}, 3: 23
5, 4, {3}: 23
6, {1}, 1: 7
6, 1, {1}: 7
6, {1}, 5: 11
6, 1, {5}: 11
6, {2}, 1: 13
6, 2, {1}: 13
6, {2}, 5: 17
6, 2, {5}: 17
6, {3}, 1: 19
6, 3, {1}: 19
6, {3}, 5: 23
6, 3, {5}: 23
6, {4}, 1: 1033
6, 4, {1}: 151
6, {4}, 5: 29
6, 4, {5}: 29
6, {5}, 1: 31
6, 5, {1}: 31
7, {1}, 1: 2801
7, 1, {1}: 2801
7, {1}, 2: 401
7, 1, {2}: 457
7, {1}, 3: 59
7, 1, {3}: 73
7, {1}, 4: 11
7, 1, {4}: 11
7, {1}, 5: 61
7, 1, {5}: 89
7, {1}, 6: 13
7, 1, {6}: 13
7, {2}, 1: 113
7, 2, {1}: 743
7, {2}, 3: 17
7, 2, {3}: 17
7, {2}, 5: 19
7, 2, {5}: 19
7, {3}, 1: 116315256993601
7, 3, {1}: 7603
7, {3}, 2: 23
7, 3, {2}: 23
7, {3}, 4: 1201
7, 3, {4}: 179
7, {3}, 5: 173
7, 3, {5}: 9203
7, {4}, 1: 29
7, 4, {1}: 29
7, {4}, 3: 31
7, 4, {3}: 31
7, {4}, 5: 229
7, 4, {5}: 1657
7, {5}, 1: 281
7, 5, {1}: 8413470255870653
7, {5}, 2: 37
7, 5, {2}: 37
7, {5}, 3: 283
7, 5, {3}: 269
7, {5}, 4: 1999
7, 5, {4}: 277
7, {5}, 6: 41
7, 5, {6}: 41
7, {6}, 1: 43
7, 6, {1}: 43
7, {6}, 5: 47
7, 6, {5}: 47
8, {1}, 1: 73
8, 1, {1}: 73
8, {1}, 3: 11
8, 1, {3}: 11
8, {1}, 5: 13
8, 1, {5}: 13
8, {1}, 7: 79
8, 1, {7}: 127
8, {2}, 1: 17
8, 2, {1}: 17
8, {2}, 3: 19
8, 2, {3}: 19
8, {2}, 5: 149
8, 2, {5}: 173
8, {2}, 7: 23
8, 2, {7}: 23
8, {3}, 1: 1753
8, 3, {1}: 1609
8, {3}, 5: 29
8, 3, {5}: 29
8, {3}, 7: 31
8, 3, {7}: 31
8, {4}, 1: 76695841
8, 4, {1}: 284694975049
8, {4}, 3: 2339
8, 4, {3}: 283
8, {4}, 5: 37
8, 4, {5}: 37
8, {4}, 7: 21870014779720278736374332149114462520188534743847615898363462279537144492484599310778624146468224150373895489844303219383829573677353011540369291867378470695590964880740521967077028064041941947533607
8, 4, {7}: 20479
8, {5}, 1: 41
8, 5, {1}: 41
8, {5}, 3: 43
8, 5, {3}: 43
8, {5}, 7: 47
8, 5, {7}: 47
8, {6}, 1: 433
8, 6, {1}: 56657856797822194249
8, {6}, 5: 53
8, 6, {5}: 53
8, {6}, 7: 439
8, 6, {7}: 3583
8, {7}, 1: 549755813881
8, 7, {1}: 457
8, {7}, 3: 59
8, 7, {3}: 59
8, {7}, 5: 61
8, 7, {5}: 61
9, {1}, 1: 0
9, 1, {1}: 0
9, {1}, 2: 11
9, 1, {2}: 11
9, {1}, 4: 13
9, 1, {4}: 13
9, {1}, 5: 0
9, 1, {5}: 131
9, {1}, 7: 97
9, 1, {7}: 151
9, {1}, 8: 17
9, 1, {8}: 17
9, {2}, 1: 19
9, 2, {1}: 19
9, {2}, 5: 23
9, 2, {5}: 23
9, {2}, 7: 14767
9, 2, {7}: 0
9, {3}, 1: 271
9, 3, {1}: 0
9, {3}, 2: 29
9, 3, {2}: 29
9, {3}, 4: 31
9, 3, {4}: 31
9, {3}, 5: 0
9, 3, {5}: 293
9, {3}, 7: 277
9, 3, {7}: 313
9, {3}, 8: 0
9, 3, {8}: 0
9, {4}, 1: 37
9, 4, {1}: 37
9, {4}, 5: 41
9, 4, {5}: 41
9, {4}, 7: 43
9, 4, {7}: 43
9, {5}, 1: 36901
9, 5, {1}: 0
9, {5}, 2: 47
9, 5, {2}: 47
9, {5}, 4: 4099
9, 5, {4}: 172595827849
9, {5}, 7: 457
9, 5, {7}: 0
9, {5}, 8: 53
9, 5, {8}: 53
9, {6}, 1: 541
9, 6, {1}: 0
9, {6}, 5: 59
9, 6, {5}: 59
9, {6}, 7: 61
9, 6, {7}: 61
9, {7}, 1: 631
9, 7, {1}: 577
9, {7}, 2: 0
9, 7, {2}: 587
9, {7}, 4: 67
9, 7, {4}: 67
9, {7}, 5: 0
9, 7, {5}: 617
9, {7}, 8: 71
9, 7, {8}: 71
9, {8}, 1: 73
9, 8, {1}: 73
9, {8}, 5: 0
9, 8, {5}: 6287
9, {8}, 7: 79
9, 8, {7}: 79
10, {1}, 1: 11
10, 1, {1}: 11
10, {1}, 3: 13
10, 1, {3}: 13
10, {1}, 7: 17
10, 1, {7}: 17
10, {1}, 9: 19
10, 1, {9}: 19
10, {2}, 1: 2221
10, 2, {1}: 211
10, {2}, 3: 23
10, 2, {3}: 23
10, {2}, 7: 227
10, 2, {7}: 277
10, {2}, 9: 29
10, 2, {9}: 29
10, {3}, 1: 31
10, 3, {1}: 31
10, {3}, 7: 37
10, 3, {7}: 37
10, {4}, 1: 41
10, 4, {1}: 41
10, {4}, 3: 43
10, 4, {3}: 43
10, {4}, 7: 47
10, 4, {7}: 47
10, {4}, 9: 449
10, 4, {9}: 499
10, {5}, 1: 555555555551
10, 5, {1}: 511111
10, {5}, 3: 53
10, 5, {3}: 53
10, {5}, 7: 557
10, 5, {7}: 577
10, {5}, 9: 59
10, 5, {9}: 59
10, {6}, 1: 61
10, 6, {1}: 61
10, {6}, 7: 67
10, 6, {7}: 67
10, {7}, 1: 71
10, 7, {1}: 71
10, {7}, 3: 73
10, 7, {3}: 73
10, {7}, 9: 79
10, 7, {9}: 79
10, {8}, 1: 881
10, 8, {1}: 811
10, {8}, 3: 83
10, 8, {3}: 83
10, {8}, 7: 887
10, 8, {7}: 877
10, {8}, 9: 89
10, 8, {9}: 89
10, {9}, 1: 991
10, 9, {1}: 911
10, {9}, 7: 97
10, 9, {7}: 97
11, {1}, 1: 50544702849929377
11, 1, {1}: 50544702849929377
11, {1}, 2: 13
11, 1, {2}: 13
11, {1}, 3: 0
11, 1, {3}: 157
11, {1}, 4: 0
11, 1, {4}: 7783884238889124073
11, {1}, 5: 137
11, 1, {5}: 181
11, {1}, 6: 17
11, 1, {6}: 17
11, {1}, 7: 139
11, 1, {7}: 24889
11, {1}, 8: 19
11, 1, {8}: 19
11, {1}, 9: 0
11, 1, {9}: 229
11, {1}, 10: 0
11, 1, {10}: 241
11, {2}, 1: 23
11, 2, {1}: 23
11, {2}, 3: 354313
11, 2, {3}: 3061
11, {2}, 5: 269
11, 2, {5}: 0
11, {2}, 7: 29
11, 2, {7}: 29
11, {2}, 9: 31
11, 2, {9}: 31
11, {3}, 1: 397
11, 3, {1}: 0
11, {3}, 2: 4391
11, 3, {2}: 4259
11, {3}, 4: 37
11, 3, {4}: 37
11, {3}, 5: 401
11, 3, {5}: 0
11, {3}, 7: 85593501187
11, 3, {7}: 0
11, {3}, 8: 41
11, 3, {8}: 41
11, {3}, 10: 43
11, 3, {10}: 43
11, {4}, 1: 29156193474041220857161146715104735751776055777
11, 4, {1}: 0
11, {4}, 3: 47
11, 4, {3}: 47
11, {4}, 5: 5857
11, 4, {5}: 724729
11, {4}, 7: 114124668247
11, 4, {7}: 0
11, {4}, 9: 53
11, 4, {9}: 53
11, {5}, 1: 661
11, 5, {1}: 617
11, {5}, 2: 0
11, 5, {2}: 9212117
11, {5}, 3: 0
11, 5, {3}: 641
11, {5}, 4: 59
11, 5, {4}: 59
11, {5}, 6: 61
11, 5, {6}: 61
11, {5}, 7: 80527
11, 5, {7}: 0
11, {5}, 8: 0
11, 5, {8}: 701
11, {5}, 9: 0
11, 5, {9}: 86381
11, {6}, 1: 67
11, 6, {1}: 67
11, {6}, 5: 71
11, 6, {5}: 71
11, {6}, 7: 73
11, 6, {7}: 73
11, {7}, 1: 42811363313890182397
11, 7, {1}: 859
11, {7}, 2: 79
11, 7, {2}: 79
11, {7}, 3: 0
11, 7, {3}: 883
11, {7}, 4: 0
11, 7, {4}: 108343
11, {7}, 5: 929
11, 7, {5}: 907
11, {7}, 6: 83
11, 7, {6}: 83
11, {7}, 8: 150051217
11, 7, {8}: 114199
11, {7}, 9: 0
11, 7, {9}: 115663
11, {7}, 10: 0
11, 7, {10}: 967
11, {8}, 1: 89
11, 8, {1}: 89
11, {8}, 3: 4447933850793785179
11, 8, {3}: 11047
11, {8}, 5: 1061
11, 8, {5}: 0
11, {8}, 7: 1063
11, 8, {7}: 11579
11, {8}, 9: 97
11, 8, {9}: 97
11, {9}, 1: 20902638977899027326901591016678209
11, 9, {1}: 0
11, {9}, 2: 101
11, 9, {2}: 101
11, {9}, 4: 103
11, 9, {4}: 103
11, {9}, 5: 1193
11, 9, {5}: 0
11, {9}, 7: 2122152919
11, 9, {7}: 0
11, {9}, 8: 107
11, 9, {8}: 107
11, {9}, 10: 109
11, 9, {10}: 109
11, {10}, 1: 1321
11, 10, {1}: 0
11, {10}, 3: 113
11, 10, {3}: 113
11, {10}, 7: 1327
11, 10, {7}: 0
11, {10}, 9: 14639
11, 10, {9}: 80662724392413945103199
12, {1}, 1: 13
12, 1, {1}: 13
12, {1}, 5: 17
12, 1, {5}: 17
12, {1}, 7: 19
12, 1, {7}: 19
12, {1}, 11: 23
12, 1, {11}: 23
12, {2}, 1: 313
12, 2, {1}: 3613
12, {2}, 5: 29
12, 2, {5}: 29
12, {2}, 7: 31
12, 2, {7}: 31
12, {2}, 11: 3779
12, 2, {11}: 431
12, {3}, 1: 37
12, 3, {1}: 37
12, {3}, 5: 41
12, 3, {5}: 41
12, {3}, 7: 43
12, 3, {7}: 43
12, {3}, 11: 47
12, 3, {11}: 47
12, {4}, 1: 7537
12, 4, {1}: 7069
12, {4}, 5: 53
12, 4, {5}: 53
12, {4}, 7: 631
12, 4, {7}: 8011
12, {4}, 11: 59
12, 4, {11}: 59
12, {5}, 1: 61
12, 5, {1}: 61
12, {5}, 7: 67
12, 5, {7}: 67
12, {5}, 11: 71
12, 5, {11}: 71
12, {6}, 1: 73
12, 6, {1}: 73
12, {6}, 5: 941
12, 6, {5}: 929
12, {6}, 7: 79
12, 6, {7}: 79
12, {6}, 11: 83
12, 6, {11}: 83
12, {7}, 1: 1093
12, 7, {1}: 1021
12, {7}, 5: 89
12, 7, {5}: 89
12, {7}, 11: 1103
12, 7, {11}: 1151
12, {8}, 1: 97
12, 8, {1}: 97
12, {8}, 5: 101
12, 8, {5}: 101
12, {8}, 7: 103
12, 8, {7}: 103
12, {8}, 11: 107
12, 8, {11}: 107
12, {9}, 1: 109
12, 9, {1}: 109
12, {9}, 5: 113
12, 9, {5}: 113
12, {9}, 7: 16963
12, 9, {7}: 16651
12, {9}, 11: 203591
12, 9, {11}: 1439
12, {10}, 1: 226201
12, 10, {1}: 1453
12, {10}, 7: 127
12, 10, {7}: 127
12, {10}, 11: 131
12, 10, {11}: 131
12, {11}, 1: 248821
12, 11, {1}: 1597
12, {11}, 5: 137
12, 11, {5}: 137
12, {11}, 7: 139
12, 11, {7}: 139
13, {1}, 1: 30941
13, 1, {1}: 30941
13, {1}, 2: 2381
13, 1, {2}: 197
13, {1}, 3: 883708283
13, 1, {3}: 211
13, {1}, 4: 17
13, 1, {4}: 17
13, {1}, 5: 0
13, 1, {5}: 239
13, {1}, 6: 19
13, 1, {6}: 19
13, {1}, 7: 0
13, 1, {7}: 253217502498750291800692183145337720992638880271493569431738157631027569095215561
13, {1}, 8: 0
13, 1, {8}: 281
13, {1}, 9: 191
13, 1, {9}: 27130132404659193376721686434661
13, {1}, 10: 23
13, 1, {10}: 23
13, {1}, 11: 193
13, 1, {11}: 820195757799727198696695842441476208994963187388611974376331352215246160014077762227387500472080168786083449145277773838858800795991958180632280974279961571375401216690067428169654353503039823288371270608465421195831630752643523277310127185899536482776399744897654945387104284638368882957170938237278749541598968947323011480936838588602269948473325034154837529102013688233548651627077006312693219100274314806851388327828512220445130387062692274032398838117351850000439212156970074507805886649022038706852334408911410638474707605620484783396663735375992701354072765197190488304749330337810593696686818871540035682031309739770552183078238960961
13, {1}, 12: 36898271981403391525359432679065451
13, 1, {12}: 337
13, {2}, 1: 4759
13, 2, {1}: 106637277112689077
13, {2}, 3: 29
13, 2, {3}: 29
13, {2}, 5: 31
13, 2, {5}: 31
13, {2}, 7: 804473
13, 2, {7}: 959173
13, {2}, 9: 373
13, 2, {9}: 0
13, {2}, 11: 37
13, 2, {11}: 37
13, {3}, 1: 547
13, 3, {1}: 521
13, {3}, 2: 41
13, 3, {2}: 41
13, {3}, 4: 43
13, 3, {4}: 43
13, {3}, 5: 6220138738168647434831423806501836269264860388724755065149565951442236175936796719067011700334477855412396475124991691344796424350196661821800919998163958711515136934394634097179800502201458093822871
13, 3, {5}: 577
13, {3}, 7: 0
13, 3, {7}: 2923035083
13, {3}, 8: 47
13, 3, {8}: 47
13, {3}, 10: 0
13, 3, {10}: 647
13, {3}, 11: 557
13, 3, {11}: 661
13, {4}, 1: 53
13, 4, {1}: 53
13, {4}, 3: 3534833123
13, 4, {3}: 9337
13, {4}, 5: 733
13, 4, {5}: 13799574804865291194219202692403868309075434457702185936718792737
13, {4}, 7: 59
13, 4, {7}: 59
13, {4}, 9: 61
13, 4, {9}: 61
13, {4}, 11: 739
13, 4, {11}: 0
13, {5}, 1: 911
13, 5, {1}: 859
13, {5}, 2: 67
13, 5, {2}: 67
13, {5}, 3: 4418541403
13, 5, {3}: 887
13, {5}, 4: 277256920492991599
13, 5, {4}: 11717
13, {5}, 6: 71
13, 5, {6}: 71
13, {5}, 7: 0
13, 5, {7}: 89921716241132417850870043495861784708189815671814466610282151602592361843582839405888101361717803180497433838124247307746249298923864405720133765156792108299126338169033221693415179426566058950783399084612596039386427021973466021161486411164221010228501523
13, {5}, 8: 73
13, 5, {8}: 73
13, {5}, 9: 919
13, 5, {9}: 971
13, {5}, 11: 3604339966408890811
13, 5, {11}: 815662743439
13, {5}, 12: 0
13, 5, {12}: 1013
13, {6}, 1: 79
13, 6, {1}: 79
13, {6}, 5: 83
13, 6, {5}: 83
13, {6}, 7: 14281
13, 6, {7}: 9624308578305020228623
13, {6}, 11: 89
13, 6, {11}: 89
13, {7}, 1: 178099219309623994113347865115830245881192672331136086644940932390281437773862542264058288467257753659801600961075130547842788841314191170557163588785806820352498775469685611756153126667905685532469070927112200362751893732722602014850726541568809237898479136618304580608401918281234930143064257920309170965969499129532757126377921069913907908471671716574775646406870993578835592914767961359736945068963155952491960373644449418976551172435322847621930721542144058540845006834135041368564051814099612330378729974668784588333951075548113502723356005589020090498660438660159816508080021058414799091444088419142837713138761057222122289170719363548408846549381700350017033339161308029642370902736513421829569951497819557491865511689513661142885392815700489495652470272195143512237819660601060635160718950091398065703223634277265560584842616957317643843802700518533058037207116004784352250055837483109124201522964355475650386943932211405298597905457035869654221665584889762003382262425915627119116388998518127939798007098237014217655662706251901683109346257926396141021838296111310372968225022079707942822795129744698997404408233889930805048836987175297692891091686536077480054995854699722419639970465336199946812908942288140516904083829095941075082576747429233198452931491612278651981884049653724609828879600848950258125404977404144280808141884961849927596265042459349498828074205132275149771178481317527349922743964367067517723323072236797216719438348669059261865669008875549670954883841127641347209610157306553559175511522826280497259599838188113180381333755455900349307408532955497746124837076056334168406604339828775363723268449080391071951364995984584279858031164184210011834081
13, 7, {1}: 0
13, {7}, 2: 2815633
13, 7, {2}: 425554642597531069476088349319265344520247822549860785329030452393239592223053102610453
13, {7}, 3: 1277
13, 7, {3}: 0
13, {7}, 4: 16657
13, 7, {4}: 16111
13, {7}, 5: 1279
13, 7, {5}: 0
13, {7}, 6: 97
13, 7, {6}: 97
13, {7}, 8: 16661
13, 7, {8}: 16843
13, {7}, 9: 1283
13, 7, {9}: 0
13, {7}, 10: 101
13, 7, {10}: 101
13, {7}, 11: 176677145512151
13, 7, {11}: 0
13, {7}, 12: 103
13, 7, {12}: 103
13, {8}, 1: 201916737728161
13, 8, {1}: 0
13, {8}, 3: 107
13, 8, {3}: 107
13, {8}, 5: 109
13, 8, {5}: 109
13, {8}, 7: 948090435833789898349839159224250448489822963231314463923791
13, 8, {7}: 91021952951
13, {8}, 9: 113
13, 8, {9}: 113
13, {8}, 11: 247531
13, 8, {11}: 107838136118779143544110144382426092596204365203552428434248370827269303781694390904131147239057991725836898042412261207750257957583868509005708271654322505648777710176276577432900281939636108415883809919452408705622432187985384594384558596142925993309255985009535307478523156581759235779167162869019703196846751228017411378948853640401871368029723240618590933942573810568367308325289
13, {9}, 1: 121215212228974701436931483878589891140303089832220953724825443452928225193998713337930372911067805791799869147809411855365238746332705480023760792811129961633638311977064834817213347105220337863208732908350674718836123175940417328579685338580206495293657407565505774954042639105332034212418635645417015477944123471063765451261909023510740139021
13, 9, {1}: 259429
13, {9}, 2: 0
13, 9, {2}: 1549
13, {9}, 4: 68799997644951462493799712001
13, 9, {4}: 36748846266526697
13, {9}, 5: 1513040167438129868806971467209988783047172742217267483154894329901717663980542315652349848778853894915555170807951322572378209558278635157172717513628242832984817118678991631720815452108000099210400379825153731084262244319493812890311789045794884092831356123242061758082732806604188005520228674939217665328388067373699901394102699076617987792004875548394303529748128202045356868637958421168851264492251414451671881645020719873271254395994874212298538928180953385418967331680881078358158544926445810098124494008083153487609262254200180462767265649895085196812223539172699394835278304012594889076972978898251549233575133572026086685197059747614441040152909798221820479642718133459387770777298554261474133030690753241864544153004060794064644652707512761735881885543957256884192628158805686561809594061924393932922840678322622784462425947093214038759553887929649916953155720669786551358215226363554462047120571340688059948977555505902980499655568680185582793237970604923030974220265795047325241015419809657491893019473031136829858500145730181190585716816328018616913990223894736777230358689553647254936178082288866348094694739653443883605310239766413123460690534793812785934073248428969751425870653389046775244015718236495145673518071665801891709920766181983586792103960637565600979310820525883807031028492116501924047338205921075081927116926896242518609770952106202341273961329572020549726342482699063714395183214771739428693004365223506346305628182421466017950778081664584660392184816346242270287808532367340974202505243
13, 9, {5}: 0
13, {9}, 7: 0
13, 9, {7}: 1619
13, {9}, 8: 21419
13, 9, {8}: 1332632087873
13, {9}, 10: 127
13, 9, {10}: 127
13, {9}, 11: 193272315039175487326233511816066059384236614936125602856112933108936878519237472904660314716980770567675038998906449425413153983914911198004481338508457524880312210951611198276223531290887108690292738026664067061295245572474703602610246125803179560118335836826468952746508374948156130103443176084512499609881640980577720176544209084145250756004293999763606271755552801884151163797064193306051490695131926580100826963694188422295198532753628423986864538180785129561486984798531474623258138617820724652921878726441978127950818375993469697424536456843865475449486163418975049457428208240361723655765920087962062440804142883445511641266703241468837936119086839611677856638014115541263266704520590558533279312304037176827720788243643056423797345536794811687465498057714283585292329995513313040502867333497441355894682583294242422882099383092926007095933296099723170595507590307152252179729407846066017402908790460912572771982724554085389116205103789017519828339606017138742770709364271683876516827264126734705389614550068455954555083870819276074206224473000405522699421186643568300931
13, 9, {11}: 188465890767567927768109
13, {10}, 1: 131
13, 10, {1}: 131
13, {10}, 3: 1823
13, 10, {3}: 0
13, {10}, 7: 137
13, 10, {7}: 137
13, {10}, 9: 139
13, 10, {9}: 139
13, {10}, 11: 1831
13, 10, {11}: 19564417634903
13, {11}, 1: 2003
13, 11, {1}: 1873
13, {11}, 2: 26171
13, 11, {2}: 24533
13, {11}, 3: 9720791083
13, 11, {3}: 1901
13, {11}, 4: 0
13, 11, {4}: 9244948171
13, {11}, 5: 16947116540528994433003374971133476766755585467759746042637787
13, 11, {5}: 55106069
13, {11}, 6: 149
13, 11, {6}: 149
13, {11}, 7: 0
13, 11, {7}: 275614952732329653238232466846964959764342234851112847239174811330605042147473869409955112827912038180459570611709057122713557738754852977301670028971129467515066229407333223877783672306282366369543871115968877415280318537960852589377566524955025517265527862915865085457853330890387002896522575722038083366786743993408179910457825505003824935898016650965949690327645173471015392053739386753124739151576511051033085127452855493526018723218218416151702223054562562122092583257452732969841804887384398911206575041813220337355043752278695720428137
13, {11}, 8: 151
13, 11, {8}: 151
13, {11}, 9: 2011
13, 11, {9}: 335591
13, {11}, 10: 52950113757237678592993805854491324369268684216704693554936717423836128996957000831399984295437037597224946915093310665683439
13, 11, {10}: 1999
13, {11}, 12: 16836900297891418080414469547118518955584357920776290786511507224819852347973193037600665289070901330976115445902783343792856149076064327963454445124840887022352433623214149015015943271257627167012185236811023315748308075343126054090560004563875124190448995227748073744916159908957819701603274854998000296763254125672206384758348891742961717040363229489213108521955314350073857925001010097317113705164622416602981584525394558649693204742511309000575073486313783914987497483013408328355077527202814535784777000148396721007194688339582681878366906510944731328876064735814127172451578146421749559114747412555063799277435883965467381
13, 11, {12}: 2027
13, {12}, 1: 157
13, 12, {1}: 157
13, {12}, 5: 1792160394029
13, 12, {5}: 0
13, {12}, 7: 163
13, 12, {7}: 163
13, {12}, 11: 167
13, 12, {11}: 167
14, {1}, 1: 211
14, 1, {1}: 211
14, {1}, 3: 17
14, 1, {3}: 17
14, {1}, 5: 19
14, 1, {5}: 19
14, {1}, 9: 23
14, 1, {9}: 23
14, {1}, 11: 41381
14, 1, {11}: 70921
14, {1}, 13: 223
14, 1, {13}: 76831
14, {2}, 1: 29
14, 2, {1}: 29
14, {2}, 3: 31
14, 2, {3}: 31
14, {2}, 5: 0
14, 2, {5}: 467
14, {2}, 9: 37
14, 2, {9}: 37
14, {2}, 11: 431
14, 2, {11}: 557
14, {2}, 13: 41
14, 2, {13}: 41
14, {3}, 1: 43
14, 3, {1}: 43
14, {3}, 5: 47
14, 3, {5}: 47
14, {3}, 11: 53
14, 3, {11}: 53
14, {3}, 13: 643
14, 3, {13}: 0
14, {4}, 1: 32434921
14, 4, {1}: 156619
14, {4}, 3: 59
14, 4, {3}: 59
14, {4}, 5: 61
14, 4, {5}: 61
14, {4}, 9: 0
14, 4, {9}: 919
14, {4}, 11: 67
14, 4, {11}: 67
14, {4}, 13: 853
14, 4, {13}: 0
14, {5}, 1: 71
14, 5, {1}: 71
14, {5}, 3: 73
14, 5, {3}: 73
14, {5}, 9: 79
14, 5, {9}: 79
14, {5}, 11: 1061
14, 5, {11}: 0
14, {5}, 13: 83
14, 5, {13}: 83
14, {6}, 1: 48652381
14, 6, {1}: 0
14, {6}, 5: 89
14, 6, {5}: 89
14, {6}, 11: 248231
14, 6, {11}: 0
14, {6}, 13: 97
14, 6, {13}: 97
14, {7}, 1: 1471
14, 7, {1}: 271867
14, {7}, 3: 101
14, 7, {3}: 101
14, {7}, 5: 103
14, 7, {5}: 103
14, {7}, 9: 107
14, 7, {9}: 107
14, {7}, 11: 109
14, 7, {11}: 109
14, {7}, 13: 1483
14, 7, {13}: 1567
14, {8}, 1: 113
14, 8, {1}: 113
14, {8}, 3: 0
14, 8, {3}: 1613
14, {8}, 5: 0
14, 8, {5}: 1527891201751406184274498849901197440218663
14, {8}, 9: 14893189141836674105809869073123801
14, 8, {9}: 333923
14, {8}, 11: 3178964943473909010162782853260586588933938914164889347525573201287387822569505763054988368562457051
14, 8, {11}: 1733
14, {8}, 13: 1693
14, 8, {13}: 0
14, {9}, 1: 127
14, 9, {1}: 127
14, {9}, 5: 131
14, 9, {5}: 131
14, {9}, 11: 137
14, 9, {11}: 137
14, {9}, 13: 139
14, 9, {13}: 139
14, {10}, 1: 81087301
14, 10, {1}: 0
14, {10}, 3: 450545636966997425132717095409966070465562893779498582938637002140743
14, 10, {3}: 5502353
14, {10}, 9: 149
14, 10, {9}: 149
14, {10}, 11: 151
14, 10, {11}: 151
14, {10}, 13: 2113
14, 10, {13}: 0
14, {11}, 1: 2311
14, 11, {1}: 0
14, {11}, 3: 157
14, 11, {3}: 157
14, {11}, 5: 0
14, 11, {5}: 437351
14, {11}, 9: 163
14, 11, {9}: 163
14, {11}, 13: 167
14, 11, {13}: 167
14, {12}, 1: 2521
14, 12, {1}: 1273070779
14, {12}, 5: 173
14, 12, {5}: 173
14, {12}, 11: 179
14, 12, {11}: 179
14, {12}, 13: 181
14, 12, {13}: 181
14, {13}, 1: 2731
14, 13, {1}: 19298779963
14, {13}, 3: 0
14, 13, {3}: 2593
14, {13}, 5: 0
14, 13, {5}: 276540164647
14, {13}, 9: 191
14, 13, {9}: 191
14, {13}, 11: 193
14, 13, {11}: 193
15, {1}, 1: 241
15, 1, {1}: 241
15, {1}, 2: 17
15, 1, {2}: 17
15, {1}, 4: 19
15, 1, {4}: 19
15, {1}, 7: 303629285816214089001622765197550636782710041318620954247
15, 1, {7}: 337
15, {1}, 8: 23
15, 1, {8}: 23
15, {1}, 11: 251
15, 1, {11}: 401
15, {1}, 13: 12204253
15, 1, {13}: 433
15, {1}, 14: 29
15, 1, {14}: 29
15, {2}, 1: 31
15, 2, {1}: 31
15, {2}, 7: 37
15, 2, {7}: 37
15, {2}, 11: 41
15, 2, {11}: 41
15, {2}, 13: 43
15, 2, {13}: 43
15, {3}, 1: 417041800362721
15, 3, {1}: 691
15, {3}, 2: 47
15, 3, {2}: 47
15, {3}, 4: 2440849
15, 3, {4}: 739
15, {3}, 7: 727
15, 3, {7}: 787
15, {3}, 8: 53
15, 3, {8}: 53
15, {3}, 11: 162731
15, 3, {11}: 491182564871651
15, {3}, 13: 733
15, 3, {13}: 883
15, {3}, 14: 59
15, 3, {14}: 59
15, {4}, 1: 61
15, 4, {1}: 61
15, {4}, 7: 67
15, 4, {7}: 67
15, {4}, 11: 71
15, 4, {11}: 71
15, {4}, 13: 73
15, 4, {13}: 73
15, {5}, 1: 1201
15, 5, {1}: 2924441266741
15, {5}, 2: 18077
15, 5, {2}: 3905357
15, {5}, 4: 79
15, 5, {4}: 79
15, {5}, 7: 13729771207
15, 5, {7}: 1237
15, {5}, 8: 83
15, 5, {8}: 83
15, {5}, 11: 271211
15, 5, {11}: 1301
15, {5}, 13: 1213
15, 5, {13}: 15194280133
15, {5}, 14: 89
15, 5, {14}: 89
15, {6}, 1: 4881691
15, 6, {1}: 1037360491
15, {6}, 7: 97
15, 6, {7}: 97
15, {6}, 11: 101
15, 6, {11}: 101
15, {6}, 13: 103
15, 6, {13}: 103
15, {7}, 1: 379681
15, 7, {1}: 80547991
15, {7}, 2: 107
15, 7, {2}: 107
15, {7}, 4: 109
15, 7, {4}: 109
15, {7}, 8: 113
15, 7, {8}: 113
15, {7}, 11: 218946945190429691
15, 7, {11}: 2588949882704871041434151
15, {7}, 13: 1693
15, 7, {13}: 1783
15, {8}, 1: 28921
15, 8, {1}: 27241
15, {8}, 7: 127
15, 8, {7}: 127
15, {8}, 11: 131
15, 8, {11}: 131
15, {8}, 13: 1933
15, 8, {13}: 30133
15, {9}, 1: 2161
15, 9, {1}: 103329241
15, {9}, 2: 137
15, 9, {2}: 137
15, {9}, 4: 139
15, 9, {4}: 139
15, {9}, 7: 24713588167
15, 9, {7}: 2137
15, {9}, 8: 1647572543
15, 9, {8}: 2153
15, {9}, 11: 488171
15, 9, {11}: 495401
15, {9}, 13: 5560557338173
15, 9, {13}: 502633
15, {9}, 14: 149
15, 9, {14}: 149
15, {10}, 1: 151
15, 10, {1}: 151
15, {10}, 7: 157
15, 10, {7}: 157
15, {10}, 11: 2411
15, 10, {11}: 8190401
15, {10}, 13: 163
15, 10, {13}: 163
15, {11}, 1: 881786829315764563424246641
15, 11, {1}: 560491
15, {11}, 2: 167
15, 11, {2}: 167
15, {11}, 4: 39769
15, 11, {4}: 2539
15, {11}, 7: 2647
15, 11, {7}: 130992187
15, {11}, 8: 173
15, 11, {8}: 173
15, {11}, 13: 596653
15, 11, {13}: 2683
15, {11}, 14: 179
15, 11, {14}: 179
15, {12}, 1: 181
15, 12, {1}: 181
15, {12}, 7: 2887
15, 12, {7}: 42187
15, {12}, 11: 191
15, 12, {11}: 191
15, {12}, 13: 193
15, 12, {13}: 193
15, {13}, 1: 3121
15, 13, {1}: 661741
15, {13}, 2: 197
15, 13, {2}: 197
15, {13}, 4: 199
15, 13, {4}: 199
15, {13}, 7: 705127
15, 13, {7}: 3037
15, {13}, 8: 10577003
15, 13, {8}: 2318805803
15, {13}, 11: 91488544954572405131
15, 13, {11}: 157027901
15, {13}, 14: 535461077009
15, 13, {14}: 408700964355468749
15, {14}, 1: 211
15, 14, {1}: 211
15, {14}, 11: 3371
15, 14, {11}: 42527645637007506364690405981881277901
15, {14}, 13: 223
15, 14, {13}: 223
16, {1}, 1: 17
16, 1, {1}: 17
16, {1}, 3: 19
16, 1, {3}: 19
16, {1}, 5: 277
16, 1, {5}: 0
16, {1}, 7: 23
16, 1, {7}: 23
16, {1}, 9: 281
16, 1, {9}: 409
16, {1}, 11: 283
16, 1, {11}: 443
16, {1}, 13: 29
16, 1, {13}: 29
16, {1}, 15: 31
16, 1, {15}: 31
16, {2}, 1: 8737
16, 2, {1}: 581714951868689
16, {2}, 3: 547
16, 2, {3}: 563
16, {2}, 5: 37
16, 2, {5}: 37
16, {2}, 7: 725935716098002055388532495854438851111
16, 2, {7}: 631
16, {2}, 9: 41
16, 2, {9}: 41
16, {2}, 11: 43
16, 2, {11}: 43
16, {2}, 13: 557
16, 2, {13}: 733
16, {2}, 15: 47
16, 2, {15}: 47
16, {3}, 1: 253530120045645880299340641073
16, 3, {1}: 13171233041
16, {3}, 5: 53
16, 3, {5}: 53
16, {3}, 7: 823
16, 3, {7}: 887
16, {3}, 11: 59
16, 3, {11}: 59
16, {3}, 13: 61
16, 3, {13}: 61
16, {4}, 1: 0
16, 4, {1}: 16657
16, {4}, 3: 67
16, 4, {3}: 67
16, {4}, 5: 1093
16, 4, {5}: 1109
16, {4}, 7: 71
16, 4, {7}: 71
16, {4}, 9: 73
16, 4, {9}: 73
16, {4}, 11: 17483
16, 4, {11}: 19387
16, {4}, 13: 0
16, 4, {13}: 444540081354816304286954136617869418478679481821
16, {4}, 15: 79
16, 4, {15}: 79
16, {5}, 1: 1361
16, 5, {1}: 1297
16, {5}, 3: 83
16, 5, {3}: 83
16, {5}, 7: 1367
16, 5, {7}: 1399
16, {5}, 9: 89
16, 5, {9}: 89
16, {5}, 11: 21851
16, 5, {11}: 1613789866474427
16, {5}, 13: 1373
16, 5, {13}: 24029
16, {6}, 1: 97
16, 6, {1}: 97
16, {6}, 5: 101
16, 6, {5}: 101
16, {6}, 7: 103
16, 6, {7}: 103
16, {6}, 11: 107
16, 6, {11}: 107
16, {6}, 13: 109
16, 6, {13}: 109
16, {7}, 1: 113
16, 7, {1}: 113
16, {7}, 3: 1907
16, 7, {3}: 0
16, {7}, 5: 125269877
16, 7, {5}: 1877
16, {7}, 9: 1913
16, 7, {9}: 498073
16, {7}, 11: 32069089147
16, 7, {11}: 1979
16, {7}, 13: 2004318077
16, 7, {13}: 9972184721795404625107398548957
16, {7}, 15: 127
16, 7, {15}: 127
16, {8}, 1: 143165569
16, 8, {1}: 0
16, {8}, 3: 131
16, 8, {3}: 131
16, {8}, 5: 34949
16, 8, {5}: 0
16, {8}, 7: 56166555556563832905556281431290897236744050880292859335632521351
16, 8, {7}: 34679
16, {8}, 9: 137
16, 8, {9}: 137
16, {8}, 11: 139
16, 8, {11}: 139
16, {8}, 13: 8947853
16, 8, {13}: 2269
16, {8}, 15: 0
16, 8, {15}: 0
16, {9}, 1: 39313
16, 9, {1}: 594193
16, {9}, 5: 149
16, 9, {5}: 149
16, {9}, 7: 151
16, 9, {7}: 151
16, {9}, 11: 2459
16, 9, {11}: 637883
16, {9}, 13: 157
16, 9, {13}: 157
16, {10}, 1: 733007751841
16, 10, {1}: 41233
16, {10}, 3: 163
16, 10, {3}: 163
16, {10}, 7: 167
16, 10, {7}: 167
16, {10}, 9: 2729
16, 10, {9}: 2713
16, {10}, 11: 2731
16, 10, {11}: 43963
16, {10}, 13: 173
16, 10, {13}: 173
16, {11}, 1: 48049
16, 11, {1}: 2833
16, {11}, 3: 179
16, 11, {3}: 179
16, {11}, 5: 181
16, 11, {5}: 181
16, {11}, 7: 2999
16, 11, {7}: 49248958327
16, {11}, 9: 3001
16, 11, {9}: 2969
16, {11}, 13: 12303293
16, 11, {13}: 3037
16, {11}, 15: 191
16, 11, {15}: 191
16, {12}, 1: 193
16, 12, {1}: 193
16, {12}, 5: 197
16, 12, {5}: 197
16, {12}, 7: 199
16, 12, {7}: 199
16, {12}, 11: 0
16, 12, {11}: 3259
16, {12}, 13: 0
16, 12, {13}: 843229
16, {13}, 1: 3722304977
16, 13, {1}: 0
16, {13}, 3: 211
16, 13, {3}: 211
16, {13}, 5: 3541
16, 13, {5}: 3413
16, {13}, 7: 908759
16, 13, {7}: 882551
16, {13}, 9: 999198637325934041
16, 13, {9}: 73749768669482915691491069321318626688914012237296060805206525525363591813836272035774910527919776180923677912554968891935394987986240496179621997893655532569315694783635803112700208508303413378891202384198997794213422176304573334413545606463639516918296257466344350258969657713796137622531892391975484473872559979575727003547753581022912486703477573912049826765132053211177341761946288632815391744689614375401028459100583268869723038988660204614984245471691470023113466364417874725337512714447532250846778586077760659205293618044147237229216306717637040861705334444181470200752974579322509544738704990857820454867202261704090678797538558326245584064671252468247095559023662993480878895077936090038599163027885118597295012047000583187251486272120581780591332114804425847265975339536610666721934463637123044596968540088909735294287236975077640291184075261370631154339959438296960077041200837369288569872632621900878776171600339056308989379011831334774153700978537796879728062464221403872190754080933322090903482454554035388405527848042146114638163297543714301665763439499883062039897590145382317298449300948509392928786633391596392577283746404680586571591203059252060135897745743650628381225179713605144028836751431506430002554673660083008864161778283325595456594614681
16, {13}, 11: 3547
16, 13, {11}: 0
16, {13}, 15: 223
16, 13, {15}: 223
16, {14}, 1: 61153
16, 14, {1}: 0
16, {14}, 3: 227
16, 14, {3}: 227
16, {14}, 5: 229
16, 14, {5}: 229
16, {14}, 9: 233
16, 14, {9}: 233
16, {14}, 11: 1300876803247619683256250232571154421182187
16, 14, {11}: 58304019973926508829195794288364830930948296694792337729075131089632305865113112154282586276837027138881157744159537913596428450753911485229682880296019616116874769754520009659
16, {14}, 13: 3821
16, 14, {13}: 15588829
16, {14}, 15: 239
16, 14, {15}: 239
16, {15}, 1: 241
16, 15, {1}: 241
16, {15}, 7: 0
16, 15, {7}: 66428827511
16, {15}, 11: 251
16, 15, {11}: 251
16, {15}, 13: 4093
16, 15, {13}: 1039837
17, {1}, 1: 307
17, 1, {1}: 307
17, {1}, 2: 19
17, 1, {2}: 19
17, {1}, 3: 0
17, 1, {3}: 99181
17, {1}, 4: 0
17, 1, {4}: 83916100750126603685919194992742897025955933234826458732651955310888606001
17, {1}, 5: 311
17, 1, {5}: 379
17, {1}, 6: 23
17, 1, {6}: 23
17, {1}, 7: 313
17, 1, {7}: 0
17, {1}, 8: 5227
17, 1, {8}: 433
17, {1}, 9: 0
17, 1, {9}: 0
17, {1}, 10: 0
17, 1, {10}: 135721
17, {1}, 11: 317
17, 1, {11}: 487
17, {1}, 12: 29
17, 1, {12}: 29
17, {1}, 13: 0
17, 1, {13}: 523
17, {1}, 14: 31
17, 1, {14}: 31
17, {1}, 15: 0
17, 1, {15}: 0
17, {1}, 16: 0
17, 1, {16}: 577
17, {2}, 1: 613
17, 2, {1}: 10133
17, {2}, 3: 37
17, 2, {3}: 37
17, {2}, 5: 617
17, 2, {5}: 0
17, {2}, 7: 41
17, 2, {7}: 41
17, {2}, 9: 43
17, 2, {9}: 43
17, {2}, 11: 1238072254113251
17, 2, {11}: 0
17, {2}, 13: 47
17, 2, {13}: 47
17, {2}, 15: 10453
17, 2, {15}: 14431
17, {3}, 1: 919
17, 3, {1}: 0
17, {3}, 2: 53
17, 3, {2}: 53
17, {3}, 4: 15661
17, 3, {4}: 202312198150541083680042289968086175246392709668351015606480937190197719141555582914093984895921012851445550526307546901116396443873602008692570890526616368088185273150378794958020557689359714900361309302539772871024478810285911676119560774228749311819445083611973024639995726017975216038442723332032205483020153342941185706273838443645335316482628378969548962919113661222993675604814390665842420911527121940927081962502185843741384735861252053394208248911431061837614993199706806589642337729249897643547266180540450077659378716054747677348448847073865940465584890988386180646046096142019029751447198046438996788148066525863948926986329748608837564706756460589811903109379913331680366765910754291118931715014909287015368864757215181388931953550080792379895988355601407167699413638955421000867341962147355447891731848892228022595327764226835424944122126463004299092971566585126472463347029693202245076617052610545287249829830436323088053040740425159393510325367055688958021542160943392588303575764092913624724403124665858417454172809264055857172288650133078210277438852308448011492659742540728194755339738118552604075349989025970187600141495195618093003594891737256114574520645691866237754189839522103102401063052231296238777096133716514735976159018372250704239420907521162127191197581548074707774582416270546628549687572701286582548353511119649632719976299664503918217812466555481567
17, {3}, 5: 44826081690965873408843
17, 3, {5}: 0
17, {3}, 7: 12954737608689137415155107
17, 3, {7}: 0
17, {3}, 8: 59
17, 3, {8}: 59
17, {3}, 10: 61
17, 3, {10}: 61
17, {3}, 11: 929
17, 3, {11}: 0
17, {3}, 13: 2059346319065197758094572502134042410628879165166500729310408311859100403174401197154162422913971035803320636990925926115809373804593425373
17, 3, {13}: 0
17, {3}, 14: 15671
17, 3, {14}: 19037
17, {3}, 16: 67
17, 3, {16}: 67
17, {4}, 1: 354961
17, 4, {1}: 0
17, {4}, 3: 71
17, 4, {3}: 71
17, {4}, 5: 73
17, 4, {5}: 73
17, {4}, 7: 1231
17, 4, {7}: 0
17, {4}, 9: 6034397
17, 4, {9}: 0
17, {4}, 11: 79
17, 4, {11}: 79
17, {4}, 13: 1237
17, 4, {13}: 0
17, {4}, 15: 83
17, 4, {15}: 83
17, {5}, 1: 1531
17, 5, {1}: 0
17, {5}, 2: 0
17, 5, {2}: 1481
17, {5}, 3: 0
17, 5, {3}: 1499
17, {5}, 4: 89
17, 5, {4}: 89
17, {5}, 6: 7542991
17, 5, {6}: 1553
17, {5}, 7: 258512581839480238807
17, 5, {7}: 1571
17, {5}, 8: 0
17, 5, {8}: 2192870942958148263240732846802917159482980658678987091029
17, {5}, 9: 0
17, 5, {9}: 1607
17, {5}, 11: 443711
17, 5, {11}: 11465965308803
17, {5}, 12: 97
17, 5, {12}: 97
17, {5}, 13: 1543
17, 5, {13}: 978696116876517899
17, {5}, 14: 0
17, 5, {14}: 1697
17, {5}, 16: 101
17, 5, {16}: 101
17, {6}, 1: 103
17, 6, {1}: 103
17, {6}, 5: 107
17, 6, {5}: 107
17, {6}, 7: 109
17, 6, {7}: 109
17, {6}, 11: 113
17, 6, {11}: 113
17, {6}, 13: 31327
17, 6, {13}: 33469
17, {7}, 1: 2143
17, 7, {1}: 0
17, {7}, 2: 881997331441
17, 7, {2}: 595087
17, {7}, 3: 0
17, 7, {3}: 600307
17, {7}, 4: 0
17, 7, {4}: 0
17, {7}, 5: 378893998333780600151918874693847240901089686373872230108378808042086874393030756381396925278352076800011918029919353266890458763951613729721003300506667989912895216810823140369059431198254813404980456907218177386238322978141949043394150307
17, 7, {5}: 2113
17, {7}, 6: 3051893879
17, 7, {6}: 2131
17, {7}, 8: 127
17, 7, {8}: 127
17, {7}, 9: 0
17, 7, {9}: 0
17, {7}, 10: 0
17, 7, {10}: 2203
17, {7}, 11: 2153
17, 7, {11}: 2221
17, {7}, 12: 131
17, 7, {12}: 131
17, {7}, 13: 41702102071970378195387359655683833039903949723250346925868452559148327207890893
17, 7, {13}: 652507
17, {7}, 15: 0
17, 7, {15}: 2293
17, {7}, 16: 0
17, 7, {16}: 2311
17, {8}, 1: 137
17, 8, {1}: 137
17, {8}, 3: 139
17, 8, {3}: 139
17, {8}, 5: 205169333
17, 8, {5}: 0
17, {8}, 7: 41759
17, 8, {7}: 41453
17, {8}, 9: 41761
17, 8, {9}: 3513524887
17, {8}, 11: 2459
17, 8, {11}: 0
17, {8}, 13: 149
17, 8, {13}: 149
17, {8}, 15: 151
17, 8, {15}: 151
17, {9}, 1: 1869203091411069355531855107713874939536131449989395150006339082674324728843097134694876907498265745273934952429109315615950710196111138285659470330945631318449307425659030790402840617341437874222598461881879133913175043742346246602676865994997400473054183576876916093555691941736178502386560694611375263595063904161089255385478117171010347061697561996379095781
17, 9, {1}: 0
17, {9}, 2: 0
17, 9, {2}: 26119610345027069111
17, {9}, 4: 157
17, 9, {4}: 157
17, {9}, 5: 318432595751153950765929260897357340270548161611474263261032439
17, 9, {5}: 0
17, {9}, 7: 798667
17, 9, {7}: 0
17, {9}, 8: 0
17, 9, {8}: 1126584826721
17, {9}, 10: 163
17, 9, {10}: 163
17, {9}, 11: 1484977798350500254413793712430566937554737931611808526186904667195181046087793905731792589996736627655426994137073129024366057707897203460483507678035133303365716593172893884937437026459910380639297977860505840938309245704441111978939720110627977640039737901597151324407412762783691381348217201857297710952740180161422146895372735157410289658892002936744181900705726584432634813253077897930046233672820880239776911825001489964358127541526621813068048456677805001704305921861565527867539831639250429173215267040916636092040704485096023324307405256389881176818589340226230788760221017153318401853554456294869299859474316434325775618344335121116055307007022846189770019398196291526515166484820726469725893644511218051827243865879539556725725237671
17, 9, {11}: 0
17, {9}, 13: 2767
17, 9, {13}: 0
17, {9}, 14: 167
17, 9, {14}: 167
17, {9}, 16: 15055951007560998946554855935799468825319902264000113359851960469680860934897071208938883683385877468609842462917887776927689487889
17, 9, {16}: 9173262075447700370870228797372658996167179560206469728679179339455142654588221994000682651957214660337554289179686628006320784508450532253801769
17, {10}, 1: 3061
17, 10, {1}: 0
17, {10}, 3: 173
17, 10, {3}: 173
17, {10}, 7: 3067
17, 10, {7}: 0
17, {10}, 9: 179
17, 10, {9}: 179
17, {10}, 11: 181
17, 10, {11}: 181
17, {10}, 13: 12479730563037202376599416073
17, 10, {13}: 0
17, {11}, 1: 568727680046856525361
17, 11, {1}: 1862679706313372777
17, {11}, 2: 0
17, 11, {2}: 929171
17, {11}, 3: 0
17, 11, {3}: 78041286371
17, {11}, 4: 191
17, 11, {4}: 191
17, {11}, 5: 3371
17, 11, {5}: 166550976124720415238207768204709708607521852138914978863354513192214473215336519626508334154498477187662381682580264286031705945239873635193075307951288200708854083160986114702227981399575722978160695129072904545350759971963253886646257214591
17, {11}, 6: 193
17, 11, {6}: 193
17, {11}, 7: 3373
17, 11, {7}: 955271
17, {11}, 8: 0
17, 11, {8}: 3323
17, {11}, 9: 0
17, 11, {9}: 965711
17, {11}, 10: 197
17, 11, {10}: 197
17, {11}, 12: 199
17, 11, {12}: 199
17, {11}, 13: 9394230696635382053176380469368734655867242678435691492562299088334773
17, 11, {13}: 3413
17, {11}, 14: 0
17, 11, {14}: 991811
17, {11}, 15: 0
17, 11, {15}: 3449
17, {11}, 16: 0
17, 11, {16}: 3467
17, {12}, 1: 3673
17, 12, {1}: 59263
17, {12}, 5: 3677
17, 12, {5}: 1378486138632359758323050626992747918650304829615263354995388341922232251095972398990848507942018139080087311
17, {12}, 7: 211
17, 12, {7}: 211
17, {12}, 11: 62639
17, 12, {11}: 434824684403093
17, {12}, 13: 88940907373
17, 12, {13}: 18191917
17, {13}, 1: 3768651696722334407412704432886748501027917638216745188121713071850567124051602405518201798458848401
17, 13, {1}: 635636818875898469533
17, {13}, 2: 223
17, 13, {2}: 223
17, {13}, 3: 0
17, 13, {3}: 1101433
17, {13}, 4: 0
17, 13, {4}: 1106653
17, {13}, 5: 27845915749943
17, 13, {5}: 3847
17, {13}, 6: 227
17, 13, {6}: 227
17, {13}, 7: 228154556301155739164141873957905004400241046704207
17, 13, {7}: 93736740613
17, {13}, 8: 229
17, 13, {8}: 229
17, {13}, 9: 0
17, 13, {9}: 3919
17, {13}, 10: 0
17, 13, {10}: 1137973
17, {13}, 11: 3989
17, 13, {11}: 1143193
17, {13}, 12: 233
17, 13, {12}: 233
17, {13}, 14: 160688404748616050182618301672566324918805941
17, 13, {14}: 5693449087
17, {13}, 15: 0
17, 13, {15}: 4027
17, {13}, 16: 0
17, 13, {16}: 1169293
17, {14}, 1: 239
17, 14, {1}: 239
17, {14}, 3: 241
17, 14, {3}: 241
17, {14}, 5: 4289
17, 14, {5}: 0
17, {14}, 9: 21120367
17, 14, {9}: 39353705070153506531713748224825668451495187254508108657151444786252940850766831171765242341910761541585017932281107
17, {14}, 11: 103764391931
17, 14, {11}: 0
17, {14}, 13: 251
17, 14, {13}: 251
17, {14}, 15: 533707265356695216704103124038332368542873946283142922292003783538887189897335797029414828786775114395339696927112171531204140647898504891296804256353517276617145553123762661125816798731498754318189024538693935214938297276615812780293
17, 14, {15}: 73387
17, {15}, 1: 4591
17, 15, {1}: 0
17, {15}, 2: 257
17, 15, {2}: 257
17, {15}, 4: 546208347402889
17, 15, {4}: 74923
17, {15}, 7: 4597
17, 15, {7}: 0
17, {15}, 8: 263
17, 15, {8}: 263
17, {15}, 11: 111176134211
17, 15, {11}: 0
17, {15}, 13: 4603
17, 15, {13}: 0
17, {15}, 14: 269
17, 15, {14}: 269
17, {15}, 16: 271
17, 15, {16}: 271
17, {16}, 1: 34271896307617
17, 16, {1}: 0
17, {16}, 3: 74443609190419550764562450397778200846849192983001551466849044370008879517232307105675227070196683723355515193456559323901778769141226118951876996802487398051974943265833071289084569071200666892787
17, 16, {3}: 3110633786280773828357619125469664392231273829727161657043436458261702025678677414669396669249
17, {16}, 5: 277
17, 16, {5}: 277
17, {16}, 7: 4903
17, 16, {7}: 0
17, {16}, 9: 281
17, 16, {9}: 281
17, {16}, 11: 283
17, 16, {11}: 283
17, {16}, 13: 4909
17, 16, {13}: 0
17, {16}, 15: 24137567
17, 16, {15}: 66886068539071498820247358361862720864806052666582265636907882027208271253
18, {1}, 1: 19
18, 1, {1}: 19
18, {1}, 5: 23
18, 1, {5}: 23
18, {1}, 7: 349
18, 1, {7}: 457
18, {1}, 11: 29
18, 1, {11}: 29
18, {1}, 13: 31
18, 1, {13}: 31
18, {1}, 17: 359
18, 1, {17}: 647
18, {2}, 1: 37
18, 2, {1}: 37
18, {2}, 5: 41
18, 2, {5}: 41
18, {2}, 7: 43
18, 2, {7}: 43
18, {2}, 11: 47
18, 2, {11}: 47
18, {2}, 13: 72025897
18, 2, {13}: 3198298525119427
18, {2}, 17: 53
18, 2, {17}: 53
18, {3}, 1: 18523
18, 3, {1}: 991
18, {3}, 5: 59
18, 3, {5}: 59
18, {3}, 7: 61
18, 3, {7}: 61
18, {3}, 11: 108038837
18, 3, {11}: 1181
18, {3}, 13: 67
18, 3, {13}: 67
18, {3}, 17: 71
18, 3, {17}: 71
18, {4}, 1: 73
18, 4, {1}: 73
18, {4}, 5: 1373
18, 4, {5}: 47321007179
18, {4}, 7: 79
18, 4, {7}: 79
18, {4}, 11: 83
18, 4, {11}: 83
18, {4}, 13: 1381
18, 4, {13}: 1543
18, {4}, 17: 89
18, 4, {17}: 89
18, {5}, 1: 30871
18, 5, {1}: 34130064295121260303
18, {5}, 7: 97
18, 5, {7}: 97
18, {5}, 11: 101
18, 5, {11}: 101
18, {5}, 13: 103
18, 5, {13}: 103
18, {5}, 17: 107
18, 5, {17}: 107
18, {6}, 1: 109
18, 6, {1}: 109
18, {6}, 5: 113
18, 6, {5}: 113
18, {6}, 7: 3889397851
18, 6, {7}: 412073923449193
18, {6}, 11: 2063
18, 6, {11}: 2153
18, {6}, 13: 37057
18, 6, {13}: 39451
18, {6}, 17: 2069
18, 6, {17}: 2267
18, {7}, 1: 127
18, 7, {1}: 127
18, {7}, 5: 131
18, 7, {5}: 131
18, {7}, 11: 137
18, 7, {11}: 137
18, {7}, 13: 139
18, 7, {13}: 139
18, {7}, 17: 2411
18, 7, {17}: 2591
18, {8}, 1: 49393
18, 8, {1}: 845983
18, {8}, 5: 149
18, 8, {5}: 149
18, {8}, 7: 151
18, 8, {7}: 151
18, {8}, 11: 889211
18, 8, {11}: 2801
18, {8}, 13: 157
18, 8, {13}: 157
18, {8}, 17: 2753
18, 8, {17}: 578415690713087
18, {9}, 1: 163
18, 9, {1}: 163
18, {9}, 5: 167
18, 9, {5}: 167
18, {9}, 7: 1000357
18, 9, {7}: 3049
18, {9}, 11: 173
18, 9, {11}: 173
18, {9}, 13: 55579
18, 9, {13}: 3163
18, {9}, 17: 179
18, 9, {17}: 179
18, {10}, 1: 181
18, 10, {1}: 181
18, {10}, 7: 3968612127339681427
18, 10, {7}: 3373
18, {10}, 11: 191
18, 10, {11}: 191
18, {10}, 13: 193
18, 10, {13}: 193
18, {10}, 17: 197
18, 10, {17}: 197
18, {11}, 1: 199
18, 11, {1}: 199
18, {11}, 5: 3767
18, 11, {5}: 3659
18, {11}, 7: 3769
18, 11, {7}: 3697
18, {11}, 13: 211
18, 11, {13}: 211
18, {11}, 17: 3779
18, 11, {17}: 132239526911
18, {12}, 1: 140018322601
18, 12, {1}: 3907
18, {12}, 5: 74093
18, 12, {5}: 71699
18, {12}, 7: 223
18, 12, {7}: 223
18, {12}, 11: 227
18, 12, {11}: 227
18, {12}, 13: 229
18, 12, {13}: 229
18, {12}, 17: 233
18, 12, {17}: 233
18, {13}, 1: 4447
18, 13, {1}: 4231
18, {13}, 5: 239
18, 13, {5}: 239
18, {13}, 7: 241
18, 13, {7}: 241
18, {13}, 11: 4457
18, 13, {11}: 4421
18, {13}, 17: 251
18, 13, {17}: 251
18, {14}, 1: 4789
18, 14, {1}: 26565103
18, {14}, 5: 257
18, 14, {5}: 257
18, {14}, 11: 263
18, 14, {11}: 263
18, {14}, 13: 4801
18, 14, {13}: 4783
18, {14}, 17: 269
18, 14, {17}: 269
18, {15}, 1: 271
18, 15, {1}: 271
18, {15}, 7: 277
18, 15, {7}: 277
18, {15}, 11: 281
18, 15, {11}: 281
18, {15}, 13: 283
18, 15, {13}: 283
18, {15}, 17: 5147
18, 15, {17}: 30233087
18, {16}, 1: 32011489
18, 16, {1}: 30344239
18, {16}, 5: 293
18, 16, {5}: 293
18, {16}, 7: 5479
18, 16, {7}: 95713
18, {16}, 11: 5483
18, 16, {11}: 5393
18, {16}, 13: 32011501
18, 16, {13}: 5431
18, {16}, 17: 98801
18, 16, {17}: 5507
18, {17}, 1: 307
18, 17, {1}: 307
18, {17}, 5: 311
18, 17, {5}: 311
18, {17}, 7: 313
18, 17, {7}: 313
18, {17}, 11: 317
18, 17, {11}: 317
18, {17}, 13: 5827
18, 17, {13}: 9770144707511081415118442597789015238720654947319882836100223544506052645981243442054558121499672250712069138857313219

[sweety439]

Originally Posted by sweety439

Consider the "simplest" families x{y} and {x}y, where x,y are base b digits

Necessary conditions are gcd(x,y) = 1, gcd(y,b) = 1

Zeros means there are no such prime of this family with <= 3000 digits (include the case that this family is ruled out to contain only composites)

For the zeros in the table: (for bases <=16)

Base 5:

{1}3: covering set {2,3}
{1}4: covering set {2,3}
3{1}: covering set {2,3}
4{1}: covering set {2,3}

Base 9:

{1}1: full algebra factors: (9^n-1)/8 = (3^n-1)*(3^n+1)/8
1{1}: full algebra factors: (9^n-1)/8 = (3^n-1)*(3^n+1)/8
{1}5: covering set {2,5}
2{7}: covering set {2,5}
3{1}: full algebra factors: (25*9^n-1)/8 = (5*3^n-1)*(5*3^n+1)/8
{3}5: covering set {2,5}
{3}8: covering set {2,5}
3{8}: full algebra factors: 4*9^n-1 = (2*9^n-1)*(2*9^n+1)
5{1}: covering set {2,5}
5{7}: covering set {2,5}
6{1}: covering set {2,5}
{7}2: covering set {2,5}
{7}5: covering set {2,5}
{8}5: full algebra factors: 9^n-4 = (3^n-2)*(3^n+2)

Base 11:

{1}3: covering set {2,3}
{1}4: covering set {2,3}
{1}9: covering set {2,3}
{1}A: covering set {2,3}
2{5}: covering set {2,3}
3{1}: covering set {2,3}
3{5}: covering set {2,3}
3{7}: covering set {2,3}
4{1}: covering set {2,3}
4{7}: covering set {2,3}
{5}2: covering set {2,3}
{5}3: covering set {2,3}
5{7}: (unsolved family) -------------------------------------------------------
{5}8: covering set {2,3}
{5}9: covering set {2,3}
{7}3: covering set {2,3}
{7}4: covering set {2,3}
{7}9: covering set {2,3}
{7}A: covering set {2,3}
8{5}: covering set {2,3}
9{1}: covering set {2,3}
9{5}: covering set {2,3}
9{7}: covering set {2,3}
A{1}: covering set {2,3}
A{7}: covering set {2,3}

Base 13:

{1}5: covering set {2,5,17}
{1}7: covering set {2,7}
{1}8: covering set {2,7}
2{9}: covering set {2,7}
{3}7: covering set {2,7}
{3}A: covering set {2,7}
4{B}: covering set {2,7}
{5}7: covering set {2,7}
{5}C: covering set {2,7}
7{1}: covering set {2,7}
7{3}: covering set {2,7}
7{5}: covering set {2,7}
7{9}: covering set {2,7}
7{B}: covering set {2,7}
8{1}: covering set {2,7}
{9}2: covering set {2,7}
9{5}: (unsolved family) -------------------------------------------------------
{9}7: covering set {2,7}
A{3}: covering set {2,7}
{B}4: covering set {2,7}
{B}7: covering set {2,7}
C{5}: covering set {2,7}

Base 14:

{2}5: covering set {3,5}
3{D}: covering set {3,5}
{4}9: covering set {3,5}
4{D}: (smallest prime is 4D₁₉₆₉₈ = 5*14^19698-1)
5{B}: covering set {3,5}
6{1}: covering set {3,5}
6{B}: covering set {3,5}
{8}3: covering set {3,5}
{8}5: covering set {3,5}
8{D}: partial algebra factors: 9*14^n-1 = (3*14^(n/2)-1)*(3*14^(n/2)+1) for even n, divisible by 5 for odd n
A{1}: covering set {3,5}
A{D}: covering set {3,5}
B{1}: partial algebra factors: (144*14^n-1)/13 = (12*14^(n/2)-1)*(12*14^(n/2)+1)/13 for even n, divisible by 5 for odd n
{B}5: covering set {3,5}
{D}3: covering set {3,5}
{D}5: partial algebra factors: 14^n-9 = (14^(n/2)-3)*(14^(n/2)+3) for even n, divisible by 5 for odd n

Base 16:

1{5}: full algebra factors: (4*16^n-1)/3 = (2*4^n-1)*(2*4^n+1)/3
{4}1: full algebra factors: (4*16^n-49)/15 = (2*4^n-7)*(2*4^n+7)/15
{4}D: covering set {3,7,13}
7{3}: full algebra factors: (36*16^n-1)/5 = (6*4^n-1)*(6*4^n+1)/5
8{1}: full algebra factors: (121*16^n-1)/15 = (11*4^n-1)*(11*4^n+1)/15
8{5}: full algebra factors: (25*16^n-1)/3 = (5*4^n-1)*(5*4^n+1)/3
{8}F: covering set {3,7,13}
8{F}: full algebra factors: 9*16^n-1 = (3*4^n-1)*(3*4^n+1)
{C}B: full algebra factors: (4*16^n-9)/5 = (2*4^n-3)*(2*4^n+3)/5
{C}D: full algebra factors: (4*16^n+1)/5 = (2*4^n-2*2^n+1)*(2*4^n+2*2^n+1)
D{1}: full algebra factors: (196*16^n-1)/15 = (14*4^n-1)*(14*4^n+1)/15
D{B}: (unsolved family) -------------------------------------------------------
E{1}: covering set {3,7,13}
{F}7: full algebra factors: 16^n-9 = (4^n-3)*(4^n+3)

[sweety439]

This file is the smallest prime (not include x or y themselves) in given simple family x{y} or {x}y (where x,y are base b digits) in given base 2<=b<=24, where gcd(x,y) = 1, gcd(y,b) = 1 (searched up to 5000 base b digits, 0 if no such prime found (include the case such that x{y} or {x}y proven composite by all or partial algebra factors)

format of file:

b,x,{y}: smallest prime of the form x{y} in base b
b,{x},y: smallest prime of the form {x}y in base b

such primes are generalized near-repdigit primes base b

already excluded families x{y} and {x}y with NUMERICAL covering set (e.g. {1}3, {1}4, 3{1}, 4{1} in base 5)

Such primes are ALWAYS minimal prime (start with 2 digits) in base b, except when the repeating digit (i.e. y in x{y}, or x in {x}y) is 1 and base b has generalized repunit primes (i.e. all digits are 1) smaller than the prime (in base b = 9, 25, 32, 49, 64, 81, 121, 125, 144, 169, 216, 225, 243, 289, 324, 343, 361, 441, 484, 529, 625, ..., no generalized repunit primes exist, thus in these bases b, such primes are always minimal primes (start with 2 digits) in base b)

Also,

* in base 35, all such primes with <= 313 digits are minimal primes (start with 2 digits)
* in base 39, all such primes with <= 349 digits are minimal primes (start with 2 digits)
* in base 47, all such primes with <= 127 digits are minimal primes (start with 2 digits)
* in base 51, all such primes with <= 4229 digits are minimal primes (start with 2 digits)
* in base 91, all such primes with <= 4421 digits are minimal primes (start with 2 digits)
* in base 92, all such primes with <= 439 digits are minimal primes (start with 2 digits)
* in base 124, all such primes with <= 599 digits are minimal primes (start with 2 digits)
* in base 135, all such primes with <= 1171 digits are minimal primes (start with 2 digits)
* in base 139, all such primes with <= 163 digits are minimal primes (start with 2 digits)
* in base 142, all such primes with <= 1231 digits are minimal primes (start with 2 digits)
* in base 152, all such primes with <= 270217 digits are minimal primes (start with 2 digits)
* in base 171, all such primes with <= 181 digits are minimal primes (start with 2 digits)
* in base 174, all such primes with <= 3251 digits are minimal primes (start with 2 digits)
* in base 182, all such primes with <= 167 digits are minimal primes (start with 2 digits)
* in base 183, all such primes with <= 223 digits are minimal primes (start with 2 digits)
* in base 184, all such primes with <= 16703 digits are minimal primes (start with 2 digits)
* in base 185, all such primes with <= 66337 (at least) digits are minimal primes (start with 2 digits)
* in base 199, all such primes with <= 577 digits are minimal primes (start with 2 digits)
* in base 200, all such primes with <= 17807 digits are minimal primes (start with 2 digits)
* in base 201, all such primes with <= 271 digits are minimal primes (start with 2 digits)

[sweety439]

In fact, all these primes are minimal primes (start with 2 digits) base b:

* Smallest generalized repunit prime base b (if exists)
* The primes for all k<b for CRUS Sierpinski conjecture base b
* The primes for all k<b for CRUS Riesel conjecture base b
* Smallest generalized near-repdigit primes base b of the form x{y} or {x}y for all (x,y) digit pair (if exists)

Since ....

* Generalized repunit numbers base b are 111...111 in base b
* k<b for CRUS Sierpinski conjecture base b are [k]000...0001 in base b
* k<b for CRUS Riesel conjecture base b are [k-1][b-1][b-1][b-1]...[b-1][b-1][b-1] in base b
* Generalized near-repdigit numbers base b of the form x{y} or {x}y are [x][y][y][y]...[y][y][y] or [x][x][x]...[x][x][x][y] in base b

[sweety439]

I think that the "minimal primes (start with 2 digits) problem" for all bases b>90 will never be proven (when searched to 1M or 1G or even 1T base b digits, and even when we allow probable primes in place of proven primes.), like CRUS S/R280, S/R511, S/R855, and S/R910 problems and the "Sierpinski/Riesel twin prime conjecture" (the conjecture that 237 is the smallest k divisible by 3 such that k*2^n+-1 are not twin primes for all n>=1), for more information, see this post

[sweety439]

Originally Posted by sweety439

This file is the smallest prime (not include x or y themselves) in given simple family x{y} or {x}y (where x,y are base b digits) in given base 2<=b<=24, where gcd(x,y) = 1, gcd(y,b) = 1 (searched up to 5000 base b digits, 0 if no such prime found (include the case such that x{y} or {x}y proven composite by all or partial algebra factors)

format of file:

b,x,{y}: smallest prime of the form x{y} in base b
b,{x},y: smallest prime of the form {x}y in base b

such primes are generalized near-repdigit primes base b

already excluded families x{y} and {x}y with NUMERICAL covering set (e.g. {1}3, {1}4, 3{1}, 4{1} in base 5)

extended data to base 36

[sweety439]

Originally Posted by sweety439

* Case (5,1):

** 51 is prime, and thus the only minimal prime in this family.

* Case (5,3):

** 53 is prime, and thus the only minimal prime in this family.

* Case (5,5):

** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes)

*** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes)

**** However, all numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime, therefore, there is no minimal primes in this family.

**** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes)

***** However, 500525 is not prime, therefore, there is no minimal primes in this family.

***** The smallest prime of the form 5{5}25 is 555555555555525

***** The smallest prime of the form 5{5}025 is 55555025

***** For the 5{5}0025 family, since 55555025 is prime, we only need to consider the numbers 50025, 550025, 5550025, 55550025 (since any digit combo 5555 between (5,0025) will produce smaller primes)

****** However, none of them are primes, therefore, there is no minimal primes in this family.

***** The smallest prime of the form 5{5}0525 is 5550525

***** The smallest prime of the form 5{5}00525 is 5500525

***** For the 5{5}05025 family, since 55555025 is prime, we only need to consider the numbers 505025, 5505025, 55505025 (since any digit combo 555 between (5,05025) will produce smaller primes)

****** However, none of them are primes, therefore, there is no minimal primes in this family.

* Case (5,7):

** 57 is prime, and thus the only minimal prime in this family.

* Case (6,3):

** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes)

*** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime.

* Case (6,5):

** 65 is prime, and thus the only minimal prime in this family.

* Case (6,7):

** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes)

*** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, we only need to consider the families 60{1,4,7}7, 6{0}17, 6{0,4}4{4}7, 6{0}77 (since any digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them will produce smaller primes)

**** All numbers of the form 6{0}17 or 6{0}77 are divisible by 3, thus cannot be prime.

**** For the 60{1,4,7}7 family, since 117, 147, 177, 417, 6477, 717, 747, 6777 are primes, we only need to consider the numbers 6017, 6047, 6077 and the family 60{4}7 (since any digit combo 11, 14, 17, 41, 47, 71, 74, 77 between (60,7) will produce smaller primes)

***** None of 6017, 6047, 6077 are primes.

**** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime.

**** For the 6{0,4}4{4}7 family, since 6007 and 407 are primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digit combo 00, 40 between (6,4{4}7) will produce smaller primes)

***** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime.

***** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime.