20201212, 10:56  #23 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
D46_{16} Posts 
* Case (5,1):
** 51 is prime, and thus the only minimal prime in this family. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes) *** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes) **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. **** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes) ***** 500525 is not prime. ***** The smallest prime of the form 5{5}25 is 555555555555525 ***** The smallest prime of the form 5{5}025 is 55555025 ***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025 (not minimal prime, since 55555025 and 555555555555525 are primes) ***** The smallest prime of the form 5{5}0525 is 5550525 ***** The smallest prime of the form 5{5}00525 is 5500525 ***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes) * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. Last fiddled with by sweety439 on 20201227 at 06:18 
20201213, 05:06  #24 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2·1,699 Posts 
* Case (6,1):
** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 111, 141, 401, 471, 701, 711, 6101, 6441 are primes, we only need to consider the families 6{0}0{0,1,4,7}1, 6{0,4}1{7}1, 6{0,7}4{1}1, 6{0,1,7}7{4,7}1 (since any digits combo 11, 14, 40, 47, 70, 71, 10, 44 between them will produce smaller primes) **** For the 6{0}0{0,1,4,7}1 family, since 6007 is prime, we only need to consider the families 6{0}0{0,1,4}1 and 60{1,4,7}7{0,1,4,7}1 (since any digits combo 1007 between (6,1) will produce smaller primes) ***** For the 6{0}0{0,1,4}1 family, since 111, 141, 401, 6101, 6441, 60411 are primes, we only need to consider the families 6{0}1, 6{0}11, 6{0}41 (since any digits combo 10, 11, 14, 40, 41, 44 between (6{0}0,1) will produce smaller primes) ****** All numbers of the form 6{0}1 are divisible by 7, thus cannot be prime. ****** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime. ****** All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime. ***** For the 60{1,4,7}7{0,1,4,7}1 family, since 701, 711, 60741 are primes, we only need to consider the family 60{1,4,7}7{7}1 (since any digits 0, 1, 4 between (60{1,4,7}7,1) will produce smaller primes) ***** Since 471, 60171 is prime, we only need to consider the family 60{7}1 (since any digits 1, 4 between (60,7{7}1) will produce smaller primes) ****** All numbers of the form 60{7}1 are divisible by 7, thus cannot be prime. **** For the 6{0,4}1{7}1 family, since 417, 471 are primes, we only need to consider the families 6{0}1{7}1 and 6{0,4}11 ***** For the 6{0}1{7}1 family, since 60171 is prime, and thus the only minimal prime in the family 6{0}1{7}1. ***** For the 6{0,4}11 family, since 401, 6441, 60411 are primes, we only need to consider the number 6411 and the family 6{0}11 ****** 6411 is not prime. ****** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime. **** For the 6{0,7}4{1}1 family, since 60411 is prime, we only need to consider the families 6{7}4{1}1 and 6{0,7}41 ***** For the 6{7}4{1}1 family, since 111, 6777 are primes, we only need to consider the numbers 641, 6411, 6741, 67411, 67741, 677411 ****** None of 641, 6411, 6741, 67411, 67741, 677411 are primes. ***** For the 6{0,7}41 family, since 701, 6777, 60741 are primes, we only need to consider the families 6{0}41 and the numbers 6741, 67741 (since any digits combo 07, 70, 777 between (6,41) will produce smaller primes) ****** All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime. ****** Neither of 6741, 67741 are primes. ***** For the 6{0,1,7}7{4,7}1 family, since 747 is prime, we only need to consider the families 6{0,1,7}7{4}1, 6{0,1,7}7{7}1, 6{0,1,7}7{7}{4}1 (since any digits combo 47 between (6{0,1,7}7,1) will produce smaller primes) ****** For the 6{0,1,7}7{4}1 family, since 6441 is prime, we only need to consider the families 6{0,1,7}71 and 6{0,1,7}741 (since any digits combo 44 between (6{0,1,7}7,1) will produce smaller primes) ******* For the 6{0,1,7}71 family, since all numbers of the form 6{0,7}71 are divisible by 7 and cannot be prime, and 111 is prime (thus, any digits combo 11 between (6,71) will produce smaller primes), we only need to consider the family 6{0,7}1{0,7}71 ******** Since 717 and 60171 are primes, we only need to consider the family 61{0,7}71 (since any digit combo 0, 7 between (6,1{0,7}71) will produce smaller primes) ********* Since 177 and 6101 are primes, we only need to consider the number 6171 (since any digit combo 0, 7 between (61,71) will produce smaller primes) ********** 6171 is not prime. ****** All numbers in the 6{0,1,7}7{7}1 or 6{0,1,7}7{7}{4}1 families are also in the 6{0,1,7}7{4}1 family, thus these two families cannot have more minimal primes. Last fiddled with by sweety439 on 20210106 at 11:26 
20201213, 05:38  #25 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2·1,699 Posts 
Upload past file, the set is not complete for bases >=7, I want to complete them.
I know some primes in the set which is not listed: * base 7: 33333333333333331 * base 8: 77774444441, 7777777777771, 555555555555525, 4_{220}7 * base 10: 555555555551 * base 11: A0000000001, A0014444444, 40000000A0041, A000000014444, A044444444441, A144444444411, 40000000000401, A0000044444441, A00000000444441, 11111111111111111, 14444444444441111, 44444444444444111, A1444444444444444, A9999999999999996, 1444444444444444444, 4000000000000000A041, A999999999999999999999, A44444444444444444444444441, 40000000000000000000000000041, 440000000000000000000000000001, 999999999999999999999999999999991, 444444444444444444444444444444444444444444441 (they are in the searching where singledigit primes are included, but this puzzle does not include singledigit primes) * base 13: 80_{32017}111 * base 14: 4D_{19698} * base 15: DE_{14} * base 17: 74_{4904} (this problem is much harder than the original minimal prime (where singledigit primes are included), see post https://mersenneforum.org/showpost.p...5&postcount=57) 
20201213, 07:53  #26 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2·1,699 Posts 
Found a minimal prime (start with 2 digits) in base 13: 7_{1504}1, which equals (7*13^150579)/12

20201213, 09:46  #27 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2·1,699 Posts 
Consider the "simplest" families x{y} and {x}y, where x,y are base b digits
Necessary conditions are gcd(x,y) = 1, gcd(y,b) = 1 Code:
b, x, y, smallest prime 2, {1}, 1: 3 2, 1, {1}: 3 3, {1}, 1: 13 3, 1, {1}: 13 3, {1}, 2: 5 3, 1, {2}: 5 3, {2}, 1: 7 3, 2, {1}: 7 4, {1}, 1: 5 4, 1, {1}: 5 4, {1}, 3: 7 4, 1, {3}: 7 4, {2}, 1: 41 4, 2, {1}: 37 4, {2}, 3: 11 4, 2, {3}: 11 4, {3}, 1: 13 4, 3, {1}: 13 5, {1}, 1: 31 5, 1, {1}: 31 5, {1}, 2: 7 5, 1, {2}: 7 5, {1}, 3: 0 5, 1, {3}: 43 5, {1}, 4: 0 5, 1, {4}: 1249 5, {2}, 1: 11 5, 2, {1}: 11 5, {2}, 3: 13 5, 2, {3}: 13 5, {3}, 1: 2341 5, 3, {1}: 0 5, {3}, 2: 17 5, 3, {2}: 17 5, {3}, 4: 19 5, 3, {4}: 19 5, {4}, 1: 3121 5, 4, {1}: 0 5, {4}, 3: 23 5, 4, {3}: 23 6, {1}, 1: 7 6, 1, {1}: 7 6, {1}, 5: 11 6, 1, {5}: 11 6, {2}, 1: 13 6, 2, {1}: 13 6, {2}, 5: 17 6, 2, {5}: 17 6, {3}, 1: 19 6, 3, {1}: 19 6, {3}, 5: 23 6, 3, {5}: 23 6, {4}, 1: 1033 6, 4, {1}: 151 6, {4}, 5: 29 6, 4, {5}: 29 6, {5}, 1: 31 6, 5, {1}: 31 7, {1}, 1: 2801 7, 1, {1}: 2801 7, {1}, 2: 401 7, 1, {2}: 457 7, {1}, 3: 59 7, 1, {3}: 73 7, {1}, 4: 11 7, 1, {4}: 11 7, {1}, 5: 61 7, 1, {5}: 89 7, {1}, 6: 13 7, 1, {6}: 13 7, {2}, 1: 113 7, 2, {1}: 743 7, {2}, 3: 17 7, 2, {3}: 17 7, {2}, 5: 19 7, 2, {5}: 19 7, {3}, 1: 116315256993601 7, 3, {1}: 7603 7, {3}, 2: 23 7, 3, {2}: 23 7, {3}, 4: 1201 7, 3, {4}: 179 7, {3}, 5: 173 7, 3, {5}: 9203 7, {4}, 1: 29 7, 4, {1}: 29 7, {4}, 3: 31 7, 4, {3}: 31 7, {4}, 5: 229 7, 4, {5}: 1657 7, {5}, 1: 281 7, 5, {1}: 8413470255870653 7, {5}, 2: 37 7, 5, {2}: 37 7, {5}, 3: 283 7, 5, {3}: 269 7, {5}, 4: 1999 7, 5, {4}: 277 7, {5}, 6: 41 7, 5, {6}: 41 7, {6}, 1: 43 7, 6, {1}: 43 7, {6}, 5: 47 7, 6, {5}: 47 8, {1}, 1: 73 8, 1, {1}: 73 8, {1}, 3: 11 8, 1, {3}: 11 8, {1}, 5: 13 8, 1, {5}: 13 8, {1}, 7: 79 8, 1, {7}: 127 8, {2}, 1: 17 8, 2, {1}: 17 8, {2}, 3: 19 8, 2, {3}: 19 8, {2}, 5: 149 8, 2, {5}: 173 8, {2}, 7: 23 8, 2, {7}: 23 8, {3}, 1: 1753 8, 3, {1}: 1609 8, {3}, 5: 29 8, 3, {5}: 29 8, {3}, 7: 31 8, 3, {7}: 31 8, {4}, 1: 76695841 8, 4, {1}: 284694975049 8, {4}, 3: 2339 8, 4, {3}: 283 8, {4}, 5: 37 8, 4, {5}: 37 8, {4}, 7: 21870014779720278736374332149114462520188534743847615898363462279537144492484599310778624146468224150373895489844303219383829573677353011540369291867378470695590964880740521967077028064041941947533607 8, 4, {7}: 20479 8, {5}, 1: 41 8, 5, {1}: 41 8, {5}, 3: 43 8, 5, {3}: 43 8, {5}, 7: 47 8, 5, {7}: 47 8, {6}, 1: 433 8, 6, {1}: 56657856797822194249 8, {6}, 5: 53 8, 6, {5}: 53 8, {6}, 7: 439 8, 6, {7}: 3583 8, {7}, 1: 549755813881 8, 7, {1}: 457 8, {7}, 3: 59 8, 7, {3}: 59 8, {7}, 5: 61 8, 7, {5}: 61 9, {1}, 1: 0 9, 1, {1}: 0 9, {1}, 2: 11 9, 1, {2}: 11 9, {1}, 4: 13 9, 1, {4}: 13 9, {1}, 5: 0 9, 1, {5}: 131 9, {1}, 7: 97 9, 1, {7}: 151 9, {1}, 8: 17 9, 1, {8}: 17 9, {2}, 1: 19 9, 2, {1}: 19 9, {2}, 5: 23 9, 2, {5}: 23 9, {2}, 7: 14767 9, 2, {7}: 0 9, {3}, 1: 271 9, 3, {1}: 0 9, {3}, 2: 29 9, 3, {2}: 29 9, {3}, 4: 31 9, 3, {4}: 31 9, {3}, 5: 0 9, 3, {5}: 293 9, {3}, 7: 277 9, 3, {7}: 313 9, {3}, 8: 0 9, 3, {8}: 0 9, {4}, 1: 37 9, 4, {1}: 37 9, {4}, 5: 41 9, 4, {5}: 41 9, {4}, 7: 43 9, 4, {7}: 43 9, {5}, 1: 36901 9, 5, {1}: 0 9, {5}, 2: 47 9, 5, {2}: 47 9, {5}, 4: 4099 9, 5, {4}: 172595827849 9, {5}, 7: 457 9, 5, {7}: 0 9, {5}, 8: 53 9, 5, {8}: 53 9, {6}, 1: 541 9, 6, {1}: 0 9, {6}, 5: 59 9, 6, {5}: 59 9, {6}, 7: 61 9, 6, {7}: 61 9, {7}, 1: 631 9, 7, {1}: 577 9, {7}, 2: 0 9, 7, {2}: 587 9, {7}, 4: 67 9, 7, {4}: 67 9, {7}, 5: 0 9, 7, {5}: 617 9, {7}, 8: 71 9, 7, {8}: 71 9, {8}, 1: 73 9, 8, {1}: 73 9, {8}, 5: 0 9, 8, {5}: 6287 9, {8}, 7: 79 9, 8, {7}: 79 10, {1}, 1: 11 10, 1, {1}: 11 10, {1}, 3: 13 10, 1, {3}: 13 10, {1}, 7: 17 10, 1, {7}: 17 10, {1}, 9: 19 10, 1, {9}: 19 10, {2}, 1: 2221 10, 2, {1}: 211 10, {2}, 3: 23 10, 2, {3}: 23 10, {2}, 7: 227 10, 2, {7}: 277 10, {2}, 9: 29 10, 2, {9}: 29 10, {3}, 1: 31 10, 3, {1}: 31 10, {3}, 7: 37 10, 3, {7}: 37 10, {4}, 1: 41 10, 4, {1}: 41 10, {4}, 3: 43 10, 4, {3}: 43 10, {4}, 7: 47 10, 4, {7}: 47 10, {4}, 9: 449 10, 4, {9}: 499 10, {5}, 1: 555555555551 10, 5, {1}: 511111 10, {5}, 3: 53 10, 5, {3}: 53 10, {5}, 7: 557 10, 5, {7}: 577 10, {5}, 9: 59 10, 5, {9}: 59 10, {6}, 1: 61 10, 6, {1}: 61 10, {6}, 7: 67 10, 6, {7}: 67 10, {7}, 1: 71 10, 7, {1}: 71 10, {7}, 3: 73 10, 7, {3}: 73 10, {7}, 9: 79 10, 7, {9}: 79 10, {8}, 1: 881 10, 8, {1}: 811 10, {8}, 3: 83 10, 8, {3}: 83 10, {8}, 7: 887 10, 8, {7}: 877 10, {8}, 9: 89 10, 8, {9}: 89 10, {9}, 1: 991 10, 9, {1}: 911 10, {9}, 7: 97 10, 9, {7}: 97 11, {1}, 1: 50544702849929377 11, 1, {1}: 50544702849929377 11, {1}, 2: 13 11, 1, {2}: 13 11, {1}, 3: 0 11, 1, {3}: 157 11, {1}, 4: 0 11, 1, {4}: 7783884238889124073 11, {1}, 5: 137 11, 1, {5}: 181 11, {1}, 6: 17 11, 1, {6}: 17 11, {1}, 7: 139 11, 1, {7}: 24889 11, {1}, 8: 19 11, 1, {8}: 19 11, {1}, 9: 0 11, 1, {9}: 229 11, {1}, 10: 0 11, 1, {10}: 241 11, {2}, 1: 23 11, 2, {1}: 23 11, {2}, 3: 354313 11, 2, {3}: 3061 11, {2}, 5: 269 11, 2, {5}: 0 11, {2}, 7: 29 11, 2, {7}: 29 11, {2}, 9: 31 11, 2, {9}: 31 11, {3}, 1: 397 11, 3, {1}: 0 11, {3}, 2: 4391 11, 3, {2}: 4259 11, {3}, 4: 37 11, 3, {4}: 37 11, {3}, 5: 401 11, 3, {5}: 0 11, {3}, 7: 85593501187 11, 3, {7}: 0 11, {3}, 8: 41 11, 3, {8}: 41 11, {3}, 10: 43 11, 3, {10}: 43 11, {4}, 1: 29156193474041220857161146715104735751776055777 11, 4, {1}: 0 11, {4}, 3: 47 11, 4, {3}: 47 11, {4}, 5: 5857 11, 4, {5}: 724729 11, {4}, 7: 114124668247 11, 4, {7}: 0 11, {4}, 9: 53 11, 4, {9}: 53 11, {5}, 1: 661 11, 5, {1}: 617 11, {5}, 2: 0 11, 5, {2}: 9212117 11, {5}, 3: 0 11, 5, {3}: 641 11, {5}, 4: 59 11, 5, {4}: 59 11, {5}, 6: 61 11, 5, {6}: 61 11, {5}, 7: 80527 11, 5, {7}: 0 11, {5}, 8: 0 11, 5, {8}: 701 11, {5}, 9: 0 11, 5, {9}: 86381 11, {6}, 1: 67 11, 6, {1}: 67 11, {6}, 5: 71 11, 6, {5}: 71 11, {6}, 7: 73 11, 6, {7}: 73 11, {7}, 1: 42811363313890182397 11, 7, {1}: 859 11, {7}, 2: 79 11, 7, {2}: 79 11, {7}, 3: 0 11, 7, {3}: 883 11, {7}, 4: 0 11, 7, {4}: 108343 11, {7}, 5: 929 11, 7, {5}: 907 11, {7}, 6: 83 11, 7, {6}: 83 11, {7}, 8: 150051217 11, 7, {8}: 114199 11, {7}, 9: 0 11, 7, {9}: 115663 11, {7}, 10: 0 11, 7, {10}: 967 11, {8}, 1: 89 11, 8, {1}: 89 11, {8}, 3: 4447933850793785179 11, 8, {3}: 11047 11, {8}, 5: 1061 11, 8, {5}: 0 11, {8}, 7: 1063 11, 8, {7}: 11579 11, {8}, 9: 97 11, 8, {9}: 97 11, {9}, 1: 20902638977899027326901591016678209 11, 9, {1}: 0 11, {9}, 2: 101 11, 9, {2}: 101 11, {9}, 4: 103 11, 9, {4}: 103 11, {9}, 5: 1193 11, 9, {5}: 0 11, {9}, 7: 2122152919 11, 9, {7}: 0 11, {9}, 8: 107 11, 9, {8}: 107 11, {9}, 10: 109 11, 9, {10}: 109 11, {10}, 1: 1321 11, 10, {1}: 0 11, {10}, 3: 113 11, 10, {3}: 113 11, {10}, 7: 1327 11, 10, {7}: 0 11, {10}, 9: 14639 11, 10, {9}: 80662724392413945103199 12, {1}, 1: 13 12, 1, {1}: 13 12, {1}, 5: 17 12, 1, {5}: 17 12, {1}, 7: 19 12, 1, {7}: 19 12, {1}, 11: 23 12, 1, {11}: 23 12, {2}, 1: 313 12, 2, {1}: 3613 12, {2}, 5: 29 12, 2, {5}: 29 12, {2}, 7: 31 12, 2, {7}: 31 12, {2}, 11: 3779 12, 2, {11}: 431 12, {3}, 1: 37 12, 3, {1}: 37 12, {3}, 5: 41 12, 3, {5}: 41 12, {3}, 7: 43 12, 3, {7}: 43 12, {3}, 11: 47 12, 3, {11}: 47 12, {4}, 1: 7537 12, 4, {1}: 7069 12, {4}, 5: 53 12, 4, {5}: 53 12, {4}, 7: 631 12, 4, {7}: 8011 12, {4}, 11: 59 12, 4, {11}: 59 12, {5}, 1: 61 12, 5, {1}: 61 12, {5}, 7: 67 12, 5, {7}: 67 12, {5}, 11: 71 12, 5, {11}: 71 12, {6}, 1: 73 12, 6, {1}: 73 12, {6}, 5: 941 12, 6, {5}: 929 12, {6}, 7: 79 12, 6, {7}: 79 12, {6}, 11: 83 12, 6, {11}: 83 12, {7}, 1: 1093 12, 7, {1}: 1021 12, {7}, 5: 89 12, 7, {5}: 89 12, {7}, 11: 1103 12, 7, {11}: 1151 12, {8}, 1: 97 12, 8, {1}: 97 12, {8}, 5: 101 12, 8, {5}: 101 12, {8}, 7: 103 12, 8, {7}: 103 12, {8}, 11: 107 12, 8, {11}: 107 12, {9}, 1: 109 12, 9, {1}: 109 12, {9}, 5: 113 12, 9, {5}: 113 12, {9}, 7: 16963 12, 9, {7}: 16651 12, {9}, 11: 203591 12, 9, {11}: 1439 12, {10}, 1: 226201 12, 10, {1}: 1453 12, {10}, 7: 127 12, 10, {7}: 127 12, {10}, 11: 131 12, 10, {11}: 131 12, {11}, 1: 248821 12, 11, {1}: 1597 12, {11}, 5: 137 12, 11, {5}: 137 12, {11}, 7: 139 12, 11, {7}: 139 13, {1}, 1: 30941 13, 1, {1}: 30941 13, {1}, 2: 2381 13, 1, {2}: 197 13, {1}, 3: 883708283 13, 1, {3}: 211 13, {1}, 4: 17 13, 1, {4}: 17 13, {1}, 5: 0 13, 1, {5}: 239 13, {1}, 6: 19 13, 1, {6}: 19 13, {1}, 7: 0 13, 1, {7}: 253217502498750291800692183145337720992638880271493569431738157631027569095215561 13, {1}, 8: 0 13, 1, {8}: 281 13, {1}, 9: 191 13, 1, {9}: 27130132404659193376721686434661 13, {1}, 10: 23 13, 1, {10}: 23 13, {1}, 11: 193 13, 1, {11}: 820195757799727198696695842441476208994963187388611974376331352215246160014077762227387500472080168786083449145277773838858800795991958180632280974279961571375401216690067428169654353503039823288371270608465421195831630752643523277310127185899536482776399744897654945387104284638368882957170938237278749541598968947323011480936838588602269948473325034154837529102013688233548651627077006312693219100274314806851388327828512220445130387062692274032398838117351850000439212156970074507805886649022038706852334408911410638474707605620484783396663735375992701354072765197190488304749330337810593696686818871540035682031309739770552183078238960961 13, {1}, 12: 36898271981403391525359432679065451 13, 1, {12}: 337 13, {2}, 1: 4759 13, 2, {1}: 106637277112689077 13, {2}, 3: 29 13, 2, {3}: 29 13, {2}, 5: 31 13, 2, {5}: 31 13, {2}, 7: 804473 13, 2, {7}: 959173 13, {2}, 9: 373 13, 2, {9}: 0 13, {2}, 11: 37 13, 2, {11}: 37 13, {3}, 1: 547 13, 3, {1}: 521 13, {3}, 2: 41 13, 3, {2}: 41 13, {3}, 4: 43 13, 3, {4}: 43 13, {3}, 5: 6220138738168647434831423806501836269264860388724755065149565951442236175936796719067011700334477855412396475124991691344796424350196661821800919998163958711515136934394634097179800502201458093822871 13, 3, {5}: 577 13, {3}, 7: 0 13, 3, {7}: 2923035083 13, {3}, 8: 47 13, 3, {8}: 47 13, {3}, 10: 0 13, 3, {10}: 647 13, {3}, 11: 557 13, 3, {11}: 661 13, {4}, 1: 53 13, 4, {1}: 53 13, {4}, 3: 3534833123 13, 4, {3}: 9337 13, {4}, 5: 733 13, 4, {5}: 13799574804865291194219202692403868309075434457702185936718792737 13, {4}, 7: 59 13, 4, {7}: 59 13, {4}, 9: 61 13, 4, {9}: 61 13, {4}, 11: 739 13, 4, {11}: 0 13, {5}, 1: 911 13, 5, {1}: 859 13, {5}, 2: 67 13, 5, {2}: 67 13, {5}, 3: 4418541403 13, 5, {3}: 887 13, {5}, 4: 277256920492991599 13, 5, {4}: 11717 13, {5}, 6: 71 13, 5, {6}: 71 13, {5}, 7: 0 13, 5, {7}: 89921716241132417850870043495861784708189815671814466610282151602592361843582839405888101361717803180497433838124247307746249298923864405720133765156792108299126338169033221693415179426566058950783399084612596039386427021973466021161486411164221010228501523 13, {5}, 8: 73 13, 5, {8}: 73 13, {5}, 9: 919 13, 5, {9}: 971 13, {5}, 11: 3604339966408890811 13, 5, {11}: 815662743439 13, {5}, 12: 0 13, 5, {12}: 1013 13, {6}, 1: 79 13, 6, {1}: 79 13, {6}, 5: 83 13, 6, {5}: 83 13, {6}, 7: 14281 13, 6, {7}: 9624308578305020228623 13, {6}, 11: 89 13, 6, {11}: 89 13, {7}, 1: 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{9}, 17: 179 18, 9, {17}: 179 18, {10}, 1: 181 18, 10, {1}: 181 18, {10}, 7: 3968612127339681427 18, 10, {7}: 3373 18, {10}, 11: 191 18, 10, {11}: 191 18, {10}, 13: 193 18, 10, {13}: 193 18, {10}, 17: 197 18, 10, {17}: 197 18, {11}, 1: 199 18, 11, {1}: 199 18, {11}, 5: 3767 18, 11, {5}: 3659 18, {11}, 7: 3769 18, 11, {7}: 3697 18, {11}, 13: 211 18, 11, {13}: 211 18, {11}, 17: 3779 18, 11, {17}: 132239526911 18, {12}, 1: 140018322601 18, 12, {1}: 3907 18, {12}, 5: 74093 18, 12, {5}: 71699 18, {12}, 7: 223 18, 12, {7}: 223 18, {12}, 11: 227 18, 12, {11}: 227 18, {12}, 13: 229 18, 12, {13}: 229 18, {12}, 17: 233 18, 12, {17}: 233 18, {13}, 1: 4447 18, 13, {1}: 4231 18, {13}, 5: 239 18, 13, {5}: 239 18, {13}, 7: 241 18, 13, {7}: 241 18, {13}, 11: 4457 18, 13, {11}: 4421 18, {13}, 17: 251 18, 13, {17}: 251 18, {14}, 1: 4789 18, 14, {1}: 26565103 18, {14}, 5: 257 18, 14, {5}: 257 18, {14}, 11: 263 18, 14, {11}: 263 18, {14}, 13: 4801 18, 14, {13}: 4783 18, {14}, 17: 269 18, 14, {17}: 269 18, {15}, 1: 271 18, 15, {1}: 271 18, {15}, 7: 277 18, 15, {7}: 277 18, {15}, 11: 281 18, 15, {11}: 281 18, {15}, 13: 283 18, 15, {13}: 283 18, {15}, 17: 5147 18, 15, {17}: 30233087 18, {16}, 1: 32011489 18, 16, {1}: 30344239 18, {16}, 5: 293 18, 16, {5}: 293 18, {16}, 7: 5479 18, 16, {7}: 95713 18, {16}, 11: 5483 18, 16, {11}: 5393 18, {16}, 13: 32011501 18, 16, {13}: 5431 18, {16}, 17: 98801 18, 16, {17}: 5507 18, {17}, 1: 307 18, 17, {1}: 307 18, {17}, 5: 311 18, 17, {5}: 311 18, {17}, 7: 313 18, 17, {7}: 313 18, {17}, 11: 317 18, 17, {11}: 317 18, {17}, 13: 5827 18, 17, {13}: 9770144707511081415118442597789015238720654947319882836100223544506052645981243442054558121499672250712069138857313219 
20201213, 10:49  #28  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
3398_{10} Posts 
Quote:
Base 5: {1}3: covering set {2,3} {1}4: covering set {2,3} 3{1}: covering set {2,3} 4{1}: covering set {2,3} Base 9: {1}1: full algebra factors: (9^n1)/8 = (3^n1)*(3^n+1)/8 1{1}: full algebra factors: (9^n1)/8 = (3^n1)*(3^n+1)/8 {1}5: covering set {2,5} 2{7}: covering set {2,5} 3{1}: full algebra factors: (25*9^n1)/8 = (5*3^n1)*(5*3^n+1)/8 {3}5: covering set {2,5} {3}8: covering set {2,5} 3{8}: full algebra factors: 4*9^n1 = (2*9^n1)*(2*9^n+1) 5{1}: covering set {2,5} 5{7}: covering set {2,5} 6{1}: covering set {2,5} {7}2: covering set {2,5} {7}5: covering set {2,5} {8}5: full algebra factors: 9^n4 = (3^n2)*(3^n+2) Base 11: {1}3: covering set {2,3} {1}4: covering set {2,3} {1}9: covering set {2,3} {1}A: covering set {2,3} 2{5}: covering set {2,3} 3{1}: covering set {2,3} 3{5}: covering set {2,3} 3{7}: covering set {2,3} 4{1}: covering set {2,3} 4{7}: covering set {2,3} {5}2: covering set {2,3} {5}3: covering set {2,3} 5{7}: (unsolved family)  {5}8: covering set {2,3} {5}9: covering set {2,3} {7}3: covering set {2,3} {7}4: covering set {2,3} {7}9: covering set {2,3} {7}A: covering set {2,3} 8{5}: covering set {2,3} 9{1}: covering set {2,3} 9{5}: covering set {2,3} 9{7}: covering set {2,3} A{1}: covering set {2,3} A{7}: covering set {2,3} Base 13: {1}5: covering set {2,5,17} {1}7: covering set {2,7} {1}8: covering set {2,7} 2{9}: covering set {2,7} {3}7: covering set {2,7} {3}A: covering set {2,7} 4{B}: covering set {2,7} {5}7: covering set {2,7} {5}C: covering set {2,7} 7{1}: covering set {2,7} 7{3}: covering set {2,7} 7{5}: covering set {2,7} 7{9}: covering set {2,7} 7{B}: covering set {2,7} 8{1}: covering set {2,7} {9}2: covering set {2,7} 9{5}: (unsolved family)  {9}7: covering set {2,7} A{3}: covering set {2,7} {B}4: covering set {2,7} {B}7: covering set {2,7} C{5}: covering set {2,7} Base 14: {2}5: covering set {3,5} 3{D}: covering set {3,5} {4}9: covering set {3,5} 4{D}: (smallest prime is 4D_{19698} = 5*14^196981) 5{B}: covering set {3,5} 6{1}: covering set {3,5} 6{B}: covering set {3,5} {8}3: covering set {3,5} {8}5: covering set {3,5} 8{D}: partial algebra factors: 9*14^n1 = (3*14^(n/2)1)*(3*14^(n/2)+1) for even n, divisible by 5 for odd n A{1}: covering set {3,5} A{D}: covering set {3,5} B{1}: partial algebra factors: (144*14^n1)/13 = (12*14^(n/2)1)*(12*14^(n/2)+1)/13 for even n, divisible by 5 for odd n {B}5: covering set {3,5} {D}3: covering set {3,5} {D}5: partial algebra factors: 14^n9 = (14^(n/2)3)*(14^(n/2)+3) for even n, divisible by 5 for odd n Base 16: 1{5}: full algebra factors: (4*16^n1)/3 = (2*4^n1)*(2*4^n+1)/3 {4}1: full algebra factors: (4*16^n49)/15 = (2*4^n7)*(2*4^n+7)/15 {4}D: covering set {3,7,13} 7{3}: full algebra factors: (36*16^n1)/5 = (6*4^n1)*(6*4^n+1)/5 8{1}: full algebra factors: (121*16^n1)/15 = (11*4^n1)*(11*4^n+1)/15 8{5}: full algebra factors: (25*16^n1)/3 = (5*4^n1)*(5*4^n+1)/3 {8}F: covering set {3,7,13} 8{F}: full algebra factors: 9*16^n1 = (3*4^n1)*(3*4^n+1) {C}B: full algebra factors: (4*16^n9)/5 = (2*4^n3)*(2*4^n+3)/5 {C}D: full algebra factors: (4*16^n+1)/5 = (2*4^n2*2^n+1)*(2*4^n+2*2^n+1) D{1}: full algebra factors: (196*16^n1)/15 = (14*4^n1)*(14*4^n+1)/15 D{B}: (unsolved family)  E{1}: covering set {3,7,13} {F}7: full algebra factors: 16^n9 = (4^n3)*(4^n+3) Last fiddled with by sweety439 on 20201213 at 11:04 

20201215, 14:04  #29 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2·1,699 Posts 
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 nearrepdigit primes base b already excluded families x{y} and {x}y with NUMERICAL covering set (e.g. {1}3, {1}4, 3{1}, 4{1} in base 5) Such primes are ALWAYS minimal prime (start with 2 digits) in base b, except when the repeating digit (i.e. y in x{y}, or x in {x}y) is 1 and base b has generalized repunit primes (i.e. all digits are 1) smaller than the prime (in base b = 9, 25, 32, 49, 64, 81, 121, 125, 144, 169, 216, 225, 243, 289, 324, 343, 361, 441, 484, 529, 625, ..., no generalized repunit primes exist, thus in these bases b, such primes are always minimal primes (start with 2 digits) in base b) Also, * in base 35, all such primes with <= 313 digits are minimal primes (start with 2 digits) * in base 39, all such primes with <= 349 digits are minimal primes (start with 2 digits) * in base 47, all such primes with <= 127 digits are minimal primes (start with 2 digits) * in base 51, all such primes with <= 4229 digits are minimal primes (start with 2 digits) * in base 91, all such primes with <= 4421 digits are minimal primes (start with 2 digits) * in base 92, all such primes with <= 439 digits are minimal primes (start with 2 digits) * in base 124, all such primes with <= 599 digits are minimal primes (start with 2 digits) * in base 135, all such primes with <= 1171 digits are minimal primes (start with 2 digits) * in base 139, all such primes with <= 163 digits are minimal primes (start with 2 digits) * in base 142, all such primes with <= 1231 digits are minimal primes (start with 2 digits) * in base 152, all such primes with <= 270217 digits are minimal primes (start with 2 digits) * in base 171, all such primes with <= 181 digits are minimal primes (start with 2 digits) * in base 174, all such primes with <= 3251 digits are minimal primes (start with 2 digits) * in base 182, all such primes with <= 167 digits are minimal primes (start with 2 digits) * in base 183, all such primes with <= 223 digits are minimal primes (start with 2 digits) * in base 184, all such primes with <= 16703 digits are minimal primes (start with 2 digits) * in base 185, all such primes with <= 66337 (at least) digits are minimal primes (start with 2 digits) * in base 199, all such primes with <= 577 digits are minimal primes (start with 2 digits) * in base 200, all such primes with <= 17807 digits are minimal primes (start with 2 digits) * in base 201, all such primes with <= 271 digits are minimal primes (start with 2 digits) Last fiddled with by sweety439 on 20201215 at 14:13 
20201215, 16:06  #30 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2×1,699 Posts 
In fact, all these primes are minimal primes (start with 2 digits) base b:
* Smallest generalized repunit prime base b (if exists) * The primes for all k<b for CRUS Sierpinski conjecture base b * The primes for all k<b for CRUS Riesel conjecture base b * Smallest generalized nearrepdigit primes base b of the form x{y} or {x}y for all (x,y) digit pair (if exists) Since .... * Generalized repunit numbers base b are 111...111 in base b * k<b for CRUS Sierpinski conjecture base b are [k]000...0001 in base b * k<b for CRUS Riesel conjecture base b are [k1][b1][b1][b1]...[b1][b1][b1] in base b * Generalized nearrepdigit numbers base b of the form x{y} or {x}y are [x][y][y][y]...[y][y][y] or [x][x][x]...[x][x][x][y] in base b 
20201215, 16:12  #31 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
110101000110_{2} Posts 
I think that the "minimal primes (start with 2 digits) problem" for all bases b>90 will never be proven (when searched to 1M or 1G or even 1T base b digits, and even when we allow probable primes in place of proven primes.), like CRUS S/R280, S/R511, S/R855, and S/R910 problems and the "Sierpinski/Riesel twin prime conjecture" (the conjecture that 237 is the smallest k divisible by 3 such that k*2^n+1 are not twin primes for all n>=1), for more information, see this post
Last fiddled with by sweety439 on 20201215 at 16:18 
20201217, 08:40  #32  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2×1,699 Posts 
Quote:


20201226, 03:14  #33  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
D46_{16} Posts 
Quote:
** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,7): ** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, we only need to consider the families 60{1,4,7}7, 6{0}17, 6{0,4}4{4}7, 6{0}77 (since any digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them will produce smaller primes) **** All numbers of the form 6{0}17 or 6{0}77 are divisible by 3, thus cannot be prime. **** For the 60{1,4,7}7 family, since 117, 147, 177, 417, 6477, 717, 747, 6777 are primes, we only need to consider the numbers 6017, 6047, 6077 and the family 60{4}7 (since any digit combo 11, 14, 17, 41, 47, 71, 74, 77 between (60,7) will produce smaller primes) ***** None of 6017, 6047, 6077 are primes. **** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime. **** For the 6{0,4}4{4}7 family, since 6007 and 407 are primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digit combo 00, 40 between (6,4{4}7) will produce smaller primes) ***** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime. ***** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime. Last fiddled with by sweety439 on 20201227 at 06:13 

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