generalized minimal (probable) primes

[sweety439]

Originally Posted by sweety439

These bases are the bases <= 1024 which is not perfect odd power (of the form m^r with odd r>1) whose "minimal prime program" have GFN or half GFN remain, for the bases <= 1024 which is perfect odd power (of the form m^r with odd r>1):

* Cubes:

** Base 8: GFN in base 2 are either 2{0}1 or 4{0}1 in base 8, however, 2 and 401 are primes, thus, base 8 does not have GFN or half GFN remain.

** Base 27: half GFN in base 3 are either 1{D}E or 4{D}E in base 27, however, D is prime, thus, base 27 does not have GFN or half GFN remain.

** Base 64: GFN in base 2 are either 4{0}1 or G{0}1 in base 64, however, 41 and G01 are primes, thus, base 64 does not have GFN or half GFN remain.

** Base 125: half GFN in base 5 are either 2:{62}:63 or 12:{62}:63 in base 125, however, 2 is prime, but the family 12:{62}:63 does not have any known (probable) prime (the only known half GFN (probable) primes in base 5 are 3, 13, 2:63), thus, base 125 has half GFN remain.

** Base 216: GFN in base 6 are either 6:{0}:1 or 36:{0}:1 in base 216, however, 6:1 is prime, but the family 36:{0}:1 does not have any known prime (the only known GFN primes in base 6 are 7, 37, 6:1), thus, base 216 has GFN remain.

** Base 343: half GFN in base 7 are either 3:{171}:172 or 24:{171}:172 in base 343, however, 3 is prime, but the family 24:{171}:172 does not have any known (probable) prime (the only known half GFN (probable) prime in base 7 is 3:172), thus, base 343 has half GFN remain.

** Base 512: GFN in base 2 are 2:{0}:1, 4:{0}:1, 16:{0}:1, 32:{0}:1, 128:{0}:1, or 256:{0}:1 in base 512, however, 2 and 128:1 are primes, but the families 4:{0}:1, 16:{0}:1, 32:{0}:1, 256:{0}:1 do not have any known prime (the only known GFN primes in base 2 are 3, 5, 17, 257, 128:1), thus, base 512 has GFN remain.

** Base 729: half GFN in base 3 are either 4:{364}:365 or 40:{364}:365 in base 729, however, 40:364:365 and 4:364:364:364:364:365 are primes, thus, base 729 does not have GFN or half GFN remain.

** Base 1000: GFN in base 10 are either 10:{0}:1 or 100:{0}:1 in base 1000, and both families do not have any known prime (the only known GFN primes in base 10 are 11 and 101), thus, base 1000 has GFN remain.

* 5th powers:

** Base 32: GFN in base 2 are 2{0}1, 4{0}1, 8{0}1, or G{0}1 in base 32, however, 2 and 81 are primes, but the families 4{0}1 and G{0}1 do not have any known prime (the only known GFN primes in base 2 are 3, 5, H, 81, 2001), thus, base 32 has GFN remain.

** Base 243: half GFN in base 3 are 1:{121}:122, 4:{121}:122, 13:{121}:122, or 40:{121}:122 in base 243, however, 1:121:121:122, 4:121, 13, 40:121:121:121:121:121:121:121:121:121:121:121:122 are primes, thus, base 243 does not have GFN or half GFN remain.

** Base 1024: GFN in base 2 are 4:{0}:1, 16:{0}:1, 64:{0}:1, or 256:{0}:1 in base 1024, however, 64:1 is prime, but the families 4:{0}:1, 16:{0}:1, 256:{0}:1 do not have any known prime (the only known GFN primes in base 2 are 3, 5, 17, 257, 64:1), thus, base 1024 has GFN remain.

* 7th powers:

** Base 128: GFN in base 2 are 2:{0}:1, 4:{0}:1, or 16:{0}:1 in base 128, however, 2 and 4:0:1 are primes, but the family 16:{0}:1 do not have any known prime (the only known GFN primes in base 2 are 3, 5, 17, 2:1, 4:0:1), thus, base 128 has GFN remain.

The smallest generalized repunit prime in base b (if exists) is always minimal prime in base b, since it is 111...111 in base b

Thus, a given base b which is not perfect power (of the form m^r with r>1) whose "minimal prime program" have generalized repunit prime remain if and only if there are no known generalized repunit prime in base b, such bases <= 1024 (perfect powers excluded) are 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015

For the bases <= 1024 which is perfect power (of the form m^r with r>1):

* Squares:

** Base 4: GRU in base 2 are 1{3} in base 4, however, 3 is prime, thus, base 4 does not have GRU remain.

** Base 9: GRU in base 3 are 1{4} in base 9, however, 14 is prime, thus, base 9 does not have GRU remain.

** Base 16: GRU in base 2 are either 1{F} or 7{F} in base 16, however, 1F and 7 are primes, thus, base 16 does not have GRU remain.

** Base 25: GRU in base 5 are 1{6} in base 25, however, 16 is prime, thus, base 25 does not have GRU remain.

** Base 36: GRU in base 6 are 1{7} in base 36, however, 7 is prime, thus, base 36 does not have GRU remain.

** Base 49: GRU in base 7 are 1:{8} in base 49, however, 1:8:8 is prime, thus, base 49 does not have GRU remain.

** Base 64: GRU in base 2 are either 1:{63} or 31:{63} in base 64, however, 1:63 and 31 are primes, thus, base 64 does not have GRU remain.

** Base 81: GRU in base 3 are either 1:{40} or 13:{40} in base 81, however, 1:40:40:40 and 13 are primes, thus, base 81 does not have GRU remain.

** Base 100: GRU in base 10 are 1:{11} in base 100, however, 11 is prime, thus, base 100 does not have GRU remain.

** Base 121: GRU in base 11 are 1:{12} in base 121, however, 1:12:12:12:12:12:12:12:12 is prime, thus, base 121 does not have GRU remain.

** Base 144: GRU in base 12 are 1:{13} in base 144, however, 13 is prime, thus, base 144 does not have GRU remain.

...

** Base 1024: GRU in base 2 are 1:{1023}, 7:{1023}, 127:{1023}, or 511:{1023} in base 1024, however, 1:1023:1023:1023, 7, 127, 511:1023 are primes, thus, base 1024 does not have GRU remain.

* Cubes:

** Base 8: GRU in base 2 are either 1{7} or 3{7} in base 8, however, 7 is prime, thus, base 8 does not have GRU remain.

** Base 27: GRU in base 3 are either 1{D} or 4{D} in base 27, however, D is prime, thus, base 27 does not have GRU remain.

** Base 64: GRU in base 2 are either 1:{63} or 31:{63} in base 64, however, 1:63 and 31 are primes, thus, base 64 does not have GRU remain.

** Base 125: GRU in base 5 are either 1:{31} or 6:{31} in base 125, however, 31 is prime, thus, base 125 does not have GRU remain.

** Base 216: GRU in base 6 are either 1:{43} or 7:{43} in base 216, however, 43 is prime, thus, base 216 does not have GRU remain.

** Base 343: GRU in base 7 are either 1:{57} or 8:{57} in base 343, however, 1:57:57:57:57 and 8:57 are primes, thus, base 343 does not have GRU remain.

** Base 512: GRU in base 2 are 1:{511}, 3:{511}, 15:{511}, 31:{511}, 127:{511}, or 255:{511} in base 512, however, 1:511:511, 3, 15:511, 31, 127, 255:511 are primes, thus, base 512 does not have GRU remain.

** Base 729: GRU in base 3 are either 1:{364} or 121:{364} in base 729, however, 1:364 and 121:364:364:364:364:364:364:364:364:364:364:364 are primes, thus, base 729 does not have GRU remain.

** Base 1000: GRU in base 10 are either 1:{111} or 11:{111} in base 1000, however, 1:111:111:111:111:111:111 and 11 are primes, thus, base 1000 does not have GRU remain.

* 5th powers:

** Base 32: GRU in base 2 are 1{V}, 3{V}, 7{V}, or F{V} in base 32, however, V is prime, thus, base 32 does not have GRU remain.

** Base 243: GRU in base 3 are 1:{121}, 4:{121}, 13:{121}, or 40:{121} in base 243, however, 1:121:121:121:121:121:121:121:121:121:121:121:121:121:121, 4:121, 13 are primes, but the family 40:{121} does not have any known (probable) prime (there are no known numbers in OEIS A028491 which is == 4 mod 5), thus, base 243 has GRU remain.

** Base 1024: GRU in base 2 are 1:{1023}, 7:{1023}, 127:{1023}, or 511:{1023} in base 1024, however, 1:1023:1023:1023, 7, 127, 511:1023 are primes, thus, base 1024 does not have GRU remain.

* 7th powers:

** Base 128: GRU in base 2 are 1:{127}, 3:{127}, 7:{127}, 15:{127}, 31:{127}, or 63:{127} in base 128, however, 127 is prime, thus, base 128 does not have GRU remain.

[sweety439]

In Sierpinski problem base b, the prime for a k-value <b is "minimal prime base b" if and only if k is not prime.

In Riesel problem base b, the prime for a k-value <b is "minimal prime base b" if and only if neither k-1 nor b-1 is prime.

However, if we exclude the single-digit primes from the set (i.e. the minimal string of the set of prime numbers >= b in base b, see problem https://mersenneforum.org/showthread.php?t=24972), then the prime for Sierpinski/Riesel problems base b for a k-value <b is always "minimal prime base b", this is why the "minimal prime problem" for the prime numbers >= b in base b is more interesting, since single-digit primes are trivial, like that in Sierpinski/Riesel problems base b, n=0 is trivial, since the corresponding number is just k+1 or k-1, and thus CRUS requires n>=1, and of course the CRUS Sierpinski/Riesel problems (requiring n>=1) is much harder than the same problem which n=0 is allowed, similarly, finding the minimal set of the strings for primes in base b with at least two digits in base b is much harder than finding the minimal set of the strings for primes (including the single-digit primes in base b) in base b, e.g.

* In base 7, the largest minimal prime is 11111, but if single-digit primes are excluded, then a much-larger prime 33333333333333331 is minimal prime.

* In base 8, the largest minimal prime is 444444441, but if single-digit primes are excluded, then a much-larger prime 7777777777771 is minimal prime.

* In base 10, the largest minimal prime is 66600049, but if single-digit primes are excluded, then a much-larger prime 555555555551 is minimal prime.

* In base 14, the largest minimal prime is 40₈₃49, but if single-digit primes are excluded, then a much-larger prime 4D₁₉₆₉₈ is minimal prime.

* In base 17, there are only 2 unsolved families when searched to length 10000, but if single-digit primes are excluded, then a large prime 74₄₉₀₄ is minimal prime.

* In base 21, there are only 3 unsolved families when searched to length 10000, but if single-digit primes are excluded, then a large prime 5D0₁₉₈₄₈1 is minimal prime.

* In base 30, the largest minimal prime is C0₁₀₂₂1, but if single-digit primes are excluded, then a much-larger prime OT₃₄₂₀₅ is minimal prime.

* In base 32, there are 78 unsolved families when searched to length 10000, but if single-digit primes are excluded, then the unsolved family S{V} is searched up to length 2000001 by CRUS with no prime found.

* In base 33, there are 33 unsolved families when searched to length 10000, but if single-digit primes are excluded, then a large prime 130₂₃₆₁₄1 is minimal prime.

* In base 35, there are only 15 unsolved families when searched to length 10000, but if single-digit primes are excluded, then a large prime 1B0₅₆₀₆₁1 is minimal prime.

* In base 37, if single-digit primes are excluded, then the unsolved family 2K{0}1 is searched up to length 1000002 by CRUS with no prime found, also another unsolved family {I}J is searched up to length 524287 with no (probable) prime found.

* In base 38, if single-digit primes are excluded, then there are four large known minimal primes 20₂₇₂₈1, V0₁₅₂₇1, Lb₁₅₇₉, ab₁₃₆₂₁₁.

* In base 42, the largest minimal prime is R₄₈₆1, but if single-digit primes are excluded, then a much-larger prime 2f₂₅₂₃ is minimal prime.

* In base 43, if single-digit primes are excluded, then the unsolved family 3b{0}1 is searched up to length 1000002 by CRUS with no prime found, also another unsolved family 2{7} is searched up to length 50001 with no (probable) prime found.

* In base 48, if single-digit primes are excluded, then there is a large known minimal prime T0₁₃₃₀₄₁1.

* In base 60, if single-digit primes are excluded, then the unsolved family Z{x} is searched up to length 100001 by CRUS with no prime found.

[sweety439]

Originally Posted by sweety439

The "minimal prime problem" is solved only in bases 2~16, 18, 20, 22~24, 30, 42, and maybe 60

b, length of largest minimal prime base b, number of minimal primes base b
2, 2, 2
3, 3, 3
4, 2, 3
5, 5, 8
6, 5, 7
7, 5, 9
8, 9, 15
9, 4, 12
10, 8, 26
11, 45, 152
12, 8, 17
13, 32021, 228
14, 86, 240
15, 107, 100
16, 3545, 483
18, 33, 50
20, 449, 651
22, 764, 1242
23, 800874, 6021
24, 100, 306
30, 1024, 220
42, 4551, 487
60, ?, ? (in theory, <2000 digits)

The lower bound of the largest minimal prime in base 60 is e₁₉₃₇1, also Q₈₉₆1 is minimal prime in base 60

The values of them are 77327010984551504786304887888912950705175204580639944878112438600952757035876727229378296619277947037764123978846596061647800547824271405420610942504398032526580100809844305805544927484087907616047490338190828917065699765520511128503381872341167737600460647597875122424469309525340927891025212547061188776538264903805527424912087179786849901547205244266878177942527870184920215328514230990454933261585699343368869812084796558411721869822394527700663131350707635393027944180439815096746293426421194665978764581162587927866393205026213755969137701220074489706959966447530447995143405265533839850435707734313891577375215804642516141551699775657946569122025380740434646135054704170314178518948917457077920299428193780905089153629086443760982953552745554032630063069920652615362123725793144110637743473716888605506966189552881692154417121864981473601912302543600701367807908042289806525645841750281185651872533226665585223241511993986508736773286351668957103552353014040845431533864448327266336789760566326440423377001353690565900244559349252519469001825116007514729851712983574622341296162969774717357252465431566117676922668834197449691552002659958493980197122256612229132710589188220351409094828130859660567041139496963915457844074136483182933611723591111024630476224702575651192615747211666681896793532612116567734947534484938472185678092597966368048609549325442879516453824590523188998379435190788947006444708909975051214459423525690882137592842255587521093881402277645744154135012098764172743649344172229163389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186401 and 44237739863175496610471318629567161227581499739702326449443211342477982035718794483071631727522073459377930102398410838554672162663612371413096490082577997717588025268103015103195710515977898833005189244382338257840944719699642630344222584794826781588580542174706455627416628292660455033430459427791568859444909759999806211055982544227289834463933517584500668147475720542564765516991468231198124803963783152542228516984157676464674424474630978346931184024801441861139072461640784439697679231361092862140120023823474610674945003533686787118333313230137931070802812541939817735865901202734962214450795186055471212270838294675126791484852523312122258393734265285929307078713993139831738349121088218059932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271161

[sweety439]

For the minimal problem in base 60, I checked these families: (x is any given digit)

* x:{0}:1
* x:{0}:49 (x not divisible by 7, x != 10)
* {x}:1
* {x}:49 (x not divisible by 7)
* x:{49} (x not divisible by 7)

All these families have known proven primes, except {40}:1, which only has known strong PRP

* x:{14}:49 (x not divisible by 7)
* x:{21}:49 (x not divisible by 7)
* x:{28}:49 (x not divisible by 7)
* x:{35}:49 (x not divisible by 7)
* x:{42}:49 (x not divisible by 7)

Only 46:{42}:49 family has no known primes, even no strong PRPs in this family are known

* {14}:x:49 (x not divisible by 7)
* {21}:x:49 (x not divisible by 7)
* {28}:x:49 (x not divisible by 7)
* {35}:x:49 (x not divisible by 7)
* {42}:x:49 (x not divisible by 7)

All these families have known proven primes, except {42}:30:49, which only has known strong PRP

[sweety439]

Originally Posted by sweety439

For the minimal problem in base 60, I checked these families: (x is any given digit)

* x:{0}:1
* x:{0}:49 (x not divisible by 7, x != 10)
* {x}:1
* {x}:49 (x not divisible by 7)
* x:{49} (x not divisible by 7)

All these families have known proven primes, except {40}:1, which only has known strong PRP

* x:{14}:49 (x not divisible by 7)
* x:{21}:49 (x not divisible by 7)
* x:{28}:49 (x not divisible by 7)
* x:{35}:49 (x not divisible by 7)
* x:{42}:49 (x not divisible by 7)

Only 46:{42}:49 family has no known primes, even no strong PRPs in this family are known

* {14}:x:49 (x not divisible by 7)
* {21}:x:49 (x not divisible by 7)
* {28}:x:49 (x not divisible by 7)
* {35}:x:49 (x not divisible by 7)
* {42}:x:49 (x not divisible by 7)

All these families have known proven primes, except {42}:30:49, which only has known strong PRP

46:{42}:49 family has trivial factor of 53, thus not need to be checked.

Also checked these families:

* 10:10:{0}:49
* 10:x:{0}:49 (x = 14, 21, 28, 35, 42, 49)
* x:10:{0}:49 (x = 14, 21, 28, 35, 42, 49)

[sweety439]

Smallest prime in given simple family in base 60:

{x}:1

1,61
2,7321
3,181
4,241
5,18301
6,21961
7,421
8,379574237281
9,541
10,601
11,661
12,208167532693722559227661016949152542372881355921
13,47581
14,51241
15,3294901
16,45548908474561
17,1021
18,65881
19,15024813541
20,1201
21,276772861
22,1321
23,1381
24,316311841
25,5491501
26,44237739863175496610471318629567161227581499739702326449443211342477982035718794483071631727522073459377930102398410838554672162663612371413096490082577997717588025268103015103195710515977898833005189244382338257840944719699642630344222584794826781588580542174706455627416628292660455033430459427791568859444909759999806211055982544227289834463933517584500668147475720542564765516991468231198124803963783152542228516984157676464674424474630978346931184024801441861139072461640784439697679231361092862140120023823474610674945003533686787118333313230137931070802812541939817735865901202734962214450795186055471212270838294675126791484852523312122258393734265285929307078713993139831738349121088218059932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271161
27,1621
28,102481
29,1741
30,1801
31,1861
32,93237738291439869343927223105084745762711864406779661016949121
33,1565743728781
34,1613190508441
35,7688101
36,2161
37,2221
38,2281
39,2341
40,77327010984551504786304887888912950705175204580639944878112438600952757035876727229378296619277947037764123978846596061647800547824271405420610942504398032526580100809844305805544927484087907616047490338190828917065699765520511128503381872341167737600460647597875122424469309525340927891025212547061188776538264903805527424912087179786849901547205244266878177942527870184920215328514230990454933261585699343368869812084796558411721869822394527700663131350707635393027944180439815096746293426421194665978764581162587927866393205026213755969137701220074489706959966447530447995143405265533839850435707734313891577375215804642516141551699775657946569122025380740434646135054704170314178518948917457077920299428193780905089153629086443760982953552745554032630063069920652615362123725793144110637743473716888605506966189552881692154417121864981473601912302543600701367807908042289806525645841750281185651872533226665585223241511993986508736773286351668957103552353014040845431533864448327266336789760566326440423377001353690565900244559349252519469001825116007514729851712983574622341296162969774717357252465431566117676922668834197449691552002659958493980197122256612229132710589188220351409094828130859660567041139496963915457844074136483182933611723591111024630476224702575651192615747211666681896793532612116567734947534484938472185678092597966368048609549325442879516453824590523188998379435190788947006444708909975051214459423525690882137592842255587521093881402277645744154135012098764172743649344172229163389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186401
41,150061
42,2521
43,122412691525381
44,9665041
45,164701
46,606264361
47,172021
48,37957423681
49,10763341
50,3001
51,3061
52,3121
53,3181
54,197641
55,3301
56,3361
57,28550279937155564567041751967467111314451825663380318242711864406779661016949152542372881355932203389830508474576271186440621
58,212281
59,3541

{x}:49

1,109
2,7369
3,229
4,52718689
5,349
6,409
8,105437329
9,12329714402004875501336027803574282441328813559322033898305084745762711864406779661016989
10,131796649
11,709
12,769
13,829
15,54949
16,1009
17,1069
18,1129
19,250413589
20,1249
22,811681549016949152569
23,1429
24,1489
25,1549
26,1609
27,1669
29,1789
30,109849
31,408569509
32,119904498831584194115132745762711864406779661016949169
33,2029
34,2089
36,3028642546542661941210719954058267766581114796637317680350572474576271186440677966101694915254237288135593220338983050847457627118644067809
37,2269
38,6490719457627129
39,2389
40,146449
41,540366109
43,157429
44,2689
45,2749
46,168409
47,172069
48,6375389621369491525423729
50,3049
51,3109
52,3169
53,3229
54,197689
55,9394462372881349
57,3469
58,3529
59,46655999989

x:{49}

1,109
2,611389
3,229
4,17389
5,349
6,409
8,411996203389
9,77035957924881355932203389
10,14590162042372436009914299567562900888905762711864406779661016949152542372881355932203389
11,709
12,769
13,829
15,56989
16,1009
17,1069
18,1129
19,71389
20,1249
22,82189
23,1429
24,1489
25,1549
26,1609
27,1669
29,1789
30,110989
31,6875389
32,118189
33,2029
34,2089
36,132589
37,2269
38,2384559087006591915952856949152542372881355932203389
39,2389
40,146989
41,150589
43,4451424541432606372881355932203389
44,2689
45,2749
46,606923389
47,133894812203389
48,10547389
50,3049
51,3109
52,3169
53,3229
54,197389
55,200989
57,3469
58,3529
59,215389

x:{0}:1

1,61
2,432001
3,181
4,241
5,2406149016991872132210992275783680000000000000000000000000000000000000000001
6,21601
7,421
8,62691331276800000000000001
9,541
10,601
11,661
12,43201
13,168480001
14,10886400001
15,54001
16,57601
17,1021
18,13996800001
19,20370649768045944216583302410995908998024578122182598718037950464000000000000000000000000000000000000000000000000000000000000000000000000000000001
20,1201
21,3527193600000001
22,1321
23,1381
24,5184001
25,90001
26,93601
27,1621
28,100801
29,1741
30,1801
31,1861
32,115201
33,118801
34,122401
35,126001
36,2161
37,2221
38,2281
39,2341
40,8640001
41,531360001
42,2521
43,2006208000001
44,965225828176625664000000000000000000001
45,125971200000001
46,165601
47,2192832000001
48,172801
49,176401
50,3001
51,3061
52,3121
53,3181
54,93551073780643988500363379682469478400000000000000000000000000000000000000000001
55,3301
56,3361
57,205201
58,(trivial factor of 59)
59,3541

x:{0}:49

1,109
2,93312000049
3,229
4,14449
5,349
6,409
8,6220800049
9,1944049
10,(trivial factor of 59)
11,709
12,769
13,829
15,54049
16,1009
17,1069
18,1129
19,68449
20,1249
22,61585920000049
23,1429
24,1489
25,1549
26,1609
27,1669
29,1789
30,6480049
31,6696049
32,115249
33,2029
34,2089
36,466560049
37,2269
38,136849
39,2389
40,1866240000049
41,1912896000049
43,154849
44,2689
45,2749
46,2146176000049
47,169249
48,172849
50,3049
51,3109
52,3169
53,3229
54,699840049
55,1625362428001224684562289212884750459919231127678405836800000000000000000000000000000000000000000000000000000000000000000000049
57,3469
58,3529
59,12744049

[sweety439]

x:{14}:49

1,109
2,8089
3,229
4,15289
5,349
6,409
8,1779289
9,33289
10,132675289
11,709
12,769
13,829
15,20055578263423416146440677966101694915289
16,1009
17,1069
18,1129
19,249315289
20,1249
22,1037502915289
23,1429
24,1489
25,1549
26,1609
27,1669
29,1789
30,6531289
31,6747289
32,116089
33,2029
34,2089
36,130489
37,2269
38,495555289
39,2389
40,144889
41,103182884390444449640082580739273165590541602171762902235022709553359196234458364272060077833460415740804281702234241751684562368942580406237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915289
43,155689
44,2689
45,2749
46,465965333694915289
47,10203289
48,37509315289
50,3049
51,3109
52,3169
53,3229
54,702915289
55,11931289
57,3469
58,3529
59,213289

x:{21}:49

1,109
2,508909
3,229
4,56452909
5,349
6,409
8,30109
9,94286240542372909
10,37309
11,709
12,769
13,829
15,42986782372909
16,1009
17,1069
18,1129
19,69709
20,1249
22,4025817357839737065999049857469767607759032979106716427631968228584180718644067796610169491525423728813559322033898305084745762711864406779661016949152542372909
23,1429
24,1489
25,1549
26,1609
27,1669
29,1789
30,66078257013152542372909
31,112909
32,6988909
33,2029
34,2089
36,101773342372909
37,2269
38,497092909
39,2389
40,31380772909
41,535972909
43,156109
44,2689
45,2749
46,166909
47,170509
48,10444909
50,3049
51,3109
52,3169
53,3229
54,195709
55,717412909
57,3469
58,3529
59,12820909

x:{28}:49

1,109
2,8929
3,229
4,57990529
5,349
6,409
8,30529
9,34129
10,2262529
11,709
12,769
13,829
15,3342529
16,1009
17,1069
18,1129
19,4206529
20,1249
22,80929
23,1429
24,1489
25,1549
26,1609
27,1669
29,1789
30,394950529
31,113329
32,116929
33,2029
34,2089
36,472710529
37,2269
38,8310529
39,2389
40,8742529
41,4212156168942609355932203389830529
43,9390529
44,2689
45,2749
46,167329
47,1041447436333870756881355932203389830529
48,13781461655186262022942372881355932203389830529
50,3049
51,3109
52,3169
53,3229
54,331946136337921041355932203389830529
55,199729
57,3469
58,3529
59,214129

x:{35}:49

1,109
2,9349
3,229
4,771484637288149
5,349
6,409
8,30949
9,34549
10,38149
11,709
12,769
13,829
15,56149
16,1009
17,1069
18,1129
19,70549
20,1249
22,81349
23,1429
24,1489
25,1549
26,1609
27,1669
29,1789
30,6608149
31,113749
32,25344488149
33,2029
34,2089
36,131749
37,2269
38,23335844534237288149
39,2389
40,526088149
41,149749
43,33898088149
44,2689
45,2749
46,10064149
47,185120325561205169675135454687984165581220276782105983361193164168670838265752820260904448577571678108045870876576322502449572547744055070802594515527926156691477915758601548938853501303980626387044701079429027352148995318212751455646534728345092572787387347410490834215458172781888350653835992601161966471095319325576754886993561238893441826145675686736477047030489460058140333352185261905102459753937730302319495540099005728377120233422356413887591971257315898743429492383226804915072909057608763313496683212160248130680520291356694770983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288149
48,10496149
50,3049
51,3109
52,3169
53,3229
54,196549
55,12008149
57,3469
58,3529
59,12872149

x:{42}:49

1,109
2,9769
3,229
4,170945053505084745769
5,349
6,409
8,1881769
9,2097769
10,38569
11,709
12,769
13,829
15,56569
16,1009
17,1069
18,1129
19,70969
20,1249
22,81769
23,1429
24,1489
25,1549
26,1609
27,1669
29,1789
30,110569
31,6849769
32,7065769
33,2029
34,2089
36,132169
37,2269
38,139369
39,2389
40,31657545769
41,150169
43,33990345769
44,2689
45,2749
46,(trivial factor of 53)
47,10305769
48,1767252099905084745769
50,3049
51,3109
52,3169
53,3229
54,11817769
55,200569
57,3469
58,3529
59,12897769

{14}:x:49

1,3074509
2,31719478566747843550804041300349830508474576271186440677966101694914569
3,11070914629
4,11070914689
5,184514749
6,11070914809
8,50929
9,50989
10,3075049
11,51109
12,51169
13,51229
15,51349
16,3075409
17,8608743701694915469
18,516524622101694915529
19,11070915589
20,184515649
22,51769
23,51829
24,664254915889
25,51949
26,52009
27,52069
29,52189
30,52249
31,39855294916309
32,52369
33,3076429
34,52489
36,52609
37,664254916669
38,30991477326101694916729
39,3076789
40,11070916849
41,664254916909
43,184517029
44,53089
45,53149
46,53690481146394350663097813232332427347712798372881355932203389830508474576271186440677966101694917209
47,53269
48,664254917329
50,86756301967596040677966101694917449
51,11070917509
52,53569
53,53629
54,3077689
55,3147493094329085095522234576271186440677966101694917749
57,30991477326101694917869
58,184517929
59,39855294917989

{21}:x:49

1,75709
2,10041238653656949152542371769
3,4611829
4,276771889
5,16606371949
6,4612009
8,76129
9,616626663337578078627630562878800705084745762711864406779661016949152542372189
10,76249
11,645936492161087054050399981653446997706215532953486454915466633416169490161247457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372309
12,76369
13,996382372429
15,4612549
16,4612609
17,4612669
18,276772729
19,4612789
20,4612849
22,4612969
23,77029
24,276773089
25,16606373149
26,4613209
27,77269
29,276773389
30,4613449
31,77509
32,77569
33,4613629
34,77689
36,16606373809
37,276773869
38,77929
39,16606373989
40,78049
41,4614109
43,78229
44,59782942374289
45,4614349
46,6071553036900241310806779661016949152542374409
47,276774469
48,61187265753757414256952240162711864406779661016949152542374529
50,78649
51,774786933152542374709
52,996382374769
53,4614829
54,78889
55,52209837304507200077039765027430616624964848906126524755206192493399488812789496523127382219681677062442951316647311186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542374949
57,4615069
58,996382375129
59,276775189

{28}:x:49

1,81583021005009885675936320216949152542372881355932203389828909
2,85460868854823834608016221386772928746969615932515587546395079909292691525423728813559322033898305084745762711864406779661016949152542372881355932203389828969
3,22141829029
4,101089
5,101149
6,101209
8,6149329
9,6149389
10,101449
11,286958123389829509
12,10410756236111524881355932203389829569
13,369029629
15,101749
16,79710589829809
17,101869
18,101929
19,1328509829989
20,6150049
22,13388318204875932203389830169
23,102229
24,6150289
25,79710589830349
26,102409
27,1328509830469
29,6150589
30,6150649
31,29828045081330194812832118462406904082062665762711864406779661016949152542372881355932203389830709
32,102769
33,102829
34,286958123389830889
36,41126317117261134141412615065804034201951786857394034478949601447152874939439946846072866224064662010793220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389831009
37,103069
38,6151129
39,1328509831189
40,369031249
41,369031309
43,286958123389831429
44,1328509831489
45,103549
46,1033049244203389831609
47,103669
48,6151729
50,1328509831849
51,286958123389831909
52,103969
53,37478722450001489572881355932203389832029
54,104089
55,104149
57,6152269
58,1328509832329
59,6152389

{35}:x:49

1,81475573154959817379255597463815552667176545024717505666213665960638224320021112459781503325113716943629003292963749951159320447728997238620350548107901686289125472897903137985503182233361374542375162900682265823294796999737443473337843435573308433456959404707058320810457306411455933006101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237286109
2,1660637286169
3,126229
4,17128518426043835517434182302188908474576271186440677966101694915254237286289
5,126349
6,36429318221401447864840677966101694915254237286409
8,77478693315254237286529
9,7686589
10,7686649
11,27677286709
12,99638237286769
13,461286829
15,126949
16,1660637287009
17,461287069
18,1660637287129
19,127189
20,127249
22,607155303690024131080677966101694915254237287369
23,131145545597045212313426440677966101694915254237287429
24,36429318221401447864840677966101694915254237287489
25,127549
26,127609
27,127669
29,27677287789
30,127849
31,358697654237287909
32,461287969
33,21521859254237288029
34,99638237288089
36,7688209
37,1660637288269
38,1660637288329
39,128389
40,128449
41,128509
43,128629
44,461288689
45,128749
46,7688809
47,358697654237288869
48,7688929
50,129049
51,7689109
52,129169
53,129229
54,129289
55,21521859254237289349
57,129469
58,129529
59,129589

{42}:x:49

1,9223309
2,293439587902707111288657562346647653394212382542384936694711075914089081641468094666645578615570550709924319586579296659981779875032195661701462202096102293874185480241052744144833349365537276799785969205297493379562404092952314178395004440542139367761173708310416661107311234222306240316209631923293822273260390016005704126405409242393725556515881932954120677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084743369
3,151429
4,260268905902788122033898305084743489
5,151549
6,151609
8,151729
9,95158435700243530652412123901049491525423728813559322033898305084743789
10,151849
11,151909
12,151969
13,152029
15,334707955121898305084744149
16,33212744209
17,9224269
18,25826231105084744329
19,152389
20,33212744449
22,1992764744569
23,152629
24,9224689
25,33212744749
26,152809
27,553544869
29,152989
30,1085381915311180600372601407525931962389847751970245597672814187895728453642389622010356871067432479240127281624918875843807895776453427504154910355073723811485480200485147288814532682450893550065346880581428503092670144866199408686377359578222051485037152838513072612496791202198960786455011981346714830632452284175840153078825137962409229301085120863398324900668527102405288924544210878811218940250587703349451751315140419049505572840115309385837985597640170169004994591035237463955094223324926699100962730596525138946332290428960176322975631221744617520009977331177155716098492037390962017969334604994823460496054237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745049
31,3452320032561365140374087785000799083572067796610169491525423728813559322033898305084745109
32,33212745169
33,1992764745229
34,9225289
36,153409
37,153469
38,153529
39,153589
40,153649
41,9225709
43,202385101230008043693559322033898305084745829
44,153889
45,153949
46,430437185084746009
47,9226069
48,1992764746129
50,553546249
51,33212746309
52,154369
53,1204948638438833898305084746429
54,95158435700243530652412123901049491525423728813559322033898305084746489
55,9226549
57,154669
58,33212746729
59,154789

[sweety439]

{49}:x:49

1,176509
2,10760569
3,176629
4,645800689
5,1807836177355932200749
6,176809
8,8369611932200929
9,176989
10,5179663972035655860472096728181925922711864406779661016949152542372881355932201049
11,177109
12,38748201169
13,645801229
15,3370877628514757222772400494683673158596409477797247719358711849483581786873706143158543146664730075705489224200224093772041630182187652701699645946277442899767500755812009374102776457086216749247209626019900398116606709317609816539188162397141065837083072359107434141070736000398492419524277805139751989637641458276851772548616402672230694356610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932203389830508474576271186440677966101694915254237288135593220338983050847457627118644067796610169491525423728813559322033898305084745762711864406779661016949152542372881355932201349
16,177409
17,10761469
18,10761529
19,177589
20,10761649
22,38748201769
23,645801829
24,177889
25,177949
26,645802009
27,178069
29,30130602955932202189
30,178249
31,645802309
32,84346404690718372881355932202369
33,10762429
34,178489
36,178609
37,1093129404791710112542372881355932202669
38,10762729
39,10762789
40,390492614308881355932202849
41,178909
43,179029
44,179089
45,38748203149
46,179209
47,179269
48,409053718397842884147674224095289021756765923247261869559322033898305084745762711864406779661016949152542372881355932203329
50,645803449
51,38748203509
52,10763569
53,10763629
54,179689
55,179749
57,10763869
58,1807836177355932203929
59,179989

10:x:{0}:49

614,477446400049
621,37309
628,8138880049
635,38149
642,38569
649,140184049

x:10:{0}:49

850,183600049
1270,76249
1690,101449
2110,3571525142839296000000000000000049
2530,151849
2950,38232000049

[sweety439]

Other possible such simple families:

10:10:{0}:49

prime is 2196049

58:58:{0}:1

prime is 212281

10:{0}:49:49

prime is 6046617600000002989

10:{0}:10:49

prime is 466560000649

58:{0}:58:1

prime is 212281

[sweety439]

14:{0}:x:49

1,24253982091278071092686802139899494400000000000000000000000000000000000000000109
2,1009
3,1069
4,1129
5,3024349
6,1249
8,50929
9,1429
10,1489
11,1549
12,1609
13,1669
15,1789
16,5361922889755312850308759177923802244504459923744908273254400000000000000000000000000000000000000000000000000000000000000000000001009
17,3025069
18,3025129
19,2029
20,2089
22,51769
23,2269
24,10886401489
25,2389
26,52009
27,52069
29,52189
30,2689
31,2749
32,52369
33,181442029
34,52489
36,3049
37,3109
38,3169
39,3229
40,39191040002449
41,3026509
43,3469
44,3529
45,53149
46,181442809
47,3709
48,3769
50,3889
51,181443109
52,53569
53,53629
54,4129
55,(trivial factor of 59)
57,3027469
58,307117308965289984000000000000000003529
59,10886403589

21:{0}:x:49

1,75709
2,1429
3,1489
4,1549
5,1609
6,1669
8,1789
9,211631616000000589
10,76249
11,272160709
12,2029
13,2089
15,16329600949
16,2269
17,4537069
18,2389
19,4537189
20,272161249
22,12697896960000001369
23,2689
24,2749
25,2753361057228796079119709444688918393883353351941059461389077445343240031237662127438976850813717433212648483052318436843042263067280107067121663921269045680514764091347878987046902409665972186282803753885626203381672952078169757610858708358660096000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001549
26,3527193600001609
27,77269
29,3049
30,3109
31,3169
32,3229
33,272162029
34,77689
36,3469
37,3529
38,77929
39,16329602389
40,3709
41,3769
43,3889
44,272162689
45,4538749
46,278553138848124030953717760000000000000000000000000002809
47,4129
48,(trivial factor of 59)
50,78649
51,12697896960000003109
52,4539169
53,979776003229
54,4549
55,58786560003349
57,4729
58,4789
59,592433080565760000000000003589

28:{0}:x:49

1,1789
2,789910774087680000000000000169
3,21772800229
4,101089
5,2029
6,2089
8,21772800529
9,2269
10,101449
11,2389
12,6048769
13,6048829
15,101749
16,2689
17,2749
18,101929
19,47394646445260800000000000001189
20,362881249
22,3049
23,3109
24,3169
25,3229
26,102409
27,78382080001669
29,3469
30,3529
31,80223303988259720914670714880000000000000000000000000000001909
32,102769
33,3709
34,3769
36,3889
37,103069
38,282175488000002329
39,6050389
40,4129
41,(trivial factor of 59)
43,1306368002629
44,2316350688374295151333383964863082569625926687057800374045900800000000000000000000000000000000000000000000000000000000000000000000000002689
45,103549
46,362882809
47,4549
48,21772802929
50,4729
51,4789
52,103969
53,4909
54,4969
55,104149
57,1039694019687845983054132464844800000000000000000000000000000000003469
58,5209
59,78382080003589

35:{0}:x:49

1,7560109
2,2269
3,126229
4,2389
5,126349
6,16456474460160000000000000409
8,453600529
9,2689
10,2749
11,5878656000000709
12,76187381760000000000769
13,97977600000829
15,3049
16,3109
17,3169
18,3229
19,127189
20,127249
22,3469
23,3529
24,27855313884812403095371776000000000000000000000000000001489
25,127549
26,3709
27,3769
29,3889
30,127849
31,7561909
32,7561969
33,4129
34,(trivial factor of 59)
36,453602209
37,1632960002269
38,7562329
39,128389
40,4549
41,128509
43,4729
44,4789
45,128749
46,4909
47,4969
48,352719360000002929
50,129049
51,5209
52,129169
53,129229
54,129289
55,5449
57,5569
58,129529
59,5689

42:{0}:x:49

1,9072109
2,2689
3,2749
4,32659200289
5,151549
6,151609
8,3049
9,3109
10,3169
11,3229
12,151969
13,152029
15,3469
16,3529
17,117573120001069
18,5614347706314368308492315310161920000000000000000000000000000000000001129
19,3709
20,3769
22,3889
23,152629
24,9073489
25,16850366041100048016273224228350264013299690448515611349014072815152914037687200443877077133470937907200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001549
26,4129
27,(trivial factor of 59)
29,152989
30,5485491486720000000001849
31,32659201909
32,544321969
33,4549
34,9074089
36,4729
37,4789
38,153529
39,4909
40,4969
41,117573120002509
43,20686430901833768712610953618533187671699305261361528832000000000000000000000000000000000000000000000000000000000000000002629
44,5209
45,153949
46,544322809
47,544322869
48,5449
50,5569
51,387215843042271258003015621585831529981559389785592810438298620409934449687415027238102244809934838667668586191368584982641890866197562403890778944989683725107200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000003109
52,5689
53,5749
54,32659203289
55,5869
57,154669
58,1959552003529
59,154789

49:{0}:x:49

1,3049
2,3109
3,3169
4,3229
5,3379480093071544116029035238523182050306352376786936240751182230865067894959768623789993478122963368981956666059054838206787747840000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000349
6,176809
8,3469
9,3529
10,10584649
11,177109
12,3709
13,3769
15,3889
16,177409
17,137168640001069
18,140390781979454511600673751040000000000000000000000000000001129
19,4129
20,(trivial factor of 59)
22,635041369
23,38102401429
24,177889
25,177949
26,4549
27,178069
29,4729
30,4789
31,10585909
32,4909
33,4969
34,178489
36,178609
37,5209
38,635042329
39,1382343854653440000000000002389
40,10586449
41,5449
43,5569
44,179089
45,5689
46,5749
47,179269
48,5869
50,137168640003049
51,29628426240000003109
52,106662334464000000003169
53,635043229
54,6229
55,179749
57,10587469
58,6469
59,6529

[sweety439]

minimal primes in base 60 up to 2^32