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

[sweety439]

Update newest data file.

[sweety439]

Found another minimal prime (start with b+1) in base b=14:

18₇₉B

Found three other minimal primes (start with b+1) in base b=15:

3330₁₄31
6330₂₆1
60₅331

[sweety439]

Length of largest minimal prime (start with b+1) in base b for b = 2, 3, 4, ..., 40:

2, 3, 3, 96, 5, >=17, 221, >=1161, 31, >=45, 42, >=32021, >=19699, >=107, >=32235, >=111334, >=300, >=110986, >=449, >=479150, >=764, >=800874, >=315, >=136967, >=8773, >=109006, >=94538, >=174240, >=34206, >=9896, >=9750, >=23617, >=9377, >=9599, >=81995, >=22023, >=136212, >=9440, >=12383

(in sequences below, 0 means no such prime exists, Italic type means either not minimal prime (start with b+1) in base b or not acceptable as the form will produce a digit >=b or <0 in base b)

Length of the smallest repunit prime (form: {1}) in base b for b = 2, 3, 4, ..., 160:

2, 3, 2, 3, 2, 5, 3, 0, 2, 17, 2, 5, 3, 3, 2, 3, 2, 19, 3, 3, 2, 5, 3, 0, 7, 3, 2, 5, 2, 7, 0, 3, 13, 313, 2, 13, 3, 349, 2, 3, 2, 5, 5, 19, 2, 127, 19, 0, 3, 4229, 2, 11, 3, 17, 7, 3, 2, 3, 2, 7, 3, 5, 0, 19, 2, 19, 5, 3, 2, 3, 2, 5, 5, 3, 41, 3, 2, 5, 3, 0, 2, 5, 17, 5, 11, 7, 2, 3, 3, 4421, 439, 7, 5, 7, 2, 17, 13, 3, 2, 3, 2, 19, 97, 3, 2, 17, 2, 17, 3, 3, 2, 23, 29, 7, 59, 3, 5, 3, 5, 0, 5, 43, 599, 0, 2, 5, 7, 5, 2, 3, 47, 13, 5, 1171, 2, 11, 2, 163, 79, 3, 1231, 3, 0, 5, 7, 3, 2, 7, 2, 13, 270217, 3, 5, 3, 2, 17, 7, 13, 7

Length of the smallest generalized Fermat prime (form: 1{0}1) in base b for b = 2, 3, 4, ..., 160:

2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 3, 0, 2, 0, 2, 0, 3, 0, 2, 0, 3, 0, 3, 0, 2, 0, 2, 0, 0, 0, 5, 0, 2, 0, (>=16777217 or 0), 0, 2, 0, 2, 0, 17, 0, 2, 0, 5, 0, (>=16777217 or 0), 0, 2, 0, 3, 0, 3, 0, 2, 0, 2, 0, (>=16777217 or 0), 0, 0, 0, 2, 0, (>=16777217 or 0), 0, 2, 0, 2, 0, 3, 0, 17, 0, 2, 0, 5, 0, 2, 0, 3, 0, (>=16777217 or 0), 0, 2, 0, 3, 0, (>=16777217 or 0), 0, 3, 0, 2, 0, (>=16777217 or 0), 0, 2, 0, 2, 0, (>=16777217 or 0), 0, 2, 0, 2, 0, 3, 0, 2, 0, 33, 0, 3, 0, 5, 0, 3, 0, (>=16777217 or 0), 0, 3, 0, 2, 0, 0, 0, 2, 0, 5, 0, 3, 0, 2, 0, 2, 0, 5, 0, 5, 0, (>=16777217 or 0), 0, 3, 0, 2, 0, 2, 0, 9, 0, 5, 0, 2, 0, 17, 0, 3

Length of the smallest generalized half Fermat prime (form: {x}y, x = (b-1)/2, y = (b+1)/2) in base b for b = 2, 3, 4, ..., 160:

0, 2, 0, 2, 0, 4, 0, 2, 0, 2, 0, 4, 0, 2, 0, 4, 0, 2, 0, 4, 0, 4, 0, 2, 0, 0, 0, 2, 0, (>=524288 or 0), 0, 8, 0, 2, 0, (>=524288 or 0), 0, 2, 0, 16, 0, 8, 0, 2, 0, 8, 0, 2, 0, 2, 0, 8, 0, (>=524288 or 0), 0, 4, 0, 2, 0, 2, 0, (>=524288 or 0), 0, 2, 0, (>=524288 or 0), 0, 2, 0, 2, 0, 4, 0, 32, 0, (>=524288 or 0), 0, 2, 0, 4, 0, (>=524288 or 0), 0, 2, 0, 16, 0, (>=524288 or 0), 0, (>=524288 or 0), 0, (>=524288 or 0), 0, 2, 0, (>=524288 or 0), 0, (>=524288 or 0), 0, 2, 0, 4, 0, 4, 0, (>=524288 or 0), 0, (>=524288 or 0), 0, 16, 0, 4, 0, 4, 0, (>=524288 or 0), 0, 4, 0, 2, 0, (>=524288 or 0), 0, 0, 0, (>=524288 or 0), 0, 4, 0, 2, 0, (>=524288 or 0), 0, (>=524288 or 0), 0, (>=524288 or 0), 0, 2, 0, 2, 0, (>=524288 or 0), 0, 2, 0, (>=524288 or 0), 0, (>=524288 or 0), 0, (>=524288 or 0), 0, 4, 0, (>=524288 or 0), 0, 16, 0, 2, 0

Length of the smallest Williams prime of the 1st kind (form: x{y}, x = b-2, y = b-1) in base b for b = 2, 3, 4, ..., 160: (the b=2 case is not acceptable, since the digit 2-2 = 0 in base 2 cannot be leading digit)

3, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 3, 2, 15, 2, 2, 3, 7, 2, 2, 2, 56, 13, 2, 134, 2, 21, 2, 3, 2, 2, 3, 16, 4, 2, 8, 136212, 2, 2, 8, 2, 8, 8, 2, 2, 2, 3, 2, 26, 2, 6, 4, 2, 2, 2, 2, 3, 4, 2, 2, 900, 4, 12, 2, 2, 2, 64, 2, 14, 2, 26, 9, 4, 3, 8, 2, 45, 3, 12, 4, 82, 21496, 2, 3, 2, 2, 4, 26, 2, 520, 78, 477, 2, 2, 3, 2, 4984, 3, 3, 2, 2, 4, 2, 4, 3, 38, 411, 7, 6, 3, 8, 286644, 3, 2, 2, 3, 3, 4, 3, 2, 4, 7, 34, 8740, 2, 2, (>2220000 or 0), 3, 9, 2, 2, 3, 4, 2, 6, 26, 3, 2, 24, 2, 2, 8, 3, 2, 2, 6, 4, 2, 2, 4, 4, 3, 2, 2, 2, 4, 128, 2, 2

Length of the smallest Williams prime of the 2nd kind (form: x{0}1, x = b-1) in base b for b = 2, 3, 4, ..., 160:

2, 2, 2, 3, 2, 2, 3, 2, 4, 11, 4, 2, 3, 2, 2, 5, 2, 30, 15, 2, 2, 15, 3, 2, 3, 5, 2, 3, 5, 6, 13, 3, 2, 3, 3, 10, 17, 2, 3, 81, 2, 3, 5, 3, 4, 17, 3, 3, 3, 2, 16, 961, 16, 2, 5, 4, 2, 15, 2, 7, 21, 2, 4, 947, 7, 2, 19, 11, 2, 5, 2, 6, 43, 5, 2, 829, 2, 2, 3, 2, 13, 3, 7, 5, 31, 4, 3023, 3, 2, 2, 9, 3, 5, 5, 3, 12, 9, 3, 2, 3, 2, 57, 3, 13, 2, 5, 6, 16, 3, 2, 2, 5, 4, 3, 17, 4, 2, 47, 2, 3, 6217, (>400000 or 0), 3, 17, 5, 166, 73, 6, 65, 15, 2, 3, 51, 3, 280, 13, 3, 2, 3, 7, 2, 5, 2, 4, 5, 5, 2, 3, 15, 2, 9, 5, 2, 7, 2, 30, 1621, 17, 6

Length of the smallest Williams prime of the 4th kind (form: 11{0}1) in base b for b = 2, 3, 4, ..., 160: (not minimal prime if there is smaller prime of the form 1{0}1)

3, 3, 0, 3, 3, 0, 3, 4, 0, 4, 3, 0, 3, 3, 0, 3, 11, 0, 3, 3, 0, 4, 3, 0, 4, 3, 0, 7, 4, 0, 7, 3, 0, 4, 5, 0, 3, 5, 0, 3, 4, 0, 4, 4, 0, 4, 8, 0, 3, 185, 0, 4, 3, 0, 4, 3, 0, 3, 23, 0, 3, 187, 0, 5, 3, 0, 4, 3, 0, 3, 122, 0, 4, 3, 0, 3, 3, 0, 3, 10, 0, 7, 11, 0, 4, 4, 0, 3, 3, 0, 4, 5, 0, 11, 16, 0, 5, 3, 0, 3, 7, 0, 7, 3, 0, 82, 400, 0, 3, 3, 0, 5, 5, 0, 46, 3, 0, 3, 4, 0, 4, 4, 0, 4, 7, 0, 5, 56, 0, 3, 56, 0, 11, 19, 0, 22, 3, 0, 4, 3, 0, 3, 5, 0, 5, 3, 0, 11, 3, 0, 4, 3, 0, 3, 5, 0, 143, 34, 0

Length of the smallest dual Williams prime of the 1st kind (form: {x}1, x = b-1) in base b for b = 2, 3, 4, ..., 160:

2, 2, 2, 5, 2, 2, 13, 2, 3, 3, 5, 2, 3, 2, 2, 11, 2, 3, 17, 2, 2, 17, 4, 2, 3, 9, 2, 33, 7, 3, 7, 4, 2, 3, 5, 67, 5, 2, 9, 3, 2, 4, 25, 3, 4, 5, 5, 24, 3, 2, 3, 21, 3, 2, 9, 3, 2, 11, 2, 5, 3, 2, 4, 19, 31, 2, 29, 4, 2, 3019, 2, 21, 51, 3, 2, 3, 2, 2, 9, 2, 169, 965, 3, 3, 29, 3, 2848, 9, 2, 2, 3, (>60000 or 0), 4, 3, 7, 6, 5, 3, 2, 3, 2, 5, 55, 4, 2, 7, 4, 4, 61, 2, 2, (>25000 or 0), 991, 4, 3, 18, 2, 9, 2, 4, 61, 17, 9, 3, 16, 18, 401, 3, 3, 25, 2, 9, 3, 13, 3, 5, 4, 2, 3, 3, 2, 281, 2, 255, 5, 3, 2, 7, 90, 2, (>25000 or 0), 6, 2, 3, 2, 6, (>25000 or 0), 6, 33

Length of the smallest dual Williams prime of the 2nd kind (form: 1{0}x, x = b-1) in base b for b = 2, 3, 4, ..., 160:

2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 3, 17, 2, 2, 5, 4, 2, 3, 2, 2, 5, 2, 4, 3, 2, 3, 11, 2, 2, 109, 4, 2, 3, 2, 2, 3, 3, 2, 3, 2, 4, 3, 2, 3, 21, 3, 2, 3, 2, 2, 3, 2, 2, 3, 2, 5, 3, 3, 8, 9, 4, 2, 3, 2, 25, 3, 2, 2, 13, 5, 4, 9, 2, 2, 5, 4, 2, 195, 4, 2, 3, 2, 3, 3, 2, 9, 3, 2, 2, 5, 3, 3, 55, 2, 2, 5, 2, 2, 3, 45, 3, 15, 4, 2, 1401, 7, 4, 5, 7, 2, 20089, 2, 2, 7, 2, 7, 5, 2, 2, 5, 64371, 4, 3, 2, 4, 505, 2, 3, 3, 2, 9, 3, 2, 2, 61, 3, 2, 3, 2, 2, 5, 3, 3, 9, 2, 3, 3, 4, 4, 21, 3, 2, 3, 2, 2, 3, 2, 3

Length of the smallest dual Williams prime of the 4th kind (form: 1{0}11) in base b for b = 2, 3, 4, ..., 160: (not minimal prime if there is smaller prime of the form 1{0}1)

2, 2, 0, 2, 2, 0, 2, 2, 0, 2, 3, 0, 2, 2, 0, 3, 2, 0, 2, 2, 0, 2, 3, 0, 2, 3, 0, 2, 2, 0, 4, 2, 0, 2, 2, 0, 3, 2, 0, 2, 4, 0, 2, 7, 0, 5, 2, 0, 2, 2, 0, 2, 2, 0, 2, 3, 0, 3, 4, 0, 3, 2, 0, 2, 3, 0, 2, 2, 0, 3, 5, 0, 2, 2, 0, 3, 2, 0, 3, 2, 0, 2, 31, 0, 2, 4, 0, 2, 2, 0, 8, 68, 0, 2, 2, 0, 2, 2, 0, 3, 4, 0, 4, 2, 0, 5, 4, 0, 3, 2, 0, 2, 2, 0, 2, 3, 0, 2, 2, 0, 13, 8, 0, 2, 4, 0, 2, 569, 0, 2, 25, 0, 2, 2, 0, 44, 2, 0, 2, 2, 0, 3, 4, 0, 2, 3, 0, 8, 3, 0, 4, 2, 0, 2, 2, 0, 2, 5, 0

Length of the smallest prime of the form 2*b^n+1 (form: 2{0}1) in base b for b = 2, 3, 4, ..., 160: (the b=2 case is not acceptable, since there is no digit 2 in base 2)

2, 2, 0, 2, 2, 0, 2, 2, 0, 2, 4, 0, 2, 2, 0, 48, 2, 0, 2, 2, 0, 2, 3, 0, 2, 3, 0, 2, 2, 0, 4, 2, 0, 2, 2, 0, 2730, 2, 0, 2, 3, 0, 2, 3, 0, 176, 2, 0, 2, 2, 0, 2, 2, 0, 2, 4, 0, 4, 4, 0, 44, 2, 0, 2, 3, 0, 2, 2, 0, 4, 3, 0, 2, 2, 0, 4, 2, 0, 12, 2, 0, 2, 5, 0, 2, 3, 0, 2, 2, 0, 4, 3, 0, 2, 2, 0, 2, 2, 0, 192276, 3, 0, 1234, 2, 0, 4, 6, 0, 52, 2, 0, 2, 2, 0, 2, 287, 0, 2, 2, 0, 756, 3, 0, 2, 5, 0, 2, 7, 0, 2, 3, 0, 2, 2, 0, 328, 2, 0, 2, 2, 0, 6, 6, 0, 2, 155, 0, 4, 4, 0, 4, 2, 0, 2, 2, 0, 2, 4, 0

Length of the smallest prime of the form 2*b^n-1 (form: 1{x}, x = b-1) in base b for b = 2, 3, 4, ..., 160:

2, 2, 2, 5, 2, 2, 3, 2, 2, 3, 2, 3, 5, 2, 2, 3, 3, 2, 11, 2, 2, 7, 2, 3, 7, 2, 3, 137, 2, 2, 7, 7, 2, 7, 2, 2, 3, 3, 2, 3, 2, 3, 5, 2, 3, 5, 5, 2, 3, 2, 2, 45, 2, 2, 3, 2, 4, 3, 6, 4, 3, 3, 2, 5, 2, 769, 5, 2, 2, 53, 35, 3, 133, 2, 2, 15, 8, 2, 3, 3, 2, 9, 2, 3, 11, 2, 25, 61, 2, 2, 3, 4, 6, 3, 2, 2, 3, 2, 2, 43, 3, 5, 69, 7, 2, 21911, 3, 3, 17, 25, 2, 3, 2, 2, 33, 2, 3, 29, 2, 2, 7, 9, 5, 3, 2, 3, 19, 2, 4, 5, 2, 5, 3, 2, 2, 3, 5, 2, 3, 2, 2, 3, 25, 13, 17, 2, 5, 5, 9, 6, 797, 3, 2, 3, 2, 2, 3, 2, 3

Length of the smallest prime of the form b^n+2 (form: 1{0}2) in base b for b = 2, 3, 4, ..., 160:

0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 12, 0, 0, 0, 2, 0, 2, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 114, 0, 0, 0, 2, 0, 8, 0, 0, 0, 2, 0, 2, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 13, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 2, 0, 256, 0, 0, 0, 9, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 3, 0, 16, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 24, 0, 0, 0, 2, 0, 2, 0, 0, 0, 5, 0, 4, 0, 0, 0, 2, 0, 2, 0, 0, 0, 4, 0, 2, 0, 0, 0, 137, 0

Length of the smallest prime of the form b^n-2 (form: {x}y, x = b-1, y = b-2) in base b for b = 2, 3, 4, ..., 160: (n>=2 is required, since single-digit primes are not acceptable in this puzzle, also (b,n) = (2,2) is also not acceptable, although 2^2-2 is prime, since 2^2-2 is not a prime which is >2, but this puzzle requires primes >b)

0, 2, 0, 2, 0, 2, 0, 2, 0, 4, 0, 2, 0, 2, 0, 6, 0, 2, 0, 2, 0, 24, 0, 7, 0, 2, 0, 2, 0, 3, 0, 2, 0, 2, 0, 2, 0, 4, 0, 4, 0, 2, 0, 11, 0, 2, 0, 2, 0, 8, 0, 4, 0, 2, 0, 12, 0, 4, 0, 2, 0, 2, 0, 8, 0, 3, 0, 2, 0, 2, 0, 4, 0, 2, 0, 2, 0, 38, 0, 130, 0, 4, 0, 4, 0, 4, 0, 2, 0, 3, 0, 2, 0, 4, 0, 747, 0, 3, 0, 4, 0, 2, 0, 10, 0, 2, 0, 3, 0, 17, 0, 10, 0, 13, 0, 2, 0, 2, 0, 2, 0, 6, 0, 42, 0, 2, 0, 3, 0, 2, 0, 6, 0, 2, 0, 10, 0, 2, 0, 4, 0, 4, 0, 2, 0, 16, 0, 50, 0, 3, 0, 9, 0, 2, 0, 22, 0, 25, 0

Length of the smallest prime of the form 3*b^n+1 (form: 3{0}1) in base b for b = 2, 3, 4, ..., 160: (the b=2 case is not acceptable, since there is no digit 3 in base 2)

2, 0, 2, 0, 2, 0, 3, 0, 2, 0, 2, 0, 2, 0, 3, 0, 4, 0, 2, 0, 2, 0, 2, 0, 2, 0, 8, 0, 4, 0, 2, 0, 2, 0, 2, 0, 4, 0, 3, 0, 2, 0, 10, 0, 2, 0, 4, 0, 2, 0, 2, 0, 2, 0, 6, 0, 3, 0, 2, 0, 13, 0, 2, 0, 2, 0, 3, 0, 2, 0, 15, 0, 2, 0, 2, 0, 3, 0, 2, 0, 3, 0, 3, 0, 3, 0, 5, 0, 2, 0, 2, 0, 2, 0, 4, 0, 3, 0, 6, 0, 2, 0, 2, 0, 4, 0, 271, 0, 2, 0, 2, 0, 13, 0, 2, 0, 47, 0, 3, 0, 2, 0, 2, 0, 2, 0, 28, 0, 22, 0, 2, 0, 5, 0, 2, 0, 9, 0, 2, 0, 3, 0, 2, 0, 2, 0, 3, 0, 4, 0, 2, 0, 2, 0, 3, 0, 5, 0, 3

Length of the smallest prime of the form 3*b^n-1 (form: 2{x}, x = b-1) in base b for b = 2, 3, 4, ..., 160: (the b=2 case is not acceptable, since there is no digit 2 in base 2)

2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 3, 0, 2, 0, 2, 0, 2, 0, 2, 0, 3, 0, 2, 0, 3, 0, 2, 0, 2, 0, 12, 0, 2, 0, 2, 0, 2, 0, 3, 0, 2524, 0, 2, 0, 2, 0, 3, 0, 2, 0, 3, 0, 3, 0, 2, 0, 2, 0, 2, 0, 60, 0, 2, 0, 2, 0, 11, 0, 3, 0, 3, 0, 3, 0, 2, 0, 2, 0, 2, 0, 15, 0, 2, 0, 2, 0, 2, 0, 2, 0, 3, 0, 2, 0, 3, 0, 2, 0, 4, 0, 4, 0, 2, 0, 2, 0, 7, 0, 3, 0, 51, 0, 64, 0, 2, 0, 2, 0, 2, 0, 3, 0, 12, 0, 51, 0, 2, 0, 2, 0, 39, 0, 2, 0, 3, 0, 3, 0, 2, 0, 27, 0, 2, 0, 4, 0, 2, 0, 2, 0, 4, 0, 2, 0, 2, 0, 3, 0, 2

Length of the smallest prime of the form b^n+3 (form: 1{0}3) in base b for b = 2, 3, 4, ..., 160: (the b=2 case is not acceptable, since there is no digit 3 in base 2)

2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 3, 0, 0, 0, 2, 0, 2, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 21, 0, 0, 0, 2, 0, 3, 0, 0, 0, 2, 0, 2, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 5, 0, 0, 0, 2, 0, 6, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 3, 0, 0, 0, 6, 0, 5, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 3, 0, 0, 0, 2, 0, 2, 0, 0, 0, 3, 0, 4, 0, 0, 0, 2, 0, 2, 0, 0, 0, 16, 0, 2, 0, 0, 0, 3, 0, 2

Length of the smallest prime of the form b^n-3 (form: {x}y, x = b-1, y = b-3) in base b for b = 2, 3, 4, ..., 160: (n>=2 is required, since single-digit primes are not acceptable in this puzzle) (the b=2 case is not acceptable, since there is no digit 2-3 = -1 in base 2)

3, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 3, 0, 0, 0, 2, 0, 6, 0, 0, 0, 2, 0, 10, 0, 0, 0, 2, 0, 2, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 21, 0, 105, 0, 0, 0, 18, 0, 2, 0, 0, 0, 5, 0, 2, 0, 0, 0, 2, 0, 5, 0, 0, 0, 3, 0, 5, 0, 0, 0, 2, 0, 13, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 204, 0, 0, 0, 2, 0, 70, 0, 0, 0, 4, 0, 3, 0, 0, 0, 2, 0, 2, 0, 0, 0, 3, 0, 2, 0, 0, 0, 6, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 7, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 346, 0, 396, 0, 0, 0, 3, 0, 21

Length of the smallest prime of the form 4*b^n+1 (form: 4{0}1) in base b for b = 2, 3, 4, ..., 160: (the b=2, 3, 4 cases are not acceptable, since there is no digit 4 in bases 2, 3, 4)

3, 2, 2, 3, 0, 2, 3, 2, 2, 0, 3, 2, 0, 2, 0, 7, 2, 4, 3, 0, 2, 343, 2, 2, 0, 2, 2, 0, 7, 0, (>=1717986919 or 0), 3, 2, 43, 0, 2, 11, 2, 6, 0, 3, 2, 0, 2, 0, 3, 2, 2, 11, 0, 11, (>1670000 or 0), 4, 3, 0, 2, 2, 0, 2, 0, 3, 3, 2, 3, 0, 2, 7, 2, 2, 0, 4, 2, 0, 3, 0, 6099, 2, 2, 3, 0, 7, 5871, 2, 3, 0, 2, 2, 0, 3, 0, 3, 2, 4, 7, 0, 2, 295, 2, 2, 0, 2, 3, 0, 2, 0, 32587, 2, 4, 11, 0, 2, 2959, 2, 2, 0, 102, 3, 0, 3, 0, 359, 7, 472, 3, 0, 2, 3, 20, 2, 0, 3, 6, 0, 2, 0, 19, 4, 2, 3, 0, 2, 11, 2, 22, 0, 4, 2, 0, 2, 0, 19, 2, 2, (>1280000 or 0), 0, 3, 875, 30, 2

Length of the smallest prime of the form 4*b^n-1 (form: 3{x}, x = b-1) in base b for b = 2, 3, 4, ..., 160: (the b=2, 3 cases are not acceptable, since there is no digit 3 in base 2, 3)

2, 2, 0, 2, 2, 0, 2, 0, 0, 2, 2, 0, 0, 2, 0, 2, 2, 0, 2, 2, 0, 6, 0, 0, 2, 2, 0, 0, 4, 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 2, 2, 0, 0, 2, 0, 1556, 2, 0, 2, 4, 0, 2, 0, 0, 2, 2, 0, 0, 2, 0, 10, 2, 0, 10, 2, 0, 2, 0, 0, 2, 1119850, 0, 0, 6, 0, 2, 2, 0, 8, 0, 0, 2, 0, 0, 6, 2, 0, 0, 2, 0, 2, 42, 0, 2, 2, 0, 4, 0, 0, 4, 14, 0, 0, 2, 0, 252, 2, 0, 2, 2, 0, 4, 0, 0, 2, 2, 0, 0, 2, 0, 2, 2, 0, 2, 2, 0, 16, 0, 0, 2, 4, 0, 0, 6, 0, 2, 14, 0, 6, 2, 0, 2, 0, 0, 6, 2, 0, 0, 2, 0, 2, 4, 0, 2, 4, 0, 2, 0, 0

Length of the smallest prime of the form b^n+4 (form: 1{0}4) in base b for b = 2, 3, 4, ..., 160: (the b=3 case is not acceptable, since there is no digit 4 in base 3)

0, 2, 0, 3, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 3, 0, 2, 0, 0, 0, 7, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 3, 0, 2, 0, 2, 0, 0, 0, 2, 0, 3, 0, 3, 0, 2, 0, 0, 0, 13403, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 3, 0, 2, 0, 2, 0, 0, 0, 3, 0, 2, 0, 47, 0, 2, 0, 0, 0, 83, 0, 2, 0, 3, 0, 0, 0, 0, 0, 2, 0, 3, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 7, 0, 2, 0, 0, 0, 10647, 0, 3, 0, 3, 0, 0, 0, 0, 0, 2, 0, 3, 0, 2, 0, 4, 0, 0, 0, 2, 0, 2, 0, 3, 0, (>25000 or 0), 0, 0, 0, 71, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 3, 0, 214, 0, 2, 0

Length of the smallest prime of the form b^n-4 (form: {x}y, x = b-1, y = b-4) in base b for b = 2, 3, 4, ..., 160: (n>=2 is required, since single-digit primes are not acceptable in this puzzle) (the b=3 case is not acceptable, since there is no digit 3-4 = -1 in base 3)

0, 2, 0, 5, 0, 0, 0, 0, 0, 3, 0, 0, 0, 3, 0, 3, 0, 0, 0, 3, 0, 3, 0, 0, 0, 7, 0, 0, 0, 0, 0, 3, 0, 13, 0, 0, 0, 0, 0, 3, 0, 0, 0, 3, 0, 65, 0, 0, 0, 3, 0, 3, 0, 0, 0, 3, 0, 0, 0, 0, 0, 3, 0, 175, 0, 0, 0, 0, 0, 45, 0, 0, 0, 13, 0, 3, 0, 0, 0, 0, 0, 3, 0, 0, 0, 29, 0, 0, 0, 0, 0, 105, 0, 45, 0, 0, 0, 0, 0, 3, 0, 0, 0, 3, 0, 7, 0, 0, 0, 13, 0, 13, 0, 0, 0, 3, 0, 0, 0, 0, 0, 299, 0, 5, 0, 0, 0, 0, 0, 3, 0, 0, 0, 165, 0, 147, 0, 0, 0, 395, 0, 23, 0, 0, 0, 3, 0, 0, 0, 0, 0, 7, 0, 3, 0, 0, 0, 0, 0

[sweety439]

Large minimal prime (start with b+1) for bases b>16 not in the list for bases 2 to 30 or list for bases 28 to 50 (because they contain single-digit primes, or because they are too large (length > 10000), primes with the latter case but not the former case are already minimal even if single-digit primes are included, and they are marked by "**") given by: (using A−Z to represent digit values 10 to 35, a−z to represent digit values 36 to 61)

Base 18: 80₂₉₈B
Base 24: 20₃₁₃7 (note: F1₉₅₇ is not minimal prime (start with b+1), since its repeating digit is 1)
Base 30: OT₃₄₂₀₅ (found by CRUS generalized Riesel conjecture base 30)
Base 33: 130₂₃₆₁₄1 (found by CRUS generalized Sierpinski conjecture base 33)
**Base 36: P₈₁₉₉₃SZ (already minimal even if single-digit primes are included, but not in the list since this prime is too large) (found by me)
**Base 37: FYa₂₂₀₂₁ (already minimal even if single-digit primes are included, but not in the list since this prime is too large) (found by CRUS generalized Riesel conjecture base 37)
Base 38: ab₁₃₆₂₁₁ (found by Williams primes search)
**Base 40: QaU₁₂₃₈₀X (already minimal even if single-digit primes are included, but not in the list since this prime is too large) (found by me)
Base 42: 2f₂₅₂₃ (found by CRUS generalized Riesel conjecture base 42)
**Base 45: O0₁₈₅₂₁1 (already minimal even if single-digit primes are included, but not in the list since this prime is too large) (found by CRUS generalized Sierpinski conjecture base 45)
Base 48: T0₁₃₃₀₄₁1 (found by CRUS generalized Sierpinski conjecture base 48)
**Base 49: SLm₅₂₆₉₈ (already minimal even if single-digit primes are included, but not in the list since this prime is too large) (found by CRUS generalized Riesel conjecture base 49)

Also see post https://mersenneforum.org/showpost.p...7&postcount=43

[sweety439]

For more lengths for the smallest prime of the form k*b^n+1 (form: k{0}1) or k*b^n-1 (form: x{y}, x = k-1, y = b-1) with k<b in base b, see CRUS

Note: This project (to solve this puzzle) requires k<b, but CRUS requires k<CK, since the CK value may be either >b or <b, thus this is neither necessary nor sufficient, but they have many intersection, e.g. 8*23^n+1 (8{0}1 in base 23), 25*30^n-1 (O{T} in base 30), 2*38^n+1 (2{0}1 in base 38), 3*42^n-1 (2{f} in base 42), 36*48^n+1 (a{0}1 in base 48), 4*53^n+1 (4{0}1 in base 53), etc.

[sweety439]

Just let you know, I know the set of the minimal primes (start with b+1) <=2^32 for all bases 2<=b<=128, and I know exactly which bases 2<=b<=1024 have these families as unsolved families (at length 25K) for the minimal primes (start with b+1) problem: (also, I know exactly which bases 2<=b<=1024 where these families are ruled out as only contain composites (except the numbers <=b)

(using A−Z to represent digit values 10 to 35, z−a to represent digit values b−1 to b−26)

(if such forms are interpretable in the bases, e.g. "C" (means 12 (twelve)) is only interpretable in bases b>=13, and "u" (means b−6) is only interpretable in bases b>=6 (if "u" appears as the first digit, then it is only interpretable in bases b>=7, since numbers cannot have leading zeros)

* {1}
* 1{0}1
* 1{0}2
* 1{0}3
* 1{0}4
* 1{0}5
* 1{0}6
* 1{0}7
* 1{0}8
* 1{0}9
* 1{0}A
* 1{0}B
* 1{0}C
* 1{0}D
* 1{0}E
* 1{0}F
* 1{0}G
* 1{0}z
* 2{0}1
* 2{0}3
* 3{0}1
* 3{0}2
* 3{0}4
* 4{0}1
* 4{0}3
* 5{0}1
* 6{0}1
* 7{0}1
* 8{0}1
* 9{0}1
* A{0}1
* B{0}1
* C{0}1
* D{0}1
* E{0}1
* F{0}1
* G{0}1
* z{0}1
* 1{2}
* 1{3}
* 1{4}
* 1{5}
* 1{6}
* 1{7}
* 1{8}
* 1{9}
* 1{A}
* 1{B}
* 1{C}
* 1{D}
* 1{E}
* 1{F}
* 1{G}
* 1{#} (for odd base b, # = (b−1)/2)
* {2}1
* {3}1
* {4}1
* {5}1
* {6}1
* {7}1
* {8}1
* {9}1
* {A}1
* {B}1
* {C}1
* {D}1
* {E}1
* {F}1
* {G}1
* {#}1 (for odd base b, # = (b−1)/2)
* 1{z}
* 2{z}
* 3{z}
* 4{z}
* 5{z}
* 6{z}
* 7{z}
* 8{z}
* 9{z}
* A{z}
* B{z}
* C{z}
* D{z}
* E{z}
* F{z}
* y{z}
* {#}$ (for odd base b, # = (b−1)/2, $ = (b+1)/2)
* ${#} (for odd base b, # = (b−1)/2, $ = (b+1)/2)
* {y}z
* {z}1
* {z}k
* {z}l
* {z}m
* {z}n
* {z}o
* {z}p
* {z}q
* {z}r
* {z}s
* {z}t
* {z}u
* {z}v
* {z}w
* {z}x
* {z}y

Also families where the smallest prime may not be minimal prime (start with b+1):

* 1{0}11 (not minimal prime (start with b+1) if there is smaller prime of the form 1{0}1)
* 11{0}1 (not minimal prime (start with b+1) if there is smaller prime of the form 1{0}1)
* 1{0}21 (not minimal prime (start with b+1) if either 21 (2*b+1) is prime or there is smaller prime of the form 1{0}1 or 1{0}2)
* 12{0}1 (not minimal prime (start with b+1) if either 12 (b+2) is prime or there is smaller prime of the form 1{0}1 or 2{0}1)
* {1}01 (not minimal prime (start with b+1) if there is smaller prime of the form {1})
* 10{1} (not minimal prime (start with b+1) if there is smaller prime of the form {1})
* {1}2 (not minimal prime (start with b+1) if there is smaller prime of the form {1})
* {1}3 (not minimal prime (start with b+1) if there is smaller prime of the form {1})
* {1}4 (not minimal prime (start with b+1) if there is smaller prime of the form {1})
* {1}z (not minimal prime (start with b+1) if there is smaller prime of the form {1})
* 2{1} (not minimal prime (start with b+1) if there is smaller prime of the form {1})
* 3{1} (not minimal prime (start with b+1) if there is smaller prime of the form {1})
* 4{1} (not minimal prime (start with b+1) if there is smaller prime of the form {1})
* z{1} (not minimal prime (start with b+1) if there is smaller prime of the form {1})
* {1}0z (not minimal prime (start with b+1) if there is smaller prime of the form {1} or {1}z)
* 10{z} (not minimal prime (start with b+1) if there is smaller prime of the form 1{z})
* 11{z} (not minimal prime (start with b+1) if either 11 (b+1) is prime or there is smaller prime of the form 1{z})
* {z}01 (not minimal prime (start with b+1) if there is smaller prime of the form {z}1)
* zy{z} (not minimal prime (start with b+1) if there is smaller prime of the form y{z})
* {z}yz (not minimal prime (start with b+1) if there is smaller prime of the form {z}y)
* {z0}z1 (almost cannot be minimal prime (start with b+1), since this is not simple family)

(in fact, there are no bases b<=1024 such that 7{0}1 is unsolved family, base 1004 is the last to drop at 54849 digits)

[sweety439]

pdf file in post #131 and text file in post #133 updated.

[sweety439]

References of the bases 2<=b<=1024 which have these unsolved families:

{1}:

https://web.archive.org/web/20021111...ds/primes.html
https://docs.google.com/document/d/e...Fp42iv53MF/pub
https://raw.githubusercontent.com/xa...iesel%20k1.txt
https://mersenneforum.org/attachment...1&d=1597771406
http://www.fermatquotient.com/PrimSerien/GenRepu.txt
http://www.primenumbers.net/Henri/us/MersFermus.htm

1{0}1:

http://jeppesn.dk/generalized-fermat.html
http://www.noprimeleftbehind.net/crus/GFN-primes.htm
http://yves.gallot.pagesperso-orange...s/results.html

2{0}1, 3{0}1, 4{0}1, 5{0}1, 6{0}1, 7{0}1, 8{0}1, 9{0}1, A{0}1, B{0}1, C{0}1:

https://www.rieselprime.de/ziki/Prot..._bases_least_n
https://mersenneforum.org/attachment...5&d=1598786719
https://mersenneforum.org/attachment...4&d=1606276304
https://mersenneforum.org/attachment...5&d=1606276304
http://www.prothsearch.com/fermat.html (2{0}1 in base 512, 4{0}1 in bases 32, 512, 1024, which are not in the first 4 references)
http://www.prothsearch.com/GFN10.html (A{0}1 in base 1000, which are not in the first 4 references)

1{z}, 2{z}, 3{z}, 4{z}, 5{z}, 6{z}, 7{z}, 8{z}, 9{z}, A{z}, B{z}:

https://www.rieselprime.de/ziki/Ries..._bases_least_n
https://mersenneforum.org/attachment...5&d=1598786719
https://mersenneforum.org/attachment...4&d=1606276304
https://mersenneforum.org/attachment...5&d=1606276304

z{0}1:

https://www.rieselprime.de/ziki/Williams_prime_MP_least
http://www.prothsearch.com/riesel1a.html (base 512)

y{z}:

https://harvey563.tripod.com/wills.txt
https://www.rieselprime.de/ziki/Williams_prime_MM_least

11{0}1: (not minimal prime (start with b+1) if there is smaller prime of the form 1{0}1)

https://www.rieselprime.de/ziki/Williams_prime_PP_least

10{z}: (not minimal prime (start with b+1) if there is smaller prime of the form 1{z})

https://www.rieselprime.de/ziki/Williams_prime_PM_least

{z}y:

https://www.primepuzzles.net/puzzles/puzz_887.htm

{#}$: (for odd base b, # = (b−1)/2, $ = (b+1)/2)

http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt
http://www.prothsearch.com/GFN05.html (base 625)

{z0}z1: (almost cannot be minimal prime (start with b+1), since this is not simple family)

http://www.fermatquotient.com/PrimSerien/GenRepuP.txt
https://cs.uwaterloo.ca/journals/JIS...NER/dubner.pdf
http://www.primenumbers.net/Henri/us/MersFermus.htm

Other references:

https://oeis.org/
http://www.primenumbers.net/prptop/prptop.php
https://primes.utm.edu/primes/
https://www.rose-hulman.edu/~rickert/Compositeseq/
http://www.noprimeleftbehind.net/crus/
https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf

[sweety439]

I tried to write a PARI/GP code that can print all minimal primes (start with b+1) up to length 1000 in given base b in <10 minutes, but not success, since the code for updating L (when L contains non-simple families) by "let w be the shortest sting in this family, if w has a subword in M, then remove the family from L, if w represents a prime, then add w to M, if the family can be proven to only contain composites, then remove the family from L" (see page of https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf) is very complex.

Thus, I only have the program that looks the small primes one-by-one, and I only checked the simple families of the from x{y}, x{y}, and x{0}y (where x,y are digits) to find the smallest primes in these simple families (or to prove that these simple families only contain composites).

The more difficult case is: Non-simple families that can be proven to only contain composites, if the gcd (greatest common divisor) of the digits in these families is >1, then these families clearly only contain composites (note: we only count the numbers > base), but there exist many non-simple families with gcd of the digits = 1 and can be proven to only contain composites (and all subsequences of all numbers in these families represent composites, when we only count the numbers > base), e.g.

{1}6{1} in base 9
{3}{0}5 in base 9
{3}{6}8 in base 9

(base 9 is the first base which has such families)

Interestingly, base 9 is also the first base with some simple families x{y} or {x}y which are ruled out as only contain composites by 2-cover (i.e. full numerical covering set with only two prime factors), since base b has such families x{y} or {x}y if and only if b+1 is not prime or prime power, and gcd(repeating digit, b+1) = 1, the first base b such that b+1 is not prime or prime power is 5, but for base 5, the only such families are 3{1}, 4{1}, {1}3, {1}4, but the smallest prime in the family whose repeating digit is 1 may not be minimal prime (start with b+1), unless base b has no repunit primes (the first such bases b are 9, 25, 32, 49, 64, ...), and base 5 has repunit prime 111 (=31 in base 10), thus base 5 has no simple families x{y} or {x}y which are ruled out as only contain composites by 2-cover (i.e. full numerical covering set with only two prime factors), and next base b such that b+1 is not prime or prime power is 9, and base 9 has these simple families: 2{7}, 5{1}, 5{7}, 6{1}, {7}2, {1}5, {3}5, {7}5, {3}8, which are ruled out as only contain composites by covering set {2,5} (also the families 5{3}, 8{3}, {1}6, but they are already ruled out as only contain composites by trivial 1-cover set {3})

Since in any base b, for a repdigit (a number whose all digits are all same) to be prime (only count numbers > base), it must be a repunit and have a prime number of digits in its base (b), and for the simple families x{y} and {x}y in base b, the only chance of their smallest primes (if exist) are not minimal primes (start with b+1) in base b is the base b repdigit is prime, thus the repeating digit in these families must be 1, and since in bases 9, 25, 32, 49, 64, ... there are no repunit primes, thus in these bases, the smallest primes (if exist) in all simple families x{y} and {x}y are always minimal primes (start with b+1) in base b

[sweety439]

Except the families *{0}1 and *{z} (where * represents any strings of digits), when the corresponding prime is large, the known primality tests for such a number are too inefficient to run (*{0}1 can be proven prime by N-1 primality test, *{z} can be proven prime by N+1 primality test). In this case one must resort to a probable primality test such as a Miller–Rabin primality test and Baillie–PSW primality test, unless a divisor of the number can be found (trial division).

[sweety439]

If the Sierpinski/Riesel CK for base b is <b (see http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm for the list of the CK for bases 2<=b<=1030), then the "minimal primes (start with b+1) base b problem" covers the Sierpinski base b problem and the Riesel base b problem, since all primes for the Sierpinski base b problem and the Riesel base b problem are minimal primes (start with b+1) base b