20211128, 17:20  #221 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
110010001000_{2} Posts 
These are all minimal primes (start with b+1) in base b=25 up to 2^32
Base 25 is a very hard base (however, of course, bases > 25 which are coprime to 6 are harder than it), we can imagine an alien force, vastly more powerful than us, landing on Earth and demanding the set of minimal primes (start with b+1) in base b=17 (or 19, 21, 22, 23, 28, 30, 36) (including primality proving of all primes in this set) or they will destroy our planet. In that case, I claim, we should marshal all our computers and all our mathematicians and attempt to find the set and to prove the primality of all numbers in this set. But suppose, instead, that they ask for the set of minimal primes (start with b+1) in base b=25 (or 26, 27, 29, 31, 32, 33, 34, 35). In that case, I believe, we should attempt to destroy the aliens. Last fiddled with by sweety439 on 20211211 at 18:48 
20211128, 17:23  #222  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2^{3}×401 Posts 
Quote:
very important note: they are minimal primes (start with b'+1) in base b'=b^2 instead of base b'=b Last fiddled with by sweety439 on 20211128 at 17:25 

20211130, 19:59  #223 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
3208_{10} Posts 
more datas

20211130, 20:41  #224  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
6210_{8} Posts 
Quote:
The string is of the form {0}S{0,2,4,5,6,8} and S is of one of these forms: * empty string * singledigit number * "gcd of its digits" <> 1 (note: gcd(0,n) = n for all n, including n=0) * X{0}Y with X+Y divisible by 3 (this includes: 2{0}1, 2{0}7, 5{0}1, 5{0}7, 8{0}1, 8{0}7) * 28{0}7 * 4{6}9 * 221 * 2021 * 2201 * 22001 * 220001 * 2200001 * (5^n)1 with n<11 * 581 * 5(0^n)27 with n<28 * 5207 * 52007 * 520007 * 649 * 6649 * 66649 * 6049 * 60049 * 600049 * 6000049 * 66049 * 660049 * 6600049 * 666049 * 6660049 * 8(5^n)1 with n<11 * 8051 * 80551 * 805551 * 8055551 * 91 * 901 * 921 * 951 * 981 * 9021 * 9051 * 9081 * 9201 * 9501 * 9581 * 9801 * 90581 * 949 * 9469 * 94669 reference: https://math.stackexchange.com/quest...enumberinit Last fiddled with by sweety439 on 20211220 at 21:44 Reason: remove 95081 from the list since 5081 is prime 

20211130, 20:52  #225 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2^{3}·401 Posts 
Update data for minimal primes, see https://sites.google.com/view/dataofminimalprimes

20211206, 10:14  #226  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2^{3}×401 Posts 
Quote:
1{0}2 b^n+2: http://www.primenumbers.net/prptop/s...&action=Search 1{0}3 b^n+3: http://www.primenumbers.net/prptop/s...&action=Search 1{0}4 b^n+4: http://www.primenumbers.net/prptop/s...&action=Search {z}w b^n4: http://www.primenumbers.net/prptop/s...&action=Search {z}x b^n3: http://www.primenumbers.net/prptop/s...&action=Search {z}y b^n2: http://www.primenumbers.net/prptop/s...&action=Search 

20211206, 13:17  #227 
"Matthew Anderson"
Dec 2010
Oregon, USA
1111110000_{2} Posts 
another message _ message transfer to small corner of net
Apprciation is felt for your effort @sweety439 and all

20211207, 14:15  #228  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
2^{3}·401 Posts 
Quote:
b=8: 1:{0}:1, ruled out as only contain composite numbers 2:{0}:1, prime at length 2 4:{0}:1, prime at length 3 b=27: {13}:14, ruled out as only contain composite numbers 1:{13}:14, prime at length 2, also 1:{13} has prime at length 3 4:{13}:14, prime at length 11, also 4:{13} has prime at length 24 b=32: 1:{0}:1, ruled out as only contain composite numbers 2:{0}:1, prime at length 4 4:{0}:1, unsolved family 8:{0}:1, prime at length 2 16:{0}:1, unsolved family b=64: 1:{0}:1, ruled out as only contain composite numbers 4:{0}:1, prime at length 2 16:{0}:1, prime at length 3 b=125: {62}:63, ruled out as only contain composite numbers 2:{62}:63, prime at length 2 12:{62}:63, unsolved family b=128: 1:{0}:1, ruled out as only contain composite numbers 2:{0}:1, prime at length 2 4:{0}:1, prime at length 3 8:{0}:1, ruled out as only contain composite numbers 16:{0}:1, unsolved family 32:{0}:1, ruled out as only contain composite numbers 64:{0}:1, ruled out as only contain composite numbers b=216: 1:{0}:1, ruled out as only contain composite numbers 6:{0}:1, prime at length 2 36:{0}:1, unsolved family b=243: {121}:122, ruled out as only contain composite numbers 1:{121}:122, prime at length 4, also 1:{121} has prime at length 15 4:{121}:122, prime at length 7, also 4:{121} has prime at length 2 13:{121}:122, although no known prime or PRP in this family, but 13:{121} has prime at length 3, thus 13:{121}:122 is still not unsolved family since the smallest prime (if exists) in this family will not be minimal prime (start with b+1) 40:{121}:122, prime at length 13, however 40:{121} has no known prime or PRP, and 40:{121} is unsolved family (however, 40:{121} is GRU family instead of GFN family, for reference of this family, see https://oeis.org/A028491, there are no single known number in https://oeis.org/A028491 which is == 4 mod 5) b=343: {171}:172, ruled out as only contain composite numbers 3:{171}:172, prime at length 2 24:{171}:172, unsolved family b=512: 1:{0}:1, ruled out as only contain composite numbers 2:{0}:1, unsolved family 4:{0}:1, unsolved family 8:{0}:1, ruled out as only contain composite numbers 16:{0}:1, unsolved family 32:{0}:1, unsolved family 64:{0}:1, ruled out as only contain composite numbers 128:{0}:1, prime at length 2 256:{0}:1, unsolved family b=729: {364}:365, ruled out as only contain composite numbers 4:{364}:365, prime at length 6 40:{364}:365, prime at length 3 b=1000: 1:{0}:1, ruled out as only contain composite numbers 10:{0}:1, unsolved family 100:{0}:1, unsolved family b=1024: 1:{0}:1, ruled out as only contain composite numbers 4:{0}:1, unsolved family 16:{0}:1, unsolved family 64:{0}:1, prime at length 2 256:{0}:1, unsolved family Last fiddled with by sweety439 on 20211211 at 15:34 

20211208, 08:21  #229  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
110010001000_{2} Posts 
Quote:
{0}S{0,2,4,5,6,8} with these sets S: * {0,3,6,9} * {0,7} * 2{0}1 * 5{0}1 * 5{0}7 * 8{0}1 * 28{0}7 * 4{6}9 * 2021 * 2200001 * (5^10)1 * 581 * 5(0^27)27 * 520007 * 6000049 * 6600049 * 6660049 * 8(5^10)1 * 8055551 * 9021 * 9201 * 9801 * 90581 * 94669 (note: {0,2,4,6,8} and {0,5} are already included in the digits after S, i.e. {0,2,4,5,6,8}) (note: 2{0}7 and 8{0}7 are already subfamilies of 28{0}7) (note: 66649 is already subsequence of 6660049) (note: 581, 9051, 9081, 9501, 9581 are already subsequences of 90581 or/and 95081) Last fiddled with by sweety439 on 20211220 at 21:45 Reason: remove 95081 from the list since 5081 is prime 

20211208, 08:29  #230 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
C88_{16} Posts 
and we have these "maximal primefree strings"
b=2: * {0}1{0} b=3: * {0}1{0}1{0} * {0,2} b=4: * {0}2{0}1{0,2} * {0}{0,3}{0,2} b=10: * {0}{0,3,6,9}{0,2,4,5,6,8} * {0}{0,7}{0,2,4,5,6,8} * {0}2{0}1{0,2,4,5,6,8} * {0}5{0}1{0,2,4,5,6,8} * {0}5{0}7{0,2,4,5,6,8} * {0}8{0}1{0,2,4,5,6,8} * {0}28{0}7{0,2,4,5,6,8} * {0}4{6}9{0,2,4,5,6,8} * {0}2021{0,2,4,5,6,8} * {0}2200001{0,2,4,5,6,8} * {0}55555555551{0,2,4,5,6,8} * {0}500000000000000000000000000027{0,2,4,5,6,8} * {0}520007{0,2,4,5,6,8} * {0}6000049{0,2,4,5,6,8} * {0}6600049{0,2,4,5,6,8} * {0}6660049{0,2,4,5,6,8} * {0}855555555551{0,2,4,5,6,8} * {0}8055551{0,2,4,5,6,8} * {0}9021{0,2,4,5,6,8} * {0}9201{0,2,4,5,6,8} * {0}9801{0,2,4,5,6,8} * {0}90581{0,2,4,5,6,8} * {0}95081{0,2,4,5,6,8} * {0}94669{0,2,4,5,6,8} Last fiddled with by sweety439 on 20211208 at 14:20 
20211210, 16:59  #231 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
C88_{16} Posts 
These factor pattern can show that such families contain no primes > base:
Reference: the divisibility rule for base b: * For prime p dividing b, the number is divisible by p if and only if the last digit of this number is divisible by p. * For prime p dividing b1, the number is divisible by p if and only if the sum of the digits of this number is divisible by p. * For prime p dividing b+1, the number is divisible by p if and only if the alternating sum of the digits of this number is divisible by p. (this can also show that all palindromic primes in any base b have an odd number of digits, the only possible exception is "11" in base b) (in these examples, only list the numbers > base) Example 1: base 10, family 4{6}9 Code:
49 = 7 * 7 469 = 7 * 67 4669 = 7 * 667 46669 = 7 * 6667 466669 = 7 * 66667 4666669 = 7 * 666667 Code:
287 = 7 * 41 2807 = 7 * 401 28007 = 7 * 4001 280007 = 7 * 40001 2800007 = 7 * 400001 28000007 = 7 * 4000001 Code:
11 = 2 * 5 111 = 7 * 14 1111 = 22 * 45 11111 = 67 * 144 111111 = 222 * 445 1111111 = 667 * 1444 11111111 = 2222 * 4445 111111111 = 6667 * 14444 1111111111 = 22222 * 44445 11111111111 = 66667 * 144444 111111111111 = 222222 * 444445 1111111111111 = 666667 * 1444444 Code:
38 = 5 * 7 388 = 18 * 21 3888 = 58 * 61 38888 = 188 * 201 388888 = 588 * 601 3888888 = 1888 * 2001 38888888 = 5888 * 6001 388888888 = 18888 * 20001 3888888888 = 58888 * 60001 38888888888 = 188888 * 200001 388888888888 = 588888 * 600001 Code:
11 = 3 * 3 101 = 5 * 15 1001 = 11 * 71 10001 = 21 * 361 100001 = 41 * 1741 1000001 = 101 * 7701 10000001 = 201 * 37601 100000001 = 401 * 177401 1000000001 = 1001 * 777001 10000000001 = 2001 * 3776001 100000000001 = 4001 * 17774001 1000000000001 = 10001 * 77770001 Code:
25 = 3 * 9 255 = 2 * 128 2555 = 3 * 919 25555 = 2 * 12828 255555 = 3 * 91919 2555555 = 2 * 1282828 25555555 = 3 * 9191919 255555555 = 2 * 128282828 2555555555 = 3 * 919191919 25555555555 = 2 * 12828282828 Code:
9B = 7 * 15 B9B = 11 * AB BB9B = B7 * 105 BBB9B = 11 * B0AB BBBB9B = BB7 * 1005 BBBBB9B = 11 * B0B0AB BBBBBB9B = BBB7 * 10005 BBBBBBB9B = 11 * B0B0B0AB BBBBBBBB9B = BBBB7 * 100005 BBBBBBBBB9B = 11 * B0B0B0B0AB Code:
B1 = 5 * 23 B01 = 3 * 395 B001 = 5 * 22B3 B0001 = 3 * 39495 B00001 = 5 * 22B2B3 B000001 = 3 * 3949495 B0000001 = 5 * 22B2B2B3 B00000001 = 3 * 394949495 B000000001 = 5 * 22B2B2B2B3 B0000000001 = 3 * 39494949495 Code:
395 = 14 * 2B 3095 = 7 * 58A 30095 = 5 * 7A71 300095 = 7 * 5758A 3000095 = 14 * 23A92B 30000095 = 7 * 575758A 300000095 = 5 * 7A527A71 3000000095 = 7 * 57575758A 30000000095 = 14 * 23A923A92B 300000000095 = 7 * 5757575758A 3000000000095 = 5 * 7A527A527A71 Code:
4D = 7 * B 44D = 3 * 16F 444D = D * 541 4444D = 7 * 9C0B 44444D = 3 * 16C16F 444444D = D * 540541 4444444D = 7 * 9C09C0B 44444444D = 3 * 16C16C16F 444444444D = D * 540540541 4444444444D = 7 * 9C09C09C0B 44444444444D = 3 * 16C16C16C16F 444444444444D = D * 540540540541 Code:
19 = 2 * D 199 = B * 27 1999 = 2 * D4D 19999 = AB * 287 199999 = 2 * D4D4D 1999999 = AAB * 2887 19999999 = 2 * D4D4D4D 199999999 = AAAB * 28887 1999999999 = 2 * D4D4D4D4D 19999999999 = AAAAB * 288887 199999999999 = 2 * D4D4D4D4D4D 1999999999999 = AAAAAB * 2888887 Code:
OZ = T * V OZZ = 4Z * 51 OZZZ = TZ * U1 OZZZZ = 4ZZ * 501 OZZZZZ = TZZ * U01 OZZZZZZ = 4ZZZ * 5001 OZZZZZZZ = TZZZ * U001 OZZZZZZZZ = 4ZZZZ * 50001 OZZZZZZZZZ = TZZZZ * U0001 OZZZZZZZZZZ = 4ZZZZZ * 500001 OZZZZZZZZZZZ = TZZZZZ * U00001 OZZZZZZZZZZZZ = 4ZZZZZZ * 5000001 http://www.worldofnumbers.com/wing.htm for: {1}0{1} (base 10): Code:
101 = 1 * 101 (the only possible prime case) 11011 = 11 * 1001 1110111 = 111 * 10001 111101111 = 1111 * 100001 11111011111 = 11111 * 1000001 1111110111111 = 111111 * 10000001 Code:
121 = 11 * 11 11211 = 101 * 111 1112111 = 1001 * 1111 111121111 = 10001 * 11111 11111211111 = 100001 * 111111 1111112111111 = 1000001 * 1111111 Code:
323 = 17 * 19 33233 = 167 * 199 3332333 = 1667 * 1999 333323333 = 16667 * 19999 33333233333 = 166667 * 199999 3333332333333 = 1666667 * 1999999 Code:
343 = 7 * 49 33433 = 67 * 499 3334333 = 667 * 4999 333343333 = 6667 * 49999 33333433333 = 66667 * 499999 3333334333333 = 666667 * 4999999 1{2}1 (base 10): Code:
11 = 11 * 1 (the only possible prime case) 121 = 11 * 11 1221 = 11 * 111 12221 = 11 * 1111 122221 = 11 * 11111 1222221 = 11 * 111111 12222221 = 11 * 1111111 Code:
77 = 11 * 7 737 = 11 * 67 7337 = 11 * 667 73337 = 11 * 6667 733337 = 11 * 66667 7333337 = 11 * 666667 73333337 = 11 * 6666667 Code:
99 = 11 * 9 979 = 11 * 89 9779 = 11 * 889 97779 = 11 * 8889 977779 = 11 * 88889 9777779 = 11 * 888889 97777779 = 11 * 8888889 Code:
99 = 11 * 9 949 = 13 * 73 9449 = 11 * 859 94449 = 3 * 31483 944449 = 11 * 85859 9444449 = 7 * 1349207 94444449 = 11 * 8585859 944444449 = 13 * 72649573 9444444449 = 11 * 858585859 94444444449 = 3 * 31481481483 944444444449 = 11 * 85858585859 9444444444449 = 7 * 1349206349207 94444444444449 = 11 * 8585858585859 37{1} (base 10): Code:
37 = 37 * 1 (the only possible prime case) 371 = 7 * 53 3711 = 3 * 1237 37111 = 37 * 1003 371111 = 13 * 28547 3711111 = 3 * 1237037 37111111 = 37 * 1003003 371111111 = 7 * 53015873 3711111111 = 3 * 1237037037 37111111111 = 37 * 1003003003 371111111111 = 13 * 28547008547 3711111111111 = 3 * 1237037037037 37111111111111 = 37 * 1003003003003 Code:
38 = 2 * 19 381 = 3 * 127 3811 = 37 * 103 38111 = 23 * 1657 381111 = 3 * 127037 3811111 = 37 * 103003 38111111 = 233 * 163567 381111111 = 3 * 127037037 3811111111 = 37 * 103003003 38111111111 = 2333 * 16335667 381111111111 = 3 * 127037037037 3811111111111 = 37 * 103003003003 38111111111111 = 23333 * 1633356667 Last fiddled with by sweety439 on 20211219 at 23:20 
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