20220717, 16:01  #364  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
3577_{10} Posts 
Quote:
Sum of all minimal primes (start with b+1) in base b Product of all minimal primes (start with b+1) in base b Base 2: 1 primes, totally 2 digits, sum, product Base 3: 3 primes, totally 7 digits, sum, product Base 4: 5 primes, totally 11 digits, sum, product Base 5: 22 primes, totally 169 digits, sum, product Base 6: 11 primes, totally 29 digits, sum, product Base 7: 71 primes, totally 288 digits, sum, product Base 8: 75 primes, totally 523 digits, sum, product Base 9: 151 primes, totally 3004 digits, sum, product Base 10: 77 primes, totally 310 digits, sum, product Base 11: 1068 primes, totally 75414 digits, sum, product Base 12: 106 primes, totally 433 digits, sum, product Base 14: 650 primes, totally 25404 digits, sum, product Base 15: 1284 primes, totally 8286 digits, sum, product Base 18: Conjecture: the sum of all minimal primes (start with b+1) base b is always in https://oeis.org/A063538, i.e. it must have a prime factor >= its square root, this has been verified for bases 2, 3, 4, 5, 6, 7, 8, 10, 12, 15, but this is very hard to prove or disprove, since proving or disproving this requires factoring large numbers. Last fiddled with by sweety439 on 20220801 at 05:52 

20220720, 08:03  #365 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
DF9_{16} Posts 
Two special cases of minimal primes (start with b+1) base b
* lovely numbers base b: Let d_1 and d_2 be digits in base b such that d_1 + d_2 = b, find the smallest prime of the form d_1*b^n+d_2 with n >= 1, this prime is always minimal prime (start with b+1) base b, this includes the special cases d_1 = 1, d_2 = b1 (which is b^n+(b1), see https://oeis.org/A076845) and d_1 = b1, d_2 = 1 (which is (b1)*b^n+1, see https://www.rieselprime.de/ziki/Williams_prime_MP_least and https://www.rieselprime.de/ziki/Williams_prime_MP_table and https://oeis.org/A305531) ** such prime is always expected to exist as there cannot be covering congruence nor algebraic factorization (since if so, then d_1, d_2, b will be all rth powers for an odd r > 1, which is impossible (by Fermat Last Theorem) * flexible numbers base b: Let d be a divisor (>1) of b1 (if b1 is prime, then d can only be b1 itself), find the smallest prime of the form ((d1)*b^n+1)/d with n >= 2, this prime is always minimal prime (start with b+1) base b ** for the case d = 2, such prime may not exist and it is widely believed that there are only finitely many such primes for fixed base b, since it is generalized half Fermat prime in base b ** for the case d > 2, such prime are usually expected to exist (as there cannot be covering congruence of this form, but there may be algebraic factorization or combine of covering congruence and algebraic factorization if d1 is indeed perfect odd power (of the form m^r with odd r > 1) or of the form 4*m^4, and if d1 is of neither of these two forms, then there must be prime of this form), but the smallest such prime may be large, e.g. Code:
b,d,smallest exponent n 13,12,564 17,8,190 23,11,3762 31,6,1026 43,14,580 70,69,555 Last fiddled with by sweety439 on 20220721 at 09:40 
20220730, 02:10  #366 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
7^{2}·73 Posts 
Quote:
(only list families which must produce minimal primes (start with b+1)) Family {1}: (not exist for bases in https://oeis.org/A096059) Base 2: 11, length 2, decimal 3, index 1 Base 3: 111, length 3, decimal 13, index 3 Base 4: 11, length 2, decimal 5, index 1 Base 5: 111, length 3, decimal 31, index 8 Base 6: 11, length 2, decimal 7, index 1 Base 7: 11111, length 5, decimal 2801, index 53 Base 8: 111, length 3, decimal 73, index 16 Base 9: not exist Base 10: 11, length 2, decimal 11, index 1 Base 11: 11111111111111111, length 17, decimal 50544702849929377, index 975 Base 12: 11, length 2, decimal 13, index 1 Base 13: 11111, length 5, decimal 30941, index 494 Base 14: 111, length 3, decimal 211, index 40 Base 15: 111, length 3, decimal 241, index 43 Base 16: 11, length 2, decimal 17, index 1 Base 17: 111, length 3, decimal 307, index 56 Base 18: 11, length 2, decimal 19, index 1 Base 19: 1111111111111111111, length 19, decimal 109912203092239643840221, index 29382 Base 20: 111, length 3, decimal 421, index 73 Base 21: 111, length 3, decimal 463, index 78 Base 22: 11, length 2, decimal 23, index 1 Family 1{0}1: (not exist for bases == 1 mod 2 and bases in https://oeis.org/A070265) Base 2: 11, length 2, decimal 3, index 1 Base 3: not exist Base 4: 11, length 2, decimal 5, index 1 Base 5: not exist Base 6: 11, length 2, decimal 7, index 1 Base 7: not exist Base 8: not exist Base 9: not exist Base 10: 11, length 2, decimal 11, index 1 Base 11: not exist Base 12: 11, length 2, decimal 13, index 1 Base 13: not exist Base 14: 101, length 3, decimal 197, index 39 Base 15: not exist Base 16: 11, length 2, decimal 17, index 1 Base 17: not exist Base 18: 11, length 2, decimal 19, index 1 Base 19: not exist Base 20: 101, length 3, decimal 401, index 71 Base 21: not exist Base 22: 11, length 2, decimal 23, index 1 Family 2{0}1: (not exist for bases == 1 mod 3) Base 2: not interpretable (base 2 has no digit "2") Base 3: 21, length 2, decimal 7, index 2 Base 4: not exist Base 5: 21, length 2, decimal 11, index 2 Base 6: 21, length 2, decimal 13, index 3 Base 7: not exist Base 8: 21, length 2, decimal 17, index 3 Base 9: 21, length 2, decimal 19, index 4 Base 10: not exist Base 11: 21, length 2, decimal 23, index 4 Base 12: 2001, length 4, decimal 3457, index 58 Base 13: not exist Base 14: 21, length 2, decimal 29, index 4 Base 15: 21, length 2, decimal 31, index 5 Base 16: not exist Base 17: 200000000000000000000000000000000000000000000001, length 48, decimal 13555929465559461990942712143872578804076607708197374744547, index 10094 Base 18: 21, length 2, decimal 37, index 5 Base 19: not exist Base 20: 21, length 2, decimal 41, index 5 Base 21: 21, length 2, decimal 43, index 6 Base 22: not exist Family 1{z}: ((conjectured) exist in all bases) Base 2: 11, length 2, decimal 3, index 1 Base 3: 12, length 2, decimal 5, index 1 Base 4: 13, length 2, decimal 7, index 2 Base 5: 14444, length 5, decimal 1249, index 16 Base 6: 15, length 2, decimal 11, index 2 Base 7: 16, length 2, decimal 13, index 2 Base 8: 177, length 3, decimal 127, index 21 Base 9: 18, length 2, decimal 17, index 3 Base 10: 19, length 2, decimal 19, index 4 Base 11: 1AA, length 3, decimal 241, index 37 Base 12: 1B, length 2, decimal 23, index 4 Base 13: 1CC, length 3, decimal 337, index 48 Base 14: 1DDDD, length 5, decimal 76831, index 233 Base 15: 1E, length 2, decimal 29, index 4 Base 16: 1F, length 2, decimal 31, index 5 Base 17: 1GG, length 3, decimal 577, index 86 Base 18: 1HH, length 3, decimal 647, index 66 Base 19: 1I, length 2, decimal 37, index 4 Base 20: 1JJJJJJJJJJ, length 11, decimal 20479999999999, index 3015 Base 21: 1K, length 2, decimal 41, index 5 Base 22: 1L, length 2, decimal 43, index 6 Family 3{0}1: (not exist for bases == 1 mod 2) Base 2: not interpretable (base 2 has no digit "3") Base 3: not interpretable (base 3 has no digit "3") Base 4: 31, length 2, decimal 13, index 4 Base 5: not exist Base 6: 31, length 2, decimal 19, index 5 Base 7: not exist Base 8: 301, length 3, decimal 193, index 24 Base 9: not exist Base 10: 31, length 2, decimal 31, index 7 Base 11: not exist Base 12: 31, length 2, decimal 37, index 7 Base 13: not exist Base 14: 31, length 2, decimal 43, index 8 Base 15: not exist Base 16: 301, length 3, decimal 769, index 69 Base 17: not exist Base 18: 3001, length 4, decimal 17497, index 195 Base 19: not exist Base 20: 31, length 2, decimal 61, index 10 Base 21: not exist Base 22: 31, length 2, decimal 67, index 11 Family 2{z}: (not exist for bases == 1 mod 2) Base 2: not interpretable (base 2 has no digit "2") Base 3: not exist Base 4: 23, length 2, decimal 11, index 3 Base 5: not exist Base 6: 25, length 2, decimal 17, index 4 Base 7: not exist Base 8: 27, length 2, decimal 23, index 5 Base 9: not exist Base 10: 29, length 2, decimal 29, index 6 Base 11: not exist Base 12: 2BB, length 3, decimal 431, index 34 Base 13: not exist Base 14: 2D, length 2, decimal 41, index 7 Base 15: not exist Base 16: 2F, length 2, decimal 47, index 9 Base 17: not exist Base 18: 2H, length 2, decimal 53, index 9 Base 19: not exist Base 20: 2J, length 2, decimal 59, index 9 Base 21: not exist Base 22: 2LL, length 3, decimal 1451, index 127 Family 4{0}1: (not exist for bases == 1 mod 5 and bases == 14 mod 15 and bases which are 4th powers) Base 2: not interpretable (base 2 has no digit "4") Base 3: not interpretable (base 3 has no digit "4") Base 4: not interpretable (base 4 has no digit "4") Base 5: 401, length 3, decimal 101, index 12 Base 6: not exist Base 7: 41, length 2, decimal 29, index 6 Base 8: 401, length 3, decimal 257, index 27 Base 9: 41, length 2, decimal 37, index 8 Base 10: 41, length 2, decimal 41, index 9 Base 11: not exist Base 12: 401, length 3, decimal 577, index 35 Base 13: 41, length 2, decimal 53, index 10 Base 14: not exist Base 15: 41, length 2, decimal 61, index 12 Base 16: not exist Base 17: 4000001, length 7, decimal 96550277, index 5138 Base 18: 41, length 2, decimal 73, index 14 Base 19: 4001, length 4, decimal 27437, index 748 Base 20: 401, length 3, decimal 1601, index 120 Base 21: not exist Base 22: 41, length 2, decimal 89, index 16 Family 3{z}: (not exist for bases == 1 mod 3 and bases == 4 mod 5 and bases which are squares) Base 2: not interpretable (base 2 has no digit "3") Base 3: not interpretable (base 3 has no digit "3") Base 4: not exist Base 5: 34, length 2, decimal 19, index 5 Base 6: 35, length 2, decimal 23, index 6 Base 7: not exist Base 8: 37, length 2, decimal 31, index 7 Base 9: not exist Base 10: not exist Base 11: 3A, length 2, decimal 43, index 9 Base 12: 3B, length 2, decimal 47, index 10 Base 13: not exist Base 14: not exist Base 15: 3E, length 2, decimal 59, index 11 Base 16: not exist Base 17: 3G, length 2, decimal 67, index 12 Base 18: 3H, length 2, decimal 71, index 13 Base 19: not exist Base 20: 3J, length 2, decimal 79, index 14 Base 21: 3K, length 2, decimal 83, index 15 Base 22: not exist Family z{0}1: ((conjectured) exist in all bases) Base 2: 11, length 2, decimal 3, index 1 Base 3: 21, length 2, decimal 7, index 2 Base 4: 31, length 2, decimal 13, index 4 Base 5: 401, length 3, decimal 101, index 12 Base 6: 51, length 2, decimal 31, index 8 Base 7: 61, length 2, decimal 43, index 10 Base 8: 701, length 3, decimal 449, index 39 Base 9: 81, length 2, decimal 73, index 17 Base 10: 9001, length 4, decimal 9001, index 56 Base 11: A0000000001, length 11, decimal 259374246011, index 905 Base 12: B001, length 4, decimal 19009, index 84 Base 13: C1, length 2, decimal 157, index 31 Base 14: D01, length 3, decimal 2549, index 120 Base 15: E1, length 2, decimal 211, index 41 Base 16: F1, length 2, decimal 241, index 47 Base 17: G0001, length 5, decimal 1336337, index 3039 Base 18: H1, length 2, decimal 307, index 56 Base 19: I00000000000000000000000000001, length 30, decimal 218336795902605993201009018384568383223, index 30322 Base 20: J00000000000001, length 15, decimal 31129600000000000001, index 3160 Base 21: K1, length 2, decimal 421, index 74 Base 22: L1, length 2, decimal 463, index 82 Family y{z}: ((conjectured) exist in all bases) Base 2: not interpretable (family should have leading zeros or trailing zeros) Base 3: 12, length 2, decimal 5, index 1 Base 4: 23, length 2, decimal 11, index 3 Base 5: 34, length 2, decimal 19, index 5 Base 6: 45, length 2, decimal 29, index 7 Base 7: 56, length 2, decimal 41, index 9 Base 8: 6777, length 4, decimal 3583, index 55 Base 9: 78, length 2, decimal 71, index 16 Base 10: 89, length 2, decimal 89, index 20 Base 11: 9A, length 2, decimal 109, index 24 Base 12: AB, length 2, decimal 131, index 27 Base 13: BCC, length 3, decimal 2027, index 176 Base 14: CD, length 2, decimal 181, index 36 Base 15: DEEEEEEEEEEEEEE, length 15, decimal 408700964355468749, index 1252 Base 16: EF, length 2, decimal 239, index 46 Base 17: FG, length 2, decimal 271, index 51 Base 18: GHH, length 3, decimal 5507, index 178 Base 19: HIIIIII, length 7, decimal 846825857, index 17286 Base 20: IJ, length 2, decimal 379, index 67 Base 21: JK, length 2, decimal 419, index 73 Base 22: KL, length 2, decimal 461, index 81 Family 1{0}2: (not exist for bases == 0 mod 2 and bases == 1 mod 3) Base 2: not interpretable (base 2 has no digit "2") Base 3: 12, length 2, decimal 5, index 1 Base 4: not exist Base 5: 12, length 2, decimal 7, index 1 Base 6: not exist Base 7: not exist Base 8: not exist Base 9: 12, length 2, decimal 11, index 1 Base 10: not exist Base 11: 12, length 2, decimal 13, index 1 Base 12: not exist Base 13: not exist Base 14: not exist Base 15: 12, length 2, decimal 17, index 1 Base 16: not exist Base 17: 12, length 2, decimal 19, index 1 Base 18: not exist Base 19: not exist Base 20: not exist Base 21: 12, length 2, decimal 23, index 1 Base 22: not exist Family 1{0}z: ((conjectured) exist in all bases) Base 2: 11, length 2, decimal 3, index 1 Base 3: 12, length 2, decimal 5, index 1 Base 4: 13, length 2, decimal 7, index 2 Base 5: 104, length 3, decimal 29, index 7 Base 6: 15, length 2, decimal 11, index 2 Base 7: 16, length 2, decimal 13, index 2 Base 8: 107, length 3, decimal 71, index 15 Base 9: 18, length 2, decimal 17, index 3 Base 10: 19, length 2, decimal 19, index 4 Base 11: 10A, length 3, decimal 131, index 26 Base 12: 1B, length 2, decimal 23, index 4 Base 13: 10C, length 3, decimal 181, index 34 Base 14: 1000000000000000D, length 17, decimal 2177953337809371149, index 606 Base 15: 1E, length 2, decimal 29, index 4 Base 16: 1F, length 2, decimal 31, index 5 Base 17: 1000G, length 5, decimal 83537, index 1348 Base 18: 100H, length 4, decimal 5849, index 185 Base 19: 1I, length 2, decimal 37, index 4 Base 20: 10J, length 3, decimal 419, index 72 Base 21: 1K, length 2, decimal 41, index 5 Base 22: 1L, length 2, decimal 43, index 6 Family {z}1: ((conjectured) exist in all bases) Base 2: 11, length 2, decimal 3, index 1 Base 3: 21, length 2, decimal 7, index 2 Base 4: 31, length 2, decimal 13, index 4 Base 5: 44441, length 5, decimal 3121, index 20 Base 6: 51, length 2, decimal 31, index 8 Base 7: 61, length 2, decimal 43, index 10 Base 8: 7777777777771, length 13, decimal 549755813881, index 73 Base 9: 81, length 2, decimal 73, index 17 Base 10: 991, length 3, decimal 991, index 44 Base 11: AA1, length 3, decimal 1321, index 111 Base 12: BBBB1, length 5, decimal 248821, index 97 Base 13: C1, length 2, decimal 157, index 31 Base 14: DD1, length 3, decimal 2731, index 131 Base 15: E1, length 2, decimal 211, index 41 Base 16: F1, length 2, decimal 241, index 47 Base 17: GGGGGGGGGG1, length 11, decimal 34271896307617, index 8834 Base 18: H1, length 2, decimal 307, index 56 Base 19: II1, length 3, decimal 6841, index 496 Base 20: JJJJJJJJJJJJJJJJ1, length 17, decimal 13107199999999999999981, index 3185 Base 21: K1, length 2, decimal 421, index 74 Base 22: L1, length 2, decimal 463, index 82 Family {z}y: (not exist for bases == 0 mod 2) Base 2: not interpretable (family should have leading zeros or trailing zeros) Base 3: 21, length 2, decimal 7, index 2 Base 4: not exist Base 5: 43, length 2, decimal 23, index 6 Base 6: not exist Base 7: 65, length 2, decimal 47, index 11 Base 8: not exist Base 9: 87, length 2, decimal 79, index 18 Base 10: not exist Base 11: AAA9, length 4, decimal 14639, index 227 Base 12: not exist Base 13: CB, length 2, decimal 167, index 33 Base 14: not exist Base 15: ED, length 2, decimal 223, index 42 Base 16: not exist Base 17: GGGGGF, length 6, decimal 24137567, index 4999 Base 18: not exist Base 19: IH, length 2, decimal 359, index 64 Base 20: not exist Base 21: KJ, length 2, decimal 439, index 77 Base 22: not exist Last fiddled with by sweety439 on 20220903 at 20:13 
20220730, 02:38  #367  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
7^{2}·73 Posts 
Quote:
Base 2: start with 1: 11 (length 2) end with 1: 11 (length 2) Base 3: start with 1: 111 (length 3) start with 2: 21 (length 2) end with 1: 111 (length 3) end with 2: 12 (length 2) Base 4: start with 1: 11, 13 (length 2) start with 2: 221 (length 3) start with 3: 31 (length 2) end with 1: 221 (length 3) end with 3: 13, 23 (length 2) Base 5: start with 1: 10_{93}13 (length 96) start with 2: 21, 23 (length 2) start with 3: 300031 (length 6) start with 4: 44441 (length 5) end with 1: 300031 (length 6) end with 2: 12, 32 (length 2) end with 3: 10_{93}13 (length 96) end with 4: 14444 (length 5) Base 6: start with 1: 11, 15 (length 2) start with 2: 21, 25 (length 2) start with 3: 31, 35 (length 2) start with 4: 40041 (length 5) start with 5: 51 (length 2) end with 1: 40041 (length 5) end with 5: 15, 25, 35, 45 (length 2) Base 7: start with 1: 1100021 (length 7) start with 2: 2111 (length 4) start with 3: 33333333333333331 (length 17) start with 4: 40054 (length 5) start with 5: 5100000001 (length 10) start with 6: 6034, 6634 (length 4) end with 1: 33333333333333331 (length 17) end with 2: 1022, 1112, 1202, 1222 (length 4) end with 3: 300053 (length 6) end with 4: 40054 (length 5) end with 5: 35555 (length 5) end with 6: 346 (length 3) Base 8: start with 1: 107, 111, 117, 141, 147, 161, 177 (length 3) start with 2: 225, 255 (length 3) start with 3: 3344441 (length 7) start with 4: 4_{220}7 (length 221) start with 5: 555555555555525 (length 15) start with 6: 60171, 60411, 60741 (length 5) start with 7: 7777777777771 (length 13) end with 1: 7777777777771 (length 13) end with 3: 4043, 4443 (length 4) end with 5: 555555555555525 (length 15) end with 7: 4_{220}7 (length 221) Base 9: start with 1: 1000000000000000000000000057 (length 28) start with 2: 27_{686}07 (length 689) start with 3: 30_{1158}11 (length 1161) start with 4: 438 (length 3) start with 5: 56111111111111111111111111111111111111 (length 38) start with 6: 631111 (length 6) start with 7: 76_{329}2 (length 331) start with 8: 8888888888888888888335 (length 22) end with 1: 30_{1158}11 (length 1161) end with 2: 76_{329}2 (length 331) end with 4: 544444444444 (length 12) end with 5: 8888888888888888888335 (length 22) end with 7: 27_{686}07 (length 689) end with 8: 33388 (length 5) Base 10: start with 1: 11, 13, 17, 19 (length 2) start with 2: 22000001 (length 8) start with 3: 349 (length 3) start with 4: 409, 449, 499 (length 3) start with 5: 5000000000000000000000000000027 (length 31) start with 6: 60000049, 66000049, 66600049 (length 8) start with 7: 727, 757, 787 (length 3) start with 8: 80555551 (length 8) start with 9: 946669 (length 6) end with 1: 555555555551 (length 12) end with 3: 13, 23, 43, 53, 73, 83 (length 2) end with 7: 5000000000000000000000000000027 (length 31) end with 9: 60000049, 66000049, 66600049 (length 8) Base 11: start with 1: 10_{125}51 (length 128) start with 2: 2888882883, 2888888883 (length 10) start with 3: 326_{122} (length 124) start with 4: 44777777777777777777777777777777777777777777777777777777777777777 (length 65) start with 5: 57_{62668} (length 62669) start with 6: 6000000000000083 (length 16) start with 7: 7_{759}44 (length 761) start with 8: 85_{220}05 (length 223) start with 9: 99777777777777777777777777777777777777777777777777777777777777777 (length 65) start with A: A_{713}58 (length 715) end with 1: 10_{125}51 (length 128) end with 2: 5555555555555555555555A52 (length 25) end with 3: 5_{119}053 (length 123) end with 4: 7_{759}44 (length 761) end with 5: 85_{220}05 (length 223) end with 6: 326_{122} (length 124) end with 7: 57_{62668} (length 62669) end with 8: A_{713}58 (length 715) end with 9: 90000000000000000000000000000000000000009799 (length 44) end with A: 5_{161}2A (length 163) Base 12: start with 1: 11, 15, 17, 1B (length 2) start with 2: 2001, 200B, 202B, 222B, 229B, 292B, 299B (length 4) start with 3: 31, 35, 37, 3B (length 2) start with 4: 400000000000000000000000000000000000000077 (length 42) start with 5: 565 (length 3) start with 6: 600A5 (length 5) start with 7: 7999B (length 5) start with 8: 81, 85, 87, 8B (length 2) start with 9: 9999B (length 5) start with A: AA000001 (length 8) start with B: B0000000000000000000000000009B (length 30) end with 1: AA000001 (length 8) end with 5: A00065 (length 6) end with 7: 400000000000000000000000000000000000000077 (length 42) end with B: B0000000000000000000000000009B (length 30) Last fiddled with by sweety439 on 20220730 at 06:14 

20220730, 02:45  #368 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
7^{2}×73 Posts 
upload data files (zipped), for bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 24, 30, 36
after unzip them, you need to rename "kernel b" to "kernel b.txt" and rename "left b" to "left b.txt", "kernel b" is the data for all known minimal primes (start with b+1) in base b, and "left b" is the data for all unsolved families in base b search limits for the unsolved families: base 13 family 9{5} at length 115000 base 13 family A{3}A at length 111000 base 16 family {3}AF at length 98000 all bases 17, 21, 36 families at length 20000 Last fiddled with by sweety439 on 20220730 at 02:51 
20220802, 07:49  #369 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
DF9_{16} Posts 
Just let you know, I know exactly what bases 2 <= b <= 1024 have these families unsolved: (at length 100000) (also exactly what bases 2 <= b <= 1024 have these families proven as only contain composites (only count the numbers > base (b)), by covering congruence, algebraic factorization, or combine of them)
{1} 1{0}1 2{0}1 3{0}1 4{0}1 5{0}1 6{0}1 7{0}1 8{0}1 9{0}1 A{0}1 B{0}1 C{0}1 1{z} 2{z} 3{z} 4{z} 5{z} 6{z} 7{z} 8{z} 9{z} A{z} B{z} z{0}1 y{z} Also these families, but only at length 20000: 1{0}2 1{0}3 1{0}4 {z}w {z}x {z}y 1{0}z {z}1 {y}z 
20220810, 18:56  #370  
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
7^{2}×73 Posts 
Quote:
Code:
4,3,2 5,4,2 6,5,2 7,3,3 7,6,2 8,7,3 9,4,2 9,8,2 10,3,2 10,9,2 11,5,2 11,10,2 12,11,2 13,3,2 13,4,2 13,6,3 13,12,564 14,13,2 15,7,2 15,14,10 16,3,3 16,5,(not exist, Aurifeuillian factorization of x^4+4*y^4) 16,15,2 17,4,9 17,8,190 17,16,2 18,17,4 19,3,2 19,6,78 19,9,13 19,18,14 20,19,2 21,4,2 21,5,2 21,10,2 21,20,2 22,3,6 22,7,3 22,21,2 23,11,3762 23,22,8 24,23,4 25,3,4 25,4,3 25,6,2 25,8,2 25,12,3 25,24,2 26,5,2 26,25,5 27,13,2 27,26,2 28,3,2 28,9,4 28,27,3 29,4,2 29,7,4 29,14,6 29,28,2 30,29,6 31,3,2 31,5,2 31,6,1026 31,10,24 31,15,99 31,30,2 32,31,2 33,4,3 33,8,2 33,16,2 33,32,252 34,3,3 34,11,2 34,33,3 35,17,2 35,34,20 36,5,45 36,7,5 36,35,2 37,3,3 37,4,6 37,6,4 37,9,2 37,12,4 37,18,12 37,36,6 38,37,4 39,19,3 39,38,2 40,3,3 40,13,3 40,39,2 41,4,3 41,5,6 41,8,2 41,10,15 41,20,2 41,40,4 42,41,2 43,3,12 43,6,38 43,7,4 43,14,580 43,21,3 43,42,24 44,43,3 45,4,28 45,11,5 45,22,2 45,44,2 46,3,3 46,5,2 46,9,7 46,15,3 46,45,2 47,23,2 47,46,2 48,47,4 49,3,2 49,4,2 49,6,3 49,8,8 49,12,26 49,16,2 49,24,4 49,48,2 50,7,2 50,49,3 51,5,2 51,10,2 51,25,5 51,50,2 52,3,3 52,17,5 52,51,17 53,4,4 53,13,2 53,26,4 53,52,24 54,53,2 55,3,2 55,6,2 55,9,2 55,18,2 55,27,6 55,54,2 56,5,78 56,11,2 56,55,2 57,4,2 57,7,5 57,8,2 57,14,6 57,28,44 57,56,2 58,3,2 58,19,2 58,57,3 59,29,2 59,58,4 60,59,2 61,3,10 61,4,2 61,5,6 61,6,3 61,10,4 61,12,6 61,15,11 61,20,70 61,30,4 61,60,2 62,61,4 63,31,3 63,62,4 64,3,2 64,7,2 64,9,(not exist, sumoftwocubes factorization) 64,21,24 64,63,11 65,4,2 65,8,2 65,16,5 65,32,2 65,64,2 66,5,15 66,13,2 66,65,2 67,3,6 67,6,6 67,11,19 67,22,(>10000) 67,33,3 67,66,2 68,67,5 69,4,2 69,17,2 69,34,2 69,68,2 70,3,4 70,23,11 70,69,555 71,5,22 71,7,3 71,10,836 71,14,14 71,35,6 71,70,2 72,71,5 73,3,4 73,4,4 73,6,2 73,8,2 73,9,28 73,12,4 73,18,9 73,24,2 73,36,85 73,72,8 74,73,10 75,37,24 75,74,12 76,3,2 76,5,2 76,15,3 76,25,3 76,75,3 77,4,2 77,19,6 77,38,4 77,76,2 78,7,8 78,11,2 78,77,3 79,3,61 79,6,162 79,13,4 79,26,8 79,39,213 79,78,6 80,79,24 81,4,3 81,5,(not exist, Aurifeuillian factorization of x^4+4*y^4) 81,8,2 81,10,29 81,16,2 81,20,5 81,40,2 81,80,4 82,3,2 82,9,10 82,27,3 82,81,6 83,41,104 83,82,680 84,83,2 85,3,2 85,4,2 85,6,13 85,7,4 85,12,6 85,14,2 85,21,57 85,28,2 85,42,10 85,84,6 86,5,6 86,17,2 86,85,2 87,43,2 87,86,2 88,3,28 88,29,2 88,87,3 89,4,6 89,8,6 89,11,288 89,22,2 89,44,2 89,88,132 90,89,2 91,3,2 91,5,9 91,6,4 91,9,5 91,10,36 91,15,4 91,18,4 91,30,8 91,45,16 91,90,140 92,7,51 92,13,4 92,91,4 93,4,156 93,23,2 93,46,2 93,92,4 94,3,51 94,31,12 94,93,2 95,47,8 95,94,2 96,5,3 96,19,2 96,95,84 97,3,7 97,4,2 97,6,2 97,8,2 97,12,??? 97,16,2 97,24,9 97,32,4 97,48,28 97,96,2 98,97,137 99,7,6 99,14,6 99,49,2 99,98,6 100,3,3 100,9,7 100,11,2 100,33,2 100,99,5 Last fiddled with by sweety439 on 20220812 at 09:00 

20220813, 13:13  #371 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
7^{2}·73 Posts 
Results for bases b>16
Base 17 is searched to length 32100 Base 18 is proven (including the primality of the primes) Base 19 is searched to length 20000 Last fiddled with by sweety439 on 20220813 at 13:19 
20220813, 13:16  #372 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
3577_{10} Posts 
Base 20 is proven (including the primality of the primes)
Base 21 is searched to length 20000 Base 22 is proven except the primality of the large strong probable prime B(K^22001)5 Base 23 is now reserved Base 24 is proven (including the primality of the primes) Last fiddled with by sweety439 on 20220813 at 13:18 
20220813, 13:19  #373 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
6771_{8} Posts 
Base 30 is proven except the primality of the large strong probable prime I(0^24608)D
Base 36 is searched to length 20000 
20220813, 17:17  #374 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
7^{2}×73 Posts 
The base 19 unsolved family 5{H}5 is very low weight (or difficulty) but eventually should yield a prime (see http://factordb.com/index.php?query=...at=1&sent=Show)
Its formula is (107*19^(n+1)233)/18 (since neither 107 nor 233 is perfect power (https://oeis.org/A001597), this family has no algebraic factors) * n == 0 mod 2: factor of 2 (also factor of 4, 5, 10, 20) this only left n == 1 mod 2 * n == 1 mod 3: factor of 3 this only left n == 3, 5 mod 6 * n == 5 mod 6: factor of 7 this only left n == 3 mod 6 * n == 9 mod 12: factor of 13 this only left n == 3 mod 12 * n == 3 mod 8: factor of 17 this only left n == 15 mod 24 the n = 15 number is divisible by many small primes (11, 29, 47, 71), but all of these primes (p) have large and nonsmooth order (znorder(Mod(19,p))), the n = 39 number is divisible by 281, and the n = 63 and 87 numbers have no small prime factors, the n = 111 number is divisible by 89 Base 26 is fully searched to length 20000, and there are 25250 known minimal (probable) primes (start with b+1) and 9 unsolved families, and base 28 is technically fully searched to length 543203 (if we allow probable primes in place of proven primes, see http://www.kurims.kyotou.ac.jp/EMIS...rs/i61/i61.pdf (see section 3: Recent Results Establishing the Mixed Sierpinski Theorem) and https://oeis.org/history?seq=A004023&start=50 (see M. F. Hasler's discussion in the pink box)), and there are 25528 known minimal (probable) primes (start with b+1), and the only one unsolved family is O{A}F (see https://github.com/xayahrainie4793/quasimepndata and https://github.com/curtisbright/mepn...a/sieve.28.txt), interestingly, in base 28 there are only 3 known minimal primes (start with b+1) (and it is likely totally 4 minimal primes (start with b+1)) with length > 5271: N(6^24051)LR, 5O(A^31238)F, O4(O^94535)9, and more interestingly, base 28 seems to be high weight base, but there are many families (which must be minimal primes (start with b+1) in all bases b, if these families are interpretable in this base b) whose smallest length (to make the number prime) (see https://docs.google.com/spreadsheets...RwmKME/pubhtml) set records: 3*b^n+1 (3{0}1, length 8, index 19858), b^n3 ({z}x, which is {R}P in base 28, length 10, index 23827), b^n5 ({z}v, which is {R}N in base 28, length 60, index 25401), (b/2)*b^n1 (#{z}, which is D{R} in base 28, length 48, index 25367), {2}1 (length 40, index 25337) Last fiddled with by sweety439 on 20220906 at 22:43 
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