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

[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.

In fact,

* the smallest prime in the 5{5}0025 family is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, which can be written as 5₁₈₃025 and equal the prime (5*8^187-20333)/7, but this prime is not minimal prime.

* the smallest prime in the 5{5}05025 family is 5555555555555555555555505025, but this prime is not minimal prime.

[sweety439]

Originally Posted by sweety439

* Case (4,1):

** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes)

*** All minimal primes in the family 4{1,4,6}1 are 4611, 444641, 444444441, see https://scholar.colorado.edu/downloads/hh63sw661 (the base 8 section)

* Case (4,3):

** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes)

*** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes)

**** However, all numbers of the form 4{0}3 are divisible by 7, and all numbers of the form 44{0}3 are divisible by 3, thus cannot be prime, therefore, there is no minimal primes in this family.

* Case (4,5):

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

* Case (4,7):

** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes)

*** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes)

**** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as 4₂₂₀7 and equal the prime (2^665+17)/7

**** The smallest prime of the form 4{4}77 is 4444477

**** The smallest prime of the form 4{7}7 is 47777

**** For the 44{7}7 family, since 47777 is prime, we only need to consider the numbers 447, 4477, 44777

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

In fact,

* the smallest prime in the 44{7}7 family is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, which can be written as 447₈₅₁ and equal the prime 37*8^851-1, but this prime is not minimal prime.

[sweety439]

Originally Posted by sweety439

* 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)

*** However, all numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime, therefore, there is no minimal primes in this family.

* 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)

**** 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)

***** However, none of 6017, 6047, 6077 are primes, and all numbers of the form 60{4}7 are divisible by 21 (octal 21, decimal 17), therefore, there is no minimal primes in this family.

**** For the 6{0}17 family, since 6007 is prime, we only need to consider the number 6017 (since any digit combo 00 between (6,17) will produce smaller primes)

***** However, 6017 is not prime, therefore, there is no minimal primes in this family.

**** 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)

***** However, all numbers of the form 6{4}7 are divisible by 3, 5, or 15 (octal 15, decimal 13), and all numbers of the form 60{4}7 are divisible by 21 (octal 21, decimal 17), therefore, there is no minimal primes in this family.

**** For the 6{0}77 family, since 6007 is prime, we only need to consider the number 6077 (since any digit combo 00 between (6,77) will produce smaller primes)

***** However, 6077 is not prime, therefore, there is no minimal primes in this family.

* Case (7,1):

** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes)

*** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes)

**** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes)

***** The smallest prime of the form 7{7}1 is 7777777777771

***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741 (not minimal prime, since 7777777777771 is prime)

***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441 (not minimal prime, since 7777777777771 is prime)

***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441 (not minimal prime, since 7777777777771 is prime)

***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441 (not minimal prime, since 7777777777771 is prime)

***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime.

***** The smallest prime of the form 7{7}4444441 is 77774444441

****** Since this prime has just 4 7's, we only need to consider the families with <=3 7's

******* The smallest prime of the form 7{4}1 is 744444441

******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime.

******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes)

[sweety439]

* Case (7,3):

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

* Case (7,5):

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

* Case (7,7):

** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes)

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

[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)

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)

extended data to base 36

search the simple families x{0}y with gcd(x,y) = 1, gcd(y,b) = 1, gcd(x+y,b-1) = 1

Such primes are ALWAYS minimal prime (start with 2 digits) in base b, except when the base (b) is prime, and x = 1 (while 10 is prime and a subsequence of the prime, but with LaurV's suggestion, the prime 10 (i.e. the prime = base) is also not counted just as the primes < base, all such primes (i.e. all smallest primes of the form x{0}y) is ALWAYS minimal prime (start with b+1) in base b)

[sweety439]

Base b minimal primes (start with 2 digits) includes:

* The smallest repunit prime base b if exists
* The smallest generalized Fermat prime base b for even b if exists
* The smallest generalized half Fermat prime (> (b+1)/2) base b for odd b if exists
* The smallest Williams prime with 1st kind base b if exists
* The smallest Williams prime with 2nd kind base b if exists
* The smallest Williams prime with 4th kind base b for bases b which no generalized Fermat primes exist (this includes all odd bases) and b is not prime (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded) if exists
* The smallest dual Williams prime with 1st kind base b if exists
* The smallest dual Williams prime with 2nd kind base b for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded) if exists
* The smallest dual Williams prime with 4th kind base b for bases b which no generalized Fermat primes exist (this includes all odd bases) and b is not prime (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded) if exists
* The smallest prime of the form 2*b^n+1 for bases b>2 if exists
* The smallest prime of the form 2*b^n-1 for bases b>2 if exists
* The smallest prime of the form b^n+2 for bases b>2 with gcd(b,2)=1 for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded) if exists
* The smallest prime of the form b^n-2 for bases b>2 with gcd(b,2)=1 if exists
* The smallest prime of the form 3*b^n+1 for bases b>3 if exists
* The smallest prime of the form 3*b^n-1 for bases b>3 if exists
* The smallest prime of the form b^n+3 for bases b>3 with gcd(b,3)=1 for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded) if exists
* The smallest prime of the form b^n-3 for bases b>3 with gcd(b,3)=1 if exists
* The smallest prime of the form 4*b^n+1 for bases b>4 if exists
* The smallest prime of the form 4*b^n-1 for bases b>4 if exists
* The smallest prime of the form b^n+4 for bases b>4 with gcd(b,4)=1 for composite bases b (this condition is not needed if as LaurV's suggestion, the prime 10 (=b) is also excluded) if exists
* The smallest prime of the form b^n-4 for bases b>4 with gcd(b,4)=1 if exists
...
* The smallest prime of the form k*b^n+1 for fixed 1<=k<=b-1 (i.e. the prime for the CRUS Sierpinski conjecture for fixed 1<=k<=b-1) if exists
* The smallest prime of the form k*b^n-1 for fixed 1<=k<=b-1 (i.e. the prime for the CRUS Riesel conjecture for fixed 1<=k<=b-1) if exists
* The smallest prime of the form b^n+k for fixed 1<=k<=b-1 if exists
* The smallest prime of the form b^n-k for fixed 1<=k<=b-1 if exists
* The smallest prime of the form (k*b^n-1)/gcd(k-1,b-1) for fixed k with 0<=(k-1)/gcd(k-1,b-1)<=b-1 and gcd(k-1,b-1) < b-1 (this reason is because if the repeating digit is 1, then this prime may not be minimal prime (start with 2 digits), unless there are no repunit primes base b (e.g. b = 9, 25, 32, 49, 64, 81, ...) (i.e. the prime for the extended Riesel conjecture for fixed k satisfying these two conditions) if exists
* The smallest prime of the form (b^n-k)/gcd(k-1,b-1) for fixed k with gcd(b,k) = 1 and 0<=k<=b-1

(see post #140 for references of these families)

[sweety439]

Originally Posted by LaurV

I found an easy way to generate those sets, and to prove that they are complete.

For the "starting from two digits" version, neither one of the exposed sets for 7 and 8 are complete. Some larger primes are still lurking in the dark there. I have the complete sets for both 8, and 7 for the both cases when the base itself is included in the set or not*, but I don't want to spoil the puzzle, this is an interesting little problem... hehe...

Hint:

gp > a=(7^17-5)/2
%1 = 116315256993601
gp > isprime(a)
%2 = 1
gp > digits(a,7)
%3 = [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1]
gp >

---------
*when the base is prime, like for 5 and 7, the sets are different; including the base results in automatic elimination of all possible extension numbers with "0 after 1" from the set, which is quite restrictive, so I also calculated the lists for the "base is not included" version, i.e. base-5 starting from 6, and base-7 starting from 8; in this case, for example, base-5 will include numbers like 104 and 10103 which are prime, and base-7 list will include 1022, 1051, 1202, .... 1100021 ... etc, they are "enriched" compared with the case when the first "10" is included. So I have the complete list for 8, and the complete two lists for 7, the normal one, and the "enriched" one. Base-5 is easy, in any case.

If even the prime "10" (i.e. prime = the base (b)) is excluded, the the minimal primes will be: (only listed prime bases, since for composite bases the set of these primes is completely the same as the set of minimal primes with >=2 digits):

2: {11}
3: {12, 21, 111}
5: {12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013}
7: {14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, ..., 33333333333333331, ...}
11: {12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 10A, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, A74, A7A, A85, AA1, AA7, 1101, 11A9, 1305, 1451, 1457, 15A7, 175A, 17A5, 17A9, 2023, 2045, 2052, 2083, 20A5, 2333, 2A05, 2A52, 3013, 3026, 3059, 3097, 3206, 3222, 3233, 3307, 3332, 3505, 4025, 4151, 4157, 4175, 4405, 4445, 4487, 450A, 4575, 5017, 5031, 5059, 5075, 5097, 5099, 5105, 515A, 517A, 520A, 5301, 5583, 5705, 577A, 5853, 5873, 5909, 5A17, 5A57, 5A77, 5A8A, 6683, 66A9, 7019, 7073, 7079, 7088, 7093, 7095, 7309, 7451, 7501, 7507, 7578, 757A, 75A7, 7787, 7804, 7844, 7848, 7853, 7877, 78A4, 7A04, 7A57, 7A79, 7A95, 8078, 8245, 8333, 8355, 8366, 8375, 8425, 8553, 8663, 8708, 8777, 878A, 8A05, 9053, 9101, 9107, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 10011, 10075, 10091, 10109, 10411, 10444, 10705, 10709, 10774, 10901, 11104, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A04, 14A11, 17045, 17704, 1774A, 17777, 177A4, 17A47, 1A091, 1A109, 1A114, 1A404, 1A411, 1A709, 20005, 20555, 22203, 25228, 25282, 25552, 25822, 28522, 30037, 30701, 30707, 31113, 33777, 35009, 35757, 39997, 40045, 4041A, 40441, 4045A, 404A1, 4111A, 411A1, 42005, 44401, 44474, 444A1, 44555, 44577, 445AA, 44744, 44A01, 47471, 47477, 47701, 5057A, 50903, 5228A, 52A22, 52A55, 52A82, 55007, 550A9, 55205, 55522, 55557, 55593, 55805, 57007, 57573, 57773, 57807, 5822A, 58307, 58505, 58A22, 59773, 59917, 59973, 59977, 59999, 5A015, 5A2A2, 5AA99, 60836, 60863, 68636, 6A609, 6A669, 6A696, 6A906, 6A966, 70048, 70103, 70471, 70583, 70714, 71474, 717A4, 71A09, 74084, 74444, 74448, 74477, 744A8, 74747, 74774, 7488A, 74A48, 75773, 77144, 77401, 77447, 77799, 77A09, 78008, 78783, 7884A, 78888, 788A8, 79939, 79993, 79999, 7A051, 7A444, 7A471, 80005, 80252, 80405, 80522, 80757, 80AA5, 83002, 84045, 85307, 86883, 88863, 8A788, 90073, 90707, 90901, 95003, 97779, 97939, 99111, 99177, 99973, A0111, A0669, A0966, A0999, A0A09, A1404, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 100019, 100079, 101113, 101119, 101911, 107003, 140004, 144011, 144404, 1A0019, 1A0141, 1A5001, 1A7005, 1A9001, 222223, 222823, 300107, 300202, 300323, 303203, 307577, 310007, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 41A011, 440A41, 441011, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 531707, 550777, 553707, 5555A9, 555A99, 557707, 55A559, 5807A7, 580A0A, 580A55, 58A0AA, 590007, 599907, 5A2228, 5A2822, 5A2AAA, 5A552A, 5AA22A, 5AAA22, 60A069, 683006, 6A0096, 6A0A96, 6A9099, 6A9909, 700778, 701074, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 710777, 717044, 717077, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A1009, 7A4408, 7A7708, 80A555, 828283, 828883, 840555, 850505, 868306, 873005, 883202, 900701, 909739, 909979, 909991, 970771, 977701, 979909, 990739, 990777, 990793, 997099, 999709, 999901, A00009, A00599, A01901, A05509, A0A058, A0A955, A10114, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 1000501, 1011141, 1030007, 1070047, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1700005, 1700474, 1A44444, ...}

[sweety439]

See posts https://mersenneforum.org/showpost.p...&postcount=306, https://mersenneforum.org/showpost.p...&postcount=325, https://mersenneforum.org/showpost.p...&postcount=326 for the proof for base 5 (when single-digit primes are excluded but 10 (i.e. base) is included)

If 10 (i.e. base) is excluded, then for the primes containing 10:

any digits before 10 cannot be 2 (because of 21)

any digits after 10 cannot be 2 (because of 12)

And we have the prime 104, and for other prime numbers, any digits after 10 cannot be 4

[sweety439]

Originally Posted by sweety439

search the simple families x{0}y with gcd(x,y) = 1, gcd(y,b) = 1, gcd(x+y,b-1) = 1

Such primes are ALWAYS minimal prime (start with 2 digits) in base b, except when the base (b) is prime, and x = 1 (while 10 is prime and a subsequence of the prime, but with LaurV's suggestion, the prime 10 (i.e. the prime = base) is also not counted just as the primes > base, all such primes (i.e. all smallest primes of the form x{0}y) is ALWAYS minimal prime (start with b+1) in base b)

Update the file for the smallest primes in these families for bases up to 36

[sweety439]

Minimal set of prime-strings with ≥2 digits in bases 2 to 12 (only bases 2 to 8 are proved to be complete)

2: {10, 11}

3: {10, 12, 21, 111}

4: {11, 13, 23, 31, 221}

5: {10, 12, 21, 23, 32, 34, 43, 111, 131, 133, 313, 401, 414, 14444, 30301, 33001, 33331, 44441, 300031}

6: {11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041}

7: {10, 14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1112, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 30011, 31111, 33001, 33311, 35555, 40054, 300053, 33333301, 33333333333333331}

8: {13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447}

9: {12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, ..., 300000000035, ..., 544444444444, ..., 2000000000007, ..., 56111111111111111111111111111111111111, ..., 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, ...}

10: {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, ..., 555555555551, ..., 5000000000000000000000000000027, ...}

11: {10, 12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, A74, A7A, A85, AA1, AA7, 11A9, 1451, 1457, 15A7, 175A, 17A5, 17A9, 2023, 2045, 2052, 2083, 20A5, 2333, 2A05, 2A52, 3013, 3026, 3059, 3097, 3206, 3222, 3233, 3307, 3332, 3505, 4025, 4151, 4157, 4175, 4405, 4445, 4487, 450A, 4575, 5017, 5031, 5059, 5075, 5097, 5099, 515A, 517A, 520A, 5301, 5583, 5705, 577A, 5853, 5873, 5909, 5A17, 5A57, 5A77, 5A8A, 6683, 66A9, 7019, 7073, 7079, 7088, 7093, 7095, 7309, 7451, 7501, 7507, 7578, 757A, 75A7, 7787, 7804, 7844, 7848, 7853, 7877, 78A4, 7A04, 7A57, 7A79, 7A95, 8078, 8245, 8333, 8355, 8366, 8375, 8425, 8553, 8663, 8708, 8777, 878A, 8A05, 9053, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A11, 1774A, 17777, 177A4, 17A47, 1A114, 1A411, 20005, 20555, 22203, 25228, 25282, 25552, 25822, 28522, 30037, 30701, 30707, 31113, 33777, 35009, 35757, 39997, 40045, 4041A, 40441, 4045A, 404A1, 4111A, 411A1, 42005, 44401, 44474, 444A1, 44555, 44577, 445AA, 44744, 44A01, 47471, 47477, 47701, 5057A, 50903, 5228A, 52A22, 52A55, 52A82, 55007, 550A9, 55205, 55522, 55557, 55593, 55805, 57007, 57573, 57773, 57807, 5822A, 58307, 58505, 58A22, 59773, 59917, 59973, 59977, 59999, 5A015, 5A2A2, 5AA99, 60836, 60863, 68636, 6A609, 6A669, 6A696, 6A906, 6A966, 70048, 70471, 70583, 70714, 71474, 717A4, 74084, 74444, 74448, 74477, 744A8, 74747, 74774, 7488A, 74A48, 75773, 77144, 77401, 77447, 77799, 77A09, 78008, 78783, 7884A, 78888, 788A8, 79939, 79993, 79999, 7A051, 7A444, 7A471, 80005, 80252, 80405, 80522, 80757, 80AA5, 83002, 84045, 85307, 86883, 88863, 8A788, 90073, 90707, 90901, 95003, 97779, 97939, 99111, 99177, 99973, A0111, A0669, A0966, A0999, A0A09, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 222223, 222823, 300202, 300323, 303203, 307577, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 440A41, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 550777, 553707, 5555A9, 555A99, 557707, 55A559, 5807A7, 580A0A, 580A55, 58A0AA, 590007, 599907, 5A2228, 5A2822, 5A2AAA, 5A552A, 5AA22A, 5AAA22, 60A069, 683006, 6A0096, 6A0A96, 6A9099, 6A9909, 700778, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A4408, 7A7708, 80A555, 828283, 828883, 840555, 850505, 868306, 873005, 883202, 900701, 909739, 909979, 909991, 970771, 977701, 979909, 990739, 990777, 990793, 997099, 999709, 999901, A00009, A00599, A05509, A0A058, A0A955, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1A44444, 2555505, 2845055, 3030023, 3333397, 4000111, 4011111, 41A1111, 4411111, 444441A, 4444771, 4470004, 4505005, 4744417, 4774441, 4777404, 4777417, 4777747, 4A11111, 4A40001, 5000093, 50005A7, 5005777, 5050553, 5055503, 5070777, 5222222, 5222AAA, 52AAAA2, 52AAAAA, 5505053, 5552AAA, 5555599, 5555A58, 5558A0A, 5558A55, 5558AAA, 55A0009, 55AAA52, 580000A, 5822222, 58AAAAA, 5A2222A, 5AA2222, 6000A69, 6000A96, 6A00069, 7000417, 7000741, 7000835, 7000857, 7007177, 7008305, 7014447, 7017444, 7074177, 7077477, 7077741, 7077747, 717444A, 7400404, 7700717, 7707778, 7774004, 777741A, 7777441, 777774A, 7777A47, 7779003, 777A008, 777A778, 777A808, 77A4777, 7900399, 8305007, 8500707, 8555707, 8883022, 8AA5222, 9000035, 9007999, 9009717, 9009777, 9009997, 9090997, 9099907, 9355555, 9790099, 9900991, 9900997, 9907909, 9909079, 9979009, 9990079, 9990091, 9990907, 9999771, 9999799, 9999979, A000696, A000991, A006906, A040041, A0AAA58, A141111, A5222A2, A600A69, A906606, A909009, A990009, A997701, AA55A52, AAA5552, AAAAA52, 22888823, 28888223, 30555777, 33000023, 40A00041, 45000055, 47040004, 50377777, 50555553, 5282AAA2, 55505003, 555A5A52, 555AAA2A, 55A5A552, 5AAAAA2A, 60A99999, 70000057, 70070474, 70074704, 70700078, 70700474, 70704704, 70777177, 74470001, 77000177, 77070477, 77470004, 77700404, 77748808, 7774A888, 77770078, 77770474, 77774704, 77777008, 77777404, 77777778, 80555055, 88828823, 88888326, 88888823, 8A522222, 90097909, 90700999, 90977777, 97000001, 97000717, 97770007, 99000001, 99000771, 99077001, 99090097, 99777707, 99900097, 99970717, 99999097, 99999707, A0000058, A0004041, A00055A9, A000A559, A5555009, A5A55552, A9700001, A9909006, A9990006, A9990606, A9999917, A9999966, 300577777, 40000A401, 447771777, 44A444441, 474000004, 477700004, 477777774, 505000003, 55555AA2A, 5555A5A2A, 700000147, 700044004, 700077774, 704000044, 704040004, 707070774, 707077704, 707770704, 707777004, 770000078, 770004704, 770070747, 770070774, 770700008, 770700084, 770707074, 777000044, 777000774, 777770477, 777770747, 7777777A4, 77A700008, 888888302, 900000091, 900090799, 970009099, 990990007, 997000077, 999999997, A0000AA58, A00990001, A05555559, A44444111, A44444777, A44477777, A66666669, A90000606, A99999006, A99999099, ..., 600000A999, ..., A000144444, ..., A900000066, ..., 33333333337, ..., 44444444447, ..., A0000000001, ..., A0014444444, ..., 40000000A0041, ..., A000000014444, ..., A044444444441, ..., A144444444411, ..., 40000000000401, ..., A0000044444441, ..., A00000000444441, ..., 11111111111111111, ..., 14444444444441111, ..., 44444444444444111, ..., A1444444444444444, ..., A9999999999999996, ..., 888888888888888883, ..., 1444444444444444444, ..., 7777777777777777771, ..., 4000000000000000A041, ..., A999999999999999999999, ..., A44444444444444444444444441, ..., 40000000000000000000000000041, ..., 440000000000000000000000000001, ..., 999999999999999999999999999999991, ..., 444444444444444444444444444444444444444444441, ...}

12: {11, 15, 17, 1B, 25, 27, 31, 35, 37, 3B, 45, 4B, 51, 57, 5B, 61, 67, 6B, 75, 81, 85, 87, 8B, 91, 95, A7, AB, B5, B7, 221, 241, 2A1, 2B1, 2BB, 401, 421, 447, 471, 497, 565, 655, 665, 701, 70B, 721, 747, 771, 77B, 797, 7A1, 7BB, 907, 90B, 9BB, A41, B21, B2B, 2001, 200B, 202B, 222B, 229B, 292B, 299B, 4441, 4707, 4777, 6A05, 6AA5, 729B, 7441, 7B41, 929B, 9777, 992B, 9947, 997B, 9997, A0A1, A201, A605, A6A5, AA65, B001, B0B1, BB01, BB41, 600A5, 7999B, 9999B, AAAA1, B04A1, B0B9B, BAA01, BAAA1, BB09B, BBBB1, 44AAA1, A00065, BBBAA1, AAA0001, B00099B, AA000001, ..., B0000000000000000000000000009B, ...}

[sweety439]

Originally Posted by sweety439

search the simple families x{0}y with gcd(x,y) = 1, gcd(y,b) = 1, gcd(x+y,b-1) = 1

Such primes are ALWAYS minimal prime (start with 2 digits) in base b, except when the base (b) is prime, and x = 1 (while 10 is prime and a subsequence of the prime, but with LaurV's suggestion, the prime 10 (i.e. the prime = base) is also not counted just as the primes < base, all such primes (i.e. all smallest primes of the form x{0}y) is ALWAYS minimal prime (start with b+1) in base b)

If as LaurV's suggestion, the prime 10 (i.e. the prime = base) is also not counted just as the primes < base, then the last digit of all primes in the set must be coprime to the base (since the last digit of all primes which do not divide the base are coprime to the base, and all primes > base do not divide the base), also, the first digit of all primes in the set (in fact, for all numbers) cannot be 0, thus, 0 can be neither the first digit nor the last digit, for the primes in this set, and 0 can only be just the middle digits, and the simple family x{0}y (where x,y are any nonzero digits in this base) ALWAYS need to test (unless this family is ruled out to only contain composites), like the simple families x{y} (y != 1) and {x}y (x != 1), which are also ALWAYS need to test (unless they are ruled out to only contain composites).