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2021-01-14, 01:21
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#133 |
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"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
5·7·83 Posts |
Last fiddled with by sweety439 on 2021-06-02 at 07:10 |
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2021-01-14, 03:53
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#134 |
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"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
B5916 Posts |
These are families I am interested: (of the form (a*b^n+c)/gcd(a+c,b-1) for fixed a>=1, b>=2, c != 0, gcd(a,c) = 1, gcd(b,c) = 1 and variable n) (although some of these families do not always product minimal primes (start with b+1))
(for bases 2<=b<=1024) * (b^n-1)/(b-1) * b^n+1 for b == 0 mod 2 * (b^n+1)/2 for b == 1 mod 2 * b^n+2 for b == 3, 5 mod 6 * (b^n+2)/3 for b == 1 mod 6 * b^n+3 for b == 2, 4 mod 6 * (b^n+3)/2 for b == 1, 5 mod 6 * b^n+4 for b == 3, 5, 7, 9 mod 10 (b not == 14 mod 15, b not perfect 4th power) * (b^n+4)/5 for b == 1 mod 10 (b not perfect 4th power) * b^n-2 for b == 1 mod 2 * b^n-3 for b == 2, 4 mod 6 * (b^n-3)/2 for b == 1, 5 mod 6 * b^n-4 for b == 3, 5 mod 6 (b not == 4 mod 5, b not perfect square) * (b^n-4)/3 for b == 1 mod 6 (b not perfect square) * 2*b^n+1 for b == 0, 2 mod 3 * (2*b^n+1)/3 for b == 1 mod 3 * 3*b^n+1 for b == 0 mod 2 * (3*b^n+1)/2 for b == 1 mod 2 * 4*b^n+1 for b == 0, 2, 3, 4 mod 5 (b not == 14 mod 15, b not perfect 4th power) * (4*b^n+1)/5 for b == 1 mod 5 (b not perfect 4th power) * 2*b^n-1 * 3*b^n-1 for b == 0 mod 2 * (3*b^n-1)/2 for b == 1 mod 2 * 4*b^n-1 for b == 0, 2 mod 3 (b not == 4 mod 5, b not perfect square) * (4*b^n-1)/3 for b == 1 mod 3 (b not == 4 mod 5, b not perfect square) * b^n+5 * b^n+6 * b^n+7 * b^n+8 * b^n+9 * b^n+10 * b^n+11 * b^n+12 * b^n+13 * b^n+14 * b^n+15 * b^n+16 * b^n-5 * b^n-6 * b^n-7 * b^n-8 * b^n-9 * b^n-10 * b^n-11 * b^n-12 * b^n-13 * b^n-14 * b^n-15 * b^n-16 * 5*b^n+1 * 6*b^n+1 * 7*b^n+1 * 8*b^n+1 * 9*b^n+1 * 10*b^n+1 * 11*b^n+1 * 12*b^n+1 * 13*b^n+1 * 14*b^n+1 * 15*b^n+1 * 16*b^n+1 * 5*b^n-1 * 6*b^n-1 * 7*b^n-1 * 8*b^n-1 * 9*b^n-1 * 10*b^n-1 * 11*b^n-1 * 12*b^n-1 * 13*b^n-1 * 14*b^n-1 * 15*b^n-1 * 16*b^n-1 * 2*b^n+3 * 2*b^n-3 * 3*b^n+2 * 3*b^n-2 * 3*b^n+4 * 3*b^n-4 * 4*b^n+3 * 4*b^n-3 * {1}2 in base b * {1}3 in base b * {1}4 in base b * {2}1 in base b * {2}3 in base b * {3}1 in base b * {3}2 in base b * {3}4 in base b * {4}1 in base b * {4}3 in base b * 1{2} in base b * 1{3} in base b * 1{4} in base b * 2{1} in base b * 2{3} in base b * 3{1} in base b * 3{2} in base b * 3{4} in base b * 4{1} in base b * 4{3} in base b * (b/2)*b^n+1 for b == 0, 2 mod 6 * (b/2)*b^n-1 for b == 0 mod 2 * (3*b/2)*b^n+1 for b == 0, 2, 4, 8 mod 10 * (3*b/2)*b^n-1 for b == 0 mod 2 * (b/3)*b^n+1 for b == 0 mod 6 * (b/3)*b^n-1 for b == 0 mod 6 * (2*b/3)*b^n+1 for b == 0, 3, 9, 12 mod 15 * (2*b/3)*b^n-1 for b == 0 mod 3 * (4*b/3)*b^n+1 for b == 0, 3, 6, 9, 12, 18 mod 21 * (4*b/3)*b^n-1 for b == 0 mod 3 * (b/4)*b^n+1 for b == 0, 4, 8, 12 mod 20 (b not == 14 mod 15, b not perfect 4th power) * (b/4)*b^n-1 for b == 0, 8 mod 12 (b not == 4 mod 5, b not perfect square) * (3*b/4)*b^n+1 for b == 0, 4, 12, 16, 20, 24 mod 28 * (3*b/4)*b^n-1 for b == 0 mod 4 * b^n+(b-1) * b^n-(b-1) * b^n+(b+1) for b == 0, 2 mod 3 * (b^n+(b+1))/3 for b == 1 mod 3 * b^n-(b+1) * (b-1)*b^n+1 * (b-1)*b^n-1 * (b+1)*b^n+1 for b == 0, 2 mod 3 * ((b+1)*b^n+1)/3 for b == 1 mod 3 * (b+1)*b^n-1 * (b^n+(b-2))/(b-1) * ((b-2)*b^n+1)/(b-1) * (b^n-(2*b-1))/(b-1) * ((2*b-1)*b^n-1)/(b-1) * (b-2)*b^n-1 for b == 0 mod 2 * (b+2)*b^n+1 for b == 0 mod 2 * (b+2)*b^n-1 for b == 0 mod 2 * (b*(b^2)^n+1)/(b+1) [this is the special case, original form is (b^n+1)/(b+1), but we should write the family as standard form ((a*b^n+c)/gcd(a+c,b-1) for fixed a>=1, b>=2, c != 0, gcd(a,c) = 1, gcd(b,c) = 1 and variable n)] Last fiddled with by sweety439 on 2021-07-15 at 10:22 |
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2021-01-14, 04:50
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#135 |
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"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
5·7·83 Posts |
Length of largest minimal prime (start with b+1) in base b for b = 2, 3, 4, ..., 40:
2, 3, 3, 96, 5, >=17, 221, >=1161, 31, >=45, 42, >=32021, >=19699, >=107, >=32235, >=111334, >=300, >=110986, >=449, >=479150, >=764, >=800874, >=315, >=136967, >=8773, >=109006, >=94538, >=174240, >=34206, >=9896, >=9750, >=23617, >=9377, >=9599, >=81995, >=22023, >=136212, >=9440, >=12383 (in sequences below, 0 means no such prime exists, Italic type means either not minimal prime (start with b+1) in base b or not acceptable as the form will produce a digit >=b or <0 in base b) Length of the smallest repunit prime (form: {1}) in base b for b = 2, 3, 4, ..., 160: 2, 3, 2, 3, 2, 5, 3, 0, 2, 17, 2, 5, 3, 3, 2, 3, 2, 19, 3, 3, 2, 5, 3, 0, 7, 3, 2, 5, 2, 7, 0, 3, 13, 313, 2, 13, 3, 349, 2, 3, 2, 5, 5, 19, 2, 127, 19, 0, 3, 4229, 2, 11, 3, 17, 7, 3, 2, 3, 2, 7, 3, 5, 0, 19, 2, 19, 5, 3, 2, 3, 2, 5, 5, 3, 41, 3, 2, 5, 3, 0, 2, 5, 17, 5, 11, 7, 2, 3, 3, 4421, 439, 7, 5, 7, 2, 17, 13, 3, 2, 3, 2, 19, 97, 3, 2, 17, 2, 17, 3, 3, 2, 23, 29, 7, 59, 3, 5, 3, 5, 0, 5, 43, 599, 0, 2, 5, 7, 5, 2, 3, 47, 13, 5, 1171, 2, 11, 2, 163, 79, 3, 1231, 3, 0, 5, 7, 3, 2, 7, 2, 13, 270217, 3, 5, 3, 2, 17, 7, 13, 7 Length of the smallest generalized Fermat prime (form: 1{0}1) in base b for b = 2, 3, 4, ..., 160: 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 3, 0, 2, 0, 2, 0, 3, 0, 2, 0, 3, 0, 3, 0, 2, 0, 2, 0, 0, 0, 5, 0, 2, 0, (>=16777217 or 0), 0, 2, 0, 2, 0, 17, 0, 2, 0, 5, 0, (>=16777217 or 0), 0, 2, 0, 3, 0, 3, 0, 2, 0, 2, 0, (>=16777217 or 0), 0, 0, 0, 2, 0, (>=16777217 or 0), 0, 2, 0, 2, 0, 3, 0, 17, 0, 2, 0, 5, 0, 2, 0, 3, 0, (>=16777217 or 0), 0, 2, 0, 3, 0, (>=16777217 or 0), 0, 3, 0, 2, 0, (>=16777217 or 0), 0, 2, 0, 2, 0, (>=16777217 or 0), 0, 2, 0, 2, 0, 3, 0, 2, 0, 33, 0, 3, 0, 5, 0, 3, 0, (>=16777217 or 0), 0, 3, 0, 2, 0, 0, 0, 2, 0, 5, 0, 3, 0, 2, 0, 2, 0, 5, 0, 5, 0, (>=16777217 or 0), 0, 3, 0, 2, 0, 2, 0, 9, 0, 5, 0, 2, 0, 17, 0, 3 Length of the smallest generalized half Fermat prime (form: {x}y, x = (b-1)/2, y = (b+1)/2) in base b for b = 2, 3, 4, ..., 160: 0, 2, 0, 2, 0, 4, 0, 2, 0, 2, 0, 4, 0, 2, 0, 4, 0, 2, 0, 4, 0, 4, 0, 2, 0, 0, 0, 2, 0, (>=524288 or 0), 0, 8, 0, 2, 0, (>=524288 or 0), 0, 2, 0, 16, 0, 8, 0, 2, 0, 8, 0, 2, 0, 2, 0, 8, 0, (>=524288 or 0), 0, 4, 0, 2, 0, 2, 0, (>=524288 or 0), 0, 2, 0, (>=524288 or 0), 0, 2, 0, 2, 0, 4, 0, 32, 0, (>=524288 or 0), 0, 2, 0, 4, 0, (>=524288 or 0), 0, 2, 0, 16, 0, (>=524288 or 0), 0, (>=524288 or 0), 0, (>=524288 or 0), 0, 2, 0, (>=524288 or 0), 0, (>=524288 or 0), 0, 2, 0, 4, 0, 4, 0, (>=524288 or 0), 0, (>=524288 or 0), 0, 16, 0, 4, 0, 4, 0, (>=524288 or 0), 0, 4, 0, 2, 0, (>=524288 or 0), 0, 0, 0, (>=524288 or 0), 0, 4, 0, 2, 0, (>=524288 or 0), 0, (>=524288 or 0), 0, (>=524288 or 0), 0, 2, 0, 2, 0, (>=524288 or 0), 0, 2, 0, (>=524288 or 0), 0, (>=524288 or 0), 0, (>=524288 or 0), 0, 4, 0, (>=524288 or 0), 0, 16, 0, 2, 0 Length of the smallest Williams prime of the 1st kind (form: x{y}, x = b-2, y = b-1) in base b for b = 2, 3, 4, ..., 160: (the b=2 case is not acceptable, since the digit 2-2 = 0 in base 2 cannot be leading digit) 3, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 3, 2, 15, 2, 2, 3, 7, 2, 2, 2, 56, 13, 2, 134, 2, 21, 2, 3, 2, 2, 3, 16, 4, 2, 8, 136212, 2, 2, 8, 2, 8, 8, 2, 2, 2, 3, 2, 26, 2, 6, 4, 2, 2, 2, 2, 3, 4, 2, 2, 900, 4, 12, 2, 2, 2, 64, 2, 14, 2, 26, 9, 4, 3, 8, 2, 45, 3, 12, 4, 82, 21496, 2, 3, 2, 2, 4, 26, 2, 520, 78, 477, 2, 2, 3, 2, 4984, 3, 3, 2, 2, 4, 2, 4, 3, 38, 411, 7, 6, 3, 8, 286644, 3, 2, 2, 3, 3, 4, 3, 2, 4, 7, 34, 8740, 2, 2, (>2220000 or 0), 3, 9, 2, 2, 3, 4, 2, 6, 26, 3, 2, 24, 2, 2, 8, 3, 2, 2, 6, 4, 2, 2, 4, 4, 3, 2, 2, 2, 4, 128, 2, 2 Length of the smallest Williams prime of the 2nd kind (form: x{0}1, x = b-1) in base b for b = 2, 3, 4, ..., 160: 2, 2, 2, 3, 2, 2, 3, 2, 4, 11, 4, 2, 3, 2, 2, 5, 2, 30, 15, 2, 2, 15, 3, 2, 3, 5, 2, 3, 5, 6, 13, 3, 2, 3, 3, 10, 17, 2, 3, 81, 2, 3, 5, 3, 4, 17, 3, 3, 3, 2, 16, 961, 16, 2, 5, 4, 2, 15, 2, 7, 21, 2, 4, 947, 7, 2, 19, 11, 2, 5, 2, 6, 43, 5, 2, 829, 2, 2, 3, 2, 13, 3, 7, 5, 31, 4, 3023, 3, 2, 2, 9, 3, 5, 5, 3, 12, 9, 3, 2, 3, 2, 57, 3, 13, 2, 5, 6, 16, 3, 2, 2, 5, 4, 3, 17, 4, 2, 47, 2, 3, 6217, (>400000 or 0), 3, 17, 5, 166, 73, 6, 65, 15, 2, 3, 51, 3, 280, 13, 3, 2, 3, 7, 2, 5, 2, 4, 5, 5, 2, 3, 15, 2, 9, 5, 2, 7, 2, 30, 1621, 17, 6 Length of the smallest Williams prime of the 4th kind (form: 11{0}1) in base b for b = 2, 3, 4, ..., 160: (not minimal prime if there is smaller prime of the form 1{0}1) 3, 3, 0, 3, 3, 0, 3, 4, 0, 4, 3, 0, 3, 3, 0, 3, 11, 0, 3, 3, 0, 4, 3, 0, 4, 3, 0, 7, 4, 0, 7, 3, 0, 4, 5, 0, 3, 5, 0, 3, 4, 0, 4, 4, 0, 4, 8, 0, 3, 185, 0, 4, 3, 0, 4, 3, 0, 3, 23, 0, 3, 187, 0, 5, 3, 0, 4, 3, 0, 3, 122, 0, 4, 3, 0, 3, 3, 0, 3, 10, 0, 7, 11, 0, 4, 4, 0, 3, 3, 0, 4, 5, 0, 11, 16, 0, 5, 3, 0, 3, 7, 0, 7, 3, 0, 82, 400, 0, 3, 3, 0, 5, 5, 0, 46, 3, 0, 3, 4, 0, 4, 4, 0, 4, 7, 0, 5, 56, 0, 3, 56, 0, 11, 19, 0, 22, 3, 0, 4, 3, 0, 3, 5, 0, 5, 3, 0, 11, 3, 0, 4, 3, 0, 3, 5, 0, 143, 34, 0 Length of the smallest dual Williams prime of the 1st kind (form: {x}1, x = b-1) in base b for b = 2, 3, 4, ..., 160: 2, 2, 2, 5, 2, 2, 13, 2, 3, 3, 5, 2, 3, 2, 2, 11, 2, 3, 17, 2, 2, 17, 4, 2, 3, 9, 2, 33, 7, 3, 7, 4, 2, 3, 5, 67, 5, 2, 9, 3, 2, 4, 25, 3, 4, 5, 5, 24, 3, 2, 3, 21, 3, 2, 9, 3, 2, 11, 2, 5, 3, 2, 4, 19, 31, 2, 29, 4, 2, 3019, 2, 21, 51, 3, 2, 3, 2, 2, 9, 2, 169, 965, 3, 3, 29, 3, 2848, 9, 2, 2, 3, (>60000 or 0), 4, 3, 7, 6, 5, 3, 2, 3, 2, 5, 55, 4, 2, 7, 4, 4, 61, 2, 2, (>25000 or 0), 991, 4, 3, 18, 2, 9, 2, 4, 61, 17, 9, 3, 16, 18, 401, 3, 3, 25, 2, 9, 3, 13, 3, 5, 4, 2, 3, 3, 2, 281, 2, 255, 5, 3, 2, 7, 90, 2, (>25000 or 0), 6, 2, 3, 2, 6, (>25000 or 0), 6, 33 Length of the smallest dual Williams prime of the 2nd kind (form: 1{0}x, x = b-1) in base b for b = 2, 3, 4, ..., 160: 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 3, 17, 2, 2, 5, 4, 2, 3, 2, 2, 5, 2, 4, 3, 2, 3, 11, 2, 2, 109, 4, 2, 3, 2, 2, 3, 3, 2, 3, 2, 4, 3, 2, 3, 21, 3, 2, 3, 2, 2, 3, 2, 2, 3, 2, 5, 3, 3, 8, 9, 4, 2, 3, 2, 25, 3, 2, 2, 13, 5, 4, 9, 2, 2, 5, 4, 2, 195, 4, 2, 3, 2, 3, 3, 2, 9, 3, 2, 2, 5, 3, 3, 55, 2, 2, 5, 2, 2, 3, 45, 3, 15, 4, 2, 1401, 7, 4, 5, 7, 2, 20089, 2, 2, 7, 2, 7, 5, 2, 2, 5, 64371, 4, 3, 2, 4, 505, 2, 3, 3, 2, 9, 3, 2, 2, 61, 3, 2, 3, 2, 2, 5, 3, 3, 9, 2, 3, 3, 4, 4, 21, 3, 2, 3, 2, 2, 3, 2, 3 Length of the smallest dual Williams prime of the 4th kind (form: 1{0}11) in base b for b = 2, 3, 4, ..., 160: (not minimal prime if there is smaller prime of the form 1{0}1) 2, 2, 0, 2, 2, 0, 2, 2, 0, 2, 3, 0, 2, 2, 0, 3, 2, 0, 2, 2, 0, 2, 3, 0, 2, 3, 0, 2, 2, 0, 4, 2, 0, 2, 2, 0, 3, 2, 0, 2, 4, 0, 2, 7, 0, 5, 2, 0, 2, 2, 0, 2, 2, 0, 2, 3, 0, 3, 4, 0, 3, 2, 0, 2, 3, 0, 2, 2, 0, 3, 5, 0, 2, 2, 0, 3, 2, 0, 3, 2, 0, 2, 31, 0, 2, 4, 0, 2, 2, 0, 8, 68, 0, 2, 2, 0, 2, 2, 0, 3, 4, 0, 4, 2, 0, 5, 4, 0, 3, 2, 0, 2, 2, 0, 2, 3, 0, 2, 2, 0, 13, 8, 0, 2, 4, 0, 2, 569, 0, 2, 25, 0, 2, 2, 0, 44, 2, 0, 2, 2, 0, 3, 4, 0, 2, 3, 0, 8, 3, 0, 4, 2, 0, 2, 2, 0, 2, 5, 0 Length of the smallest prime of the form 2*b^n+1 (form: 2{0}1) in base b for b = 2, 3, 4, ..., 160: (the b=2 case is not acceptable, since there is no digit 2 in base 2) 2, 2, 0, 2, 2, 0, 2, 2, 0, 2, 4, 0, 2, 2, 0, 48, 2, 0, 2, 2, 0, 2, 3, 0, 2, 3, 0, 2, 2, 0, 4, 2, 0, 2, 2, 0, 2730, 2, 0, 2, 3, 0, 2, 3, 0, 176, 2, 0, 2, 2, 0, 2, 2, 0, 2, 4, 0, 4, 4, 0, 44, 2, 0, 2, 3, 0, 2, 2, 0, 4, 3, 0, 2, 2, 0, 4, 2, 0, 12, 2, 0, 2, 5, 0, 2, 3, 0, 2, 2, 0, 4, 3, 0, 2, 2, 0, 2, 2, 0, 192276, 3, 0, 1234, 2, 0, 4, 6, 0, 52, 2, 0, 2, 2, 0, 2, 287, 0, 2, 2, 0, 756, 3, 0, 2, 5, 0, 2, 7, 0, 2, 3, 0, 2, 2, 0, 328, 2, 0, 2, 2, 0, 6, 6, 0, 2, 155, 0, 4, 4, 0, 4, 2, 0, 2, 2, 0, 2, 4, 0 Length of the smallest prime of the form 2*b^n-1 (form: 1{x}, x = b-1) in base b for b = 2, 3, 4, ..., 160: 2, 2, 2, 5, 2, 2, 3, 2, 2, 3, 2, 3, 5, 2, 2, 3, 3, 2, 11, 2, 2, 7, 2, 3, 7, 2, 3, 137, 2, 2, 7, 7, 2, 7, 2, 2, 3, 3, 2, 3, 2, 3, 5, 2, 3, 5, 5, 2, 3, 2, 2, 45, 2, 2, 3, 2, 4, 3, 6, 4, 3, 3, 2, 5, 2, 769, 5, 2, 2, 53, 35, 3, 133, 2, 2, 15, 8, 2, 3, 3, 2, 9, 2, 3, 11, 2, 25, 61, 2, 2, 3, 4, 6, 3, 2, 2, 3, 2, 2, 43, 3, 5, 69, 7, 2, 21911, 3, 3, 17, 25, 2, 3, 2, 2, 33, 2, 3, 29, 2, 2, 7, 9, 5, 3, 2, 3, 19, 2, 4, 5, 2, 5, 3, 2, 2, 3, 5, 2, 3, 2, 2, 3, 25, 13, 17, 2, 5, 5, 9, 6, 797, 3, 2, 3, 2, 2, 3, 2, 3 Length of the smallest prime of the form b^n+2 (form: 1{0}2) in base b for b = 2, 3, 4, ..., 160: 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 12, 0, 0, 0, 2, 0, 2, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 114, 0, 0, 0, 2, 0, 8, 0, 0, 0, 2, 0, 2, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 13, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 2, 0, 256, 0, 0, 0, 9, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 3, 0, 16, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 24, 0, 0, 0, 2, 0, 2, 0, 0, 0, 5, 0, 4, 0, 0, 0, 2, 0, 2, 0, 0, 0, 4, 0, 2, 0, 0, 0, 137, 0 Length of the smallest prime of the form b^n-2 (form: {x}y, x = b-1, y = b-2) in base b for b = 2, 3, 4, ..., 160: (n>=2 is required, since single-digit primes are not acceptable in this puzzle, also (b,n) = (2,2) is also not acceptable, although 2^2-2 is prime, since 2^2-2 is not a prime which is >2, but this puzzle requires primes >b) 0, 2, 0, 2, 0, 2, 0, 2, 0, 4, 0, 2, 0, 2, 0, 6, 0, 2, 0, 2, 0, 24, 0, 7, 0, 2, 0, 2, 0, 3, 0, 2, 0, 2, 0, 2, 0, 4, 0, 4, 0, 2, 0, 11, 0, 2, 0, 2, 0, 8, 0, 4, 0, 2, 0, 12, 0, 4, 0, 2, 0, 2, 0, 8, 0, 3, 0, 2, 0, 2, 0, 4, 0, 2, 0, 2, 0, 38, 0, 130, 0, 4, 0, 4, 0, 4, 0, 2, 0, 3, 0, 2, 0, 4, 0, 747, 0, 3, 0, 4, 0, 2, 0, 10, 0, 2, 0, 3, 0, 17, 0, 10, 0, 13, 0, 2, 0, 2, 0, 2, 0, 6, 0, 42, 0, 2, 0, 3, 0, 2, 0, 6, 0, 2, 0, 10, 0, 2, 0, 4, 0, 4, 0, 2, 0, 16, 0, 50, 0, 3, 0, 9, 0, 2, 0, 22, 0, 25, 0 Length of the smallest prime of the form 3*b^n+1 (form: 3{0}1) in base b for b = 2, 3, 4, ..., 160: (the b=2 case is not acceptable, since there is no digit 3 in base 2) 2, 0, 2, 0, 2, 0, 3, 0, 2, 0, 2, 0, 2, 0, 3, 0, 4, 0, 2, 0, 2, 0, 2, 0, 2, 0, 8, 0, 4, 0, 2, 0, 2, 0, 2, 0, 4, 0, 3, 0, 2, 0, 10, 0, 2, 0, 4, 0, 2, 0, 2, 0, 2, 0, 6, 0, 3, 0, 2, 0, 13, 0, 2, 0, 2, 0, 3, 0, 2, 0, 15, 0, 2, 0, 2, 0, 3, 0, 2, 0, 3, 0, 3, 0, 3, 0, 5, 0, 2, 0, 2, 0, 2, 0, 4, 0, 3, 0, 6, 0, 2, 0, 2, 0, 4, 0, 271, 0, 2, 0, 2, 0, 13, 0, 2, 0, 47, 0, 3, 0, 2, 0, 2, 0, 2, 0, 28, 0, 22, 0, 2, 0, 5, 0, 2, 0, 9, 0, 2, 0, 3, 0, 2, 0, 2, 0, 3, 0, 4, 0, 2, 0, 2, 0, 3, 0, 5, 0, 3 Length of the smallest prime of the form 3*b^n-1 (form: 2{x}, x = b-1) in base b for b = 2, 3, 4, ..., 160: (the b=2 case is not acceptable, since there is no digit 2 in base 2) 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 3, 0, 2, 0, 2, 0, 2, 0, 2, 0, 3, 0, 2, 0, 3, 0, 2, 0, 2, 0, 12, 0, 2, 0, 2, 0, 2, 0, 3, 0, 2524, 0, 2, 0, 2, 0, 3, 0, 2, 0, 3, 0, 3, 0, 2, 0, 2, 0, 2, 0, 60, 0, 2, 0, 2, 0, 11, 0, 3, 0, 3, 0, 3, 0, 2, 0, 2, 0, 2, 0, 15, 0, 2, 0, 2, 0, 2, 0, 2, 0, 3, 0, 2, 0, 3, 0, 2, 0, 4, 0, 4, 0, 2, 0, 2, 0, 7, 0, 3, 0, 51, 0, 64, 0, 2, 0, 2, 0, 2, 0, 3, 0, 12, 0, 51, 0, 2, 0, 2, 0, 39, 0, 2, 0, 3, 0, 3, 0, 2, 0, 27, 0, 2, 0, 4, 0, 2, 0, 2, 0, 4, 0, 2, 0, 2, 0, 3, 0, 2 Length of the smallest prime of the form b^n+3 (form: 1{0}3) in base b for b = 2, 3, 4, ..., 160: (the b=2 case is not acceptable, since there is no digit 3 in base 2) 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 3, 0, 0, 0, 2, 0, 2, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 21, 0, 0, 0, 2, 0, 3, 0, 0, 0, 2, 0, 2, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 5, 0, 0, 0, 2, 0, 6, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 3, 0, 0, 0, 6, 0, 5, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 3, 0, 0, 0, 2, 0, 2, 0, 0, 0, 3, 0, 4, 0, 0, 0, 2, 0, 2, 0, 0, 0, 16, 0, 2, 0, 0, 0, 3, 0, 2 Length of the smallest prime of the form b^n-3 (form: {x}y, x = b-1, y = b-3) in base b for b = 2, 3, 4, ..., 160: (n>=2 is required, since single-digit primes are not acceptable in this puzzle) (the b=2 case is not acceptable, since there is no digit 2-3 = -1 in base 2) 3, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 3, 0, 0, 0, 2, 0, 6, 0, 0, 0, 2, 0, 10, 0, 0, 0, 2, 0, 2, 0, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 21, 0, 105, 0, 0, 0, 18, 0, 2, 0, 0, 0, 5, 0, 2, 0, 0, 0, 2, 0, 5, 0, 0, 0, 3, 0, 5, 0, 0, 0, 2, 0, 13, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 204, 0, 0, 0, 2, 0, 70, 0, 0, 0, 4, 0, 3, 0, 0, 0, 2, 0, 2, 0, 0, 0, 3, 0, 2, 0, 0, 0, 6, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 7, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 346, 0, 396, 0, 0, 0, 3, 0, 21 Length of the smallest prime of the form 4*b^n+1 (form: 4{0}1) in base b for b = 2, 3, 4, ..., 160: (the b=2, 3, 4 cases are not acceptable, since there is no digit 4 in bases 2, 3, 4) 3, 2, 2, 3, 0, 2, 3, 2, 2, 0, 3, 2, 0, 2, 0, 7, 2, 4, 3, 0, 2, 343, 2, 2, 0, 2, 2, 0, 7, 0, (>=1717986919 or 0), 3, 2, 43, 0, 2, 11, 2, 6, 0, 3, 2, 0, 2, 0, 3, 2, 2, 11, 0, 11, (>1670000 or 0), 4, 3, 0, 2, 2, 0, 2, 0, 3, 3, 2, 3, 0, 2, 7, 2, 2, 0, 4, 2, 0, 3, 0, 6099, 2, 2, 3, 0, 7, 5871, 2, 3, 0, 2, 2, 0, 3, 0, 3, 2, 4, 7, 0, 2, 295, 2, 2, 0, 2, 3, 0, 2, 0, 32587, 2, 4, 11, 0, 2, 2959, 2, 2, 0, 102, 3, 0, 3, 0, 359, 7, 472, 3, 0, 2, 3, 20, 2, 0, 3, 6, 0, 2, 0, 19, 4, 2, 3, 0, 2, 11, 2, 22, 0, 4, 2, 0, 2, 0, 19, 2, 2, (>1280000 or 0), 0, 3, 875, 30, 2 Length of the smallest prime of the form 4*b^n-1 (form: 3{x}, x = b-1) in base b for b = 2, 3, 4, ..., 160: (the b=2, 3 cases are not acceptable, since there is no digit 3 in base 2, 3) 2, 2, 0, 2, 2, 0, 2, 0, 0, 2, 2, 0, 0, 2, 0, 2, 2, 0, 2, 2, 0, 6, 0, 0, 2, 2, 0, 0, 4, 0, 2, 2, 0, 2, 0, 0, 2, 0, 0, 2, 2, 0, 0, 2, 0, 1556, 2, 0, 2, 4, 0, 2, 0, 0, 2, 2, 0, 0, 2, 0, 10, 2, 0, 10, 2, 0, 2, 0, 0, 2, 1119850, 0, 0, 6, 0, 2, 2, 0, 8, 0, 0, 2, 0, 0, 6, 2, 0, 0, 2, 0, 2, 42, 0, 2, 2, 0, 4, 0, 0, 4, 14, 0, 0, 2, 0, 252, 2, 0, 2, 2, 0, 4, 0, 0, 2, 2, 0, 0, 2, 0, 2, 2, 0, 2, 2, 0, 16, 0, 0, 2, 4, 0, 0, 6, 0, 2, 14, 0, 6, 2, 0, 2, 0, 0, 6, 2, 0, 0, 2, 0, 2, 4, 0, 2, 4, 0, 2, 0, 0 Length of the smallest prime of the form b^n+4 (form: 1{0}4) in base b for b = 2, 3, 4, ..., 160: (the b=3 case is not acceptable, since there is no digit 4 in base 3) 0, 2, 0, 3, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 3, 0, 2, 0, 0, 0, 7, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 3, 0, 2, 0, 2, 0, 0, 0, 2, 0, 3, 0, 3, 0, 2, 0, 0, 0, 13403, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 3, 0, 2, 0, 2, 0, 0, 0, 3, 0, 2, 0, 47, 0, 2, 0, 0, 0, 83, 0, 2, 0, 3, 0, 0, 0, 0, 0, 2, 0, 3, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 7, 0, 2, 0, 0, 0, 10647, 0, 3, 0, 3, 0, 0, 0, 0, 0, 2, 0, 3, 0, 2, 0, 4, 0, 0, 0, 2, 0, 2, 0, 3, 0, (>25000 or 0), 0, 0, 0, 71, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 3, 0, 214, 0, 2, 0 Length of the smallest prime of the form b^n-4 (form: {x}y, x = b-1, y = b-4) in base b for b = 2, 3, 4, ..., 160: (n>=2 is required, since single-digit primes are not acceptable in this puzzle) (the b=3 case is not acceptable, since there is no digit 3-4 = -1 in base 3) 0, 2, 0, 5, 0, 0, 0, 0, 0, 3, 0, 0, 0, 3, 0, 3, 0, 0, 0, 3, 0, 3, 0, 0, 0, 7, 0, 0, 0, 0, 0, 3, 0, 13, 0, 0, 0, 0, 0, 3, 0, 0, 0, 3, 0, 65, 0, 0, 0, 3, 0, 3, 0, 0, 0, 3, 0, 0, 0, 0, 0, 3, 0, 175, 0, 0, 0, 0, 0, 45, 0, 0, 0, 13, 0, 3, 0, 0, 0, 0, 0, 3, 0, 0, 0, 29, 0, 0, 0, 0, 0, 105, 0, 45, 0, 0, 0, 0, 0, 3, 0, 0, 0, 3, 0, 7, 0, 0, 0, 13, 0, 13, 0, 0, 0, 3, 0, 0, 0, 0, 0, 299, 0, 5, 0, 0, 0, 0, 0, 3, 0, 0, 0, 165, 0, 147, 0, 0, 0, 395, 0, 23, 0, 0, 0, 3, 0, 0, 0, 0, 0, 7, 0, 3, 0, 0, 0, 0, 0 Last fiddled with by sweety439 on 2021-02-06 at 21:16 |
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2021-01-14, 04:55
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#136 |
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"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
5×7×83 Posts |
Large minimal prime (start with b+1) for bases b>16 not in the list for bases 2 to 30 or list for bases 28 to 50 (because they contain single-digit primes, or because they are too large (length > 10000), primes with the latter case but not the former case are already minimal even if single-digit primes are included, and they are marked by "**") given by: (using A−Z to represent digit values 10 to 35, a−z to represent digit values 36 to 61)
Base 18: 80298B Base 24: 203137 (note: F1957 is not minimal prime (start with b+1), since its repeating digit is 1) Base 30: OT34205 (found by CRUS generalized Riesel conjecture base 30) Base 33: 130236141 (found by CRUS generalized Sierpinski conjecture base 33) **Base 36: P81993SZ (already minimal even if single-digit primes are included, but not in the list since this prime is too large) (found by me) **Base 37: FYa22021 (already minimal even if single-digit primes are included, but not in the list since this prime is too large) (found by CRUS generalized Riesel conjecture base 37) Base 38: ab136211 (found by Williams primes search) **Base 40: QaU12380X (already minimal even if single-digit primes are included, but not in the list since this prime is too large) (found by me) Base 42: 2f2523 (found by CRUS generalized Riesel conjecture base 42) **Base 45: O0185211 (already minimal even if single-digit primes are included, but not in the list since this prime is too large) (found by CRUS generalized Sierpinski conjecture base 45) Base 48: T01330411 (found by CRUS generalized Sierpinski conjecture base 48) **Base 49: SLm52698 (already minimal even if single-digit primes are included, but not in the list since this prime is too large) (found by CRUS generalized Riesel conjecture base 49) Also see post https://mersenneforum.org/showpost.p...7&postcount=43 Last fiddled with by sweety439 on 2021-02-07 at 00:16 Reason: AO(0^44790)1 is not minimal prime in base 45, since O(0^18521)1 is prime in base 45 |
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2021-01-14, 19:17
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#137 |
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"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
55318 Posts |
For more lengths for the smallest prime of the form k*b^n+1 (form: k{0}1) or k*b^n-1 (form: x{y}, x = k-1, y = b-1) with k<b in base b, see CRUS
Note: The requiring of this project (to solve this puzzle) is that k<b, but the requiring of CRUS is that k<CK, since the CK value may be either >b or <b, thus this is neither necessary nor sufficient, but they have many intersection (when k<min(b,CK)), e.g. 8*23^n+1 (8{0}1 in base 23), 25*30^n-1 (O{T} in base 30), 2*38^n+1 (2{0}1 in base 38), 3*42^n-1 (2{f} in base 42), 36*48^n+1 (a{0}1 in base 48), 4*53^n+1 (4{0}1 in base 53), etc., they corresponding to the minimal primes (start with b+1) 801192141 in base 23 (8*23^119215+1), OT34205 in base 30 (25*30^34205-1), 2027281 in base 38 (2*38^2729+1), 2f2523 in base 42 (3*42^2523-1), unsolved family a{0}1 in base 48 searched to length 500001 (36*48^n+1 searched to n=500000), unsolved family 4{0}1 in base 53 searched to length 1700001 (4*53^n+1 searched to n=1700000), etc. Also this project (to solve this puzzle) includes unsolved family 4{0}1 in base 32 searched to length (2^33-2)/5 = 1717986918 (since all primes of the form 4{0}1 in base 32 must be Fermat primes, and none of the known Fermat primes (F0 to F4) are of the form 4{0}1 in base 32 (their base 32 forms are 3, 5, H, 81, 2001), and all Fermat numbers F5 to F32 are known to be composite, see http://www.prothsearch.com/fermat.html, thus, the smallest possible prime of the form 4{0}1 in base 32 is F33 = 4017179869171 in base 32, which has length 1717986919), which is excluded in CRUS, since CRUS excluded k's that make GFNs, i.e. q^m*b^n+1 where b is the base, m>=0, and q is a root of the base, and 4*32^n+1 = (2^2)*32^n+1, and 2 is a root of 32 (32^(1/5)). Reference: https://arxiv.org/pdf/1605.01371.pdf (this reference shows that the property of the existence of a Fermat prime > F4 is at most 10^(-9), and thus base 32 (also bases 128, 512, 1024) is virtually impossible to solve with current knowledge and technology, for the similar problem to other bases, it is excepted that the number of primes of the form b^(2^n)+1 (for fixed even base b) or (b^(2^n)+1)/2 (for fixed odd base b) is finite (such forms are called GFN (generalized Fermat numbers, i.e. b^(2^n)+1 (for even base b)) or half GFN (generalized half Fermat numbers, i.e. (b^(2^n)+1)/2 (for odd base b)), and the families which all possible primes are GFN or half GFN are called GFN families or half GFN families, see https://mersenneforum.org/showthread.php?t=20427), but this is undecidable at this point in time, and thus in some bases there exist families which are excepted as contain no primes, but undecidable at this point in time (they are exactly the GFN families or half GFN families in bases 2<=b<=1024 with no known (probable) primes), thus these bases are impossible to solve at this time. Bases 2<=b<=1024 which I am aware of with this problem are 31, 32, 37, 38, 50, 55, 62, 63, 67, 68, 77, 83, 86, 89, 91, 92, 93, 97, 98, 99, 104, 107, 109, 117, 122, 123, 125, 127, 128, 133, 135, 137, 143, 144, 147, 149, 151, 155, 161, 168, 177, 179, 182, 183, 186, 189, 193, 197, 200, 202, 207, 211, 212, 213, 214, 215, 216, 217, 218, 223, 225, 227, 233, 235, 241, 244, 246, 247, 249, 252, 255, 257, 258, 263, 265, 269, 273, 277, 281, 283, 285, 286, 287, 291, 293, 294, 297, 298, 302, 303, 304, 307, 308, 311, 319, 322, 324, 327, 338, 343, 344, 347, 351, 354, 355, 356, 357, 359, 361, 362, 367, 368, 369, 377, 380, 381, 383, 385, 387, 389, 390, 393, 394, 397, 398, 401, 402, 404, 407, 410, 411, 413, 416, 417, 421, 422, 423, 424, 437, 439, 443, 446, 447, 450, 454, 457, 458, 465, 467, 468, 469, 473, 475, 480, 481, 482, 483, 484, 489, 493, 495, 497, 500, 509, 511, 512, 514, 515, 518, 524, 528, 530, 533, 534, 538, 541, 547, 549, 552, 555, 558, 563, 564, 572, 574, 578, 580, 590, 591, 593, 597, 601, 602, 603, 604, 608, 611, 615, 619, 620, 621, 622, 625, 626, 627, 629, 632, 633, 635, 637, 638, 645, 647, 648, 650, 651, 653, 655, 659, 662, 663, 666, 667, 668, 670, 671, 673, 675, 678, 679, 683, 684, 687, 691, 692, 693, 694, 697, 698, 706, 707, 709, 712, 717, 720, 722, 724, 731, 733, 734, 735, 737, 741, 743, 744, 746, 749, 752, 753, 754, 755, 757, 759, 762, 765, 766, 767, 770, 771, 773, 775, 777, 783, 785, 787, 792, 793, 794, 797, 801, 802, 806, 807, 809, 812, 813, 814, 817, 818, 823, 825, 836, 840, 842, 844, 848, 849, 851, 853, 854, 865, 867, 868, 870, 872, 873, 877, 878, 887, 888, 889, 893, 896, 897, 899, 902, 903, 904, 907, 908, 911, 915, 922, 923, 924, 926, 927, 932, 933, 937, 938, 939, 941, 942, 943, 944, 945, 947, 948, 953, 954, 957, 958, 961, 964, 967, 968, 974, 975, 977, 978, 980, 983, 987, 988, 993, 994, 998, 999, 1000, 1002, 1003, 1005, 1006, 1009, 1014, 1016, 1017, 1024 (totally 369 such bases of the 1023 bases 2<=b<=1024, thus there are 1023-369=654 bases of the 1023 bases 2<=b<=1024 which might be solved at this time)) The families which are excepted as contain no primes, but undecidable at this point in time, for these 369 bases are: (totally 377 families) * 4:{0}:1, 16:{0}:1 for b = 32 * 12:{62}:63 for b = 125 * 16:{0}:1 for b = 128 * 36:{0}:1 for b = 216 * 24:{171}:172 for b = 343 * 2:{0}:1, 4:{0}:1, 16:{0}:1, 32:{0}:1, 256:{0}:1 for b = 512 * 10:{0}:1, 100:{0}:1 for b = 1000 * 4:{0}:1, 16:{0}:1, 256:{0}:1 for b = 1024 * 1:{0}:1 for other even bases b * {((b-1)/2)}:((b+1)/2) for other odd bases b Note: families 8:{0}:1, 32:{0}:1, 64:{0}:1 in base 128 can be ruled out as contain no primes, since if 2^n+1 is prime, then n must be power of 2, but 7*n+3, 7*n+5, 7*n+6 cannot be powers of 2, all powers of 2 are == 1, 2, 4 mod 7 For the "minimal prime (start with b+1) problem in base b": A base is solved if there are no unsolved families for this base and all minimal primes (start with b+1) are proven primes. A base is weakly solved if there are no unsolved families for this base but some minimal primes (start with b+1) are only probable primes. A base is almost solved if all unsolved families for this base are GFN families or half GFN families. e.g. * base 31 is almost solved if the only unsolved family is {F}G * base 32 is almost solved if the only two unsolved families are 4{0}1 and G{0}1 * base 37 is almost solved if the only unsolved family is {I}J * base 38 is almost solved if the only unsolved family is 1{0}1 * base 50 is almost solved if the only unsolved family is 1{0}1 * base 55 is almost solved if the only unsolved family is {R}S * base 62 is almost solved if the only unsolved family is 1{0}1 * base 63 is almost solved if the only unsolved family is {V}W etc. Last fiddled with by sweety439 on 2021-07-05 at 18:30 |
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2021-02-15, 16:31
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#138 |
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"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
55318 Posts |
Just let you know, I know the set of the minimal primes (start with b+1) <=2^32 for all bases 2<=b<=128, and I know exactly which bases 2<=b<=1024 have these families as unsolved families (at length 25K) for the minimal primes (start with b+1) problem: (also, I know exactly which bases 2<=b<=1024 where these families are ruled out as contain no primes >b)
(using A−Z to represent digit values 10 to 35, z−a to represent digit values b−1 to b−26) (if such forms are interpretable in the bases, e.g. "C" (means 12 (twelve)) is only interpretable in bases b>=13, and "u" (means b−6) is only interpretable in bases b>=6 (if "u" appears as the first digit, then it is only interpretable in bases b>=7, since numbers cannot have leading zeros) * {1} * 1{0}1 * 1{0}2 * 1{0}3 * 1{0}4 * 1{0}5 * 1{0}6 * 1{0}7 * 1{0}8 * 1{0}9 * 1{0}A * 1{0}B * 1{0}C * 1{0}D * 1{0}E * 1{0}F * 1{0}G * 1{0}z * 2{0}1 * 2{0}3 * 3{0}1 * 3{0}2 * 3{0}4 * 4{0}1 * 4{0}3 * 5{0}1 * 6{0}1 * 7{0}1 * 8{0}1 * 9{0}1 * A{0}1 * B{0}1 * C{0}1 * D{0}1 * E{0}1 * F{0}1 * G{0}1 * z{0}1 * 1{2} * 1{3} * 1{4} * 1{5} * 1{6} * 1{7} * 1{8} * 1{9} * 1{A} * 1{B} * 1{C} * 1{D} * 1{E} * 1{F} * 1{G} * 1{#} (for odd base b, # = (b−1)/2) * {2}1 * {3}1 * {4}1 * {5}1 * {6}1 * {7}1 * {8}1 * {9}1 * {A}1 * {B}1 * {C}1 * {D}1 * {E}1 * {F}1 * {G}1 * {#}1 (for odd base b, # = (b−1)/2) * 1{z} * 2{z} * 3{z} * 4{z} * 5{z} * 6{z} * 7{z} * 8{z} * 9{z} * A{z} * B{z} * C{z} * D{z} * E{z} * F{z} * y{z} * {#}$ (for odd base b, # = (b−1)/2, $ = (b+1)/2) * ${#} (for odd base b, # = (b−1)/2, $ = (b+1)/2) * {y}z * {z}1 * {z}k * {z}l * {z}m * {z}n * {z}o * {z}p * {z}q * {z}r * {z}s * {z}t * {z}u * {z}v * {z}w * {z}x * {z}y Also families where the smallest prime may not be minimal prime (start with b+1): * 1{0}11 (not minimal prime (start with b+1) if there is smaller prime of the form 1{0}1) * 11{0}1 (not minimal prime (start with b+1) if there is smaller prime of the form 1{0}1) * 1{0}21 (not minimal prime (start with b+1) if either 21 (2*b+1) is prime or there is smaller prime of the form 1{0}1 or 1{0}2) * 12{0}1 (not minimal prime (start with b+1) if either 12 (b+2) is prime or there is smaller prime of the form 1{0}1 or 2{0}1) * {1}01 (not minimal prime (start with b+1) if there is smaller prime of the form {1}) * 10{1} (not minimal prime (start with b+1) if there is smaller prime of the form {1}) * {1}2 (not minimal prime (start with b+1) if there is smaller prime of the form {1}) * {1}3 (not minimal prime (start with b+1) if there is smaller prime of the form {1}) * {1}4 (not minimal prime (start with b+1) if there is smaller prime of the form {1}) * {1}z (not minimal prime (start with b+1) if there is smaller prime of the form {1}) * 2{1} (not minimal prime (start with b+1) if there is smaller prime of the form {1}) * 3{1} (not minimal prime (start with b+1) if there is smaller prime of the form {1}) * 4{1} (not minimal prime (start with b+1) if there is smaller prime of the form {1}) * z{1} (not minimal prime (start with b+1) if there is smaller prime of the form {1}) * {1}0z (not minimal prime (start with b+1) if there is smaller prime of the form {1} or {1}z) * 10{z} (not minimal prime (start with b+1) if there is smaller prime of the form 1{z}) * 11{z} (not minimal prime (start with b+1) if either 11 (b+1) is prime or there is smaller prime of the form 1{z}) * {z}01 (not minimal prime (start with b+1) if there is smaller prime of the form {z}1) * zy{z} (not minimal prime (start with b+1) if there is smaller prime of the form y{z}) * {z}yz (not minimal prime (start with b+1) if there is smaller prime of the form {z}y) * {z0}z1 (almost cannot be minimal prime (start with b+1), since this is not simple family) (in fact, there are no bases 2<=b<=1024 such that 7{0}1 is unsolved family, base 1004 is the last to drop at length 54849, also there are no bases 2<=b<=1024 such that {z}x is unsolved family, base 542 is the last to drop at length 1944) Last fiddled with by sweety439 on 2021-05-03 at 22:13 |
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2021-02-15, 16:46
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#139 |
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"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
5×7×83 Posts |
Currently this project only has bases 2<=b<=16, I have plan to extend bases to 36 when all bases 2<=b<=16 have searched to length >=100K and all unsolved families are also found, and after extending bases to 36 and finding all minimal primes (start with b+1) with length <=100K and all unsolved families for all bases 2<=b<=36, I will extend bases to 64, then to 256 and 1024
The final goal of this project is solving all bases 2<=b<=1024 (i.e. finding all minimal primes (start with b+1) in all bases 2<=b<=1024 and proving that they are all such primes and proving the primality for all of them). Many of these primes have already been found but much more work is needed to find additional primes (the smallest primes in the unsolved families). Solving all bases 2<=b<=1024 (i.e. finding all minimal primes (start with b+1) in all bases 2<=b<=1024 and proving that they are all such primes and proving the primality for all of them) is not possible but we aim to find many minimal primes (start with b+1) in bases 2<=b<=1024 (including all such primes with length <= 25K) and find all unsolved families in all bases 2<=b<=1024 and prove that all such primes not in current list (these primes should have length > 25K) for bases 2<=b<=1024 are in one of these unsolved families for the corresponding base b and proving the primality for many of the minimal primes (start with b+1) in bases 2<=b<=1024 (special forms (where * represents string of digits with length <= (1/3)*(length of the number)): *{0}1 can be proven prime by N-1 primality test, *{z} can be proven prime by N+1 primality test, for other forms, we can only use Primo with ECPP primality test to prove the primality, and if the number is very large (say > 2^65536), we can only resort to a probable primality test such as Miller–Rabin primality test and Baillie–PSW primality test, unless a divisor of the number can be found. Since we are testing many numbers in an exponential sequence, it is possible to use a sieving process to find divisors rather than using trial division. To do this, we made use of Geoffrey Reynolds’ srsieve software. This program uses the baby-step giant-step algorithm to find all primes p which divide a*b^n+c where p and n lie in a specified range. Since this program cannot handle the general case (a*b^n+c)/gcd(a+c,b-1) when gcd(a+c,b-1) > 1 we only used it to sieve the sequence a*b^n+c for primes p not dividing gcd(a+c,b-1), and initialized the list of candidates to not include n for which there is some prime p dividing gcd(a+c,b-1) for which p divides (a*b^n+c)/gcd(a+c,b-1). The program had to be modified slightly to remove a check which would prevent it from running in the case when a, b, and c were all odd (since then 2 divides a*b^n+c, but 2 may not divide (a*b^n+c)/gcd(a+c,b-1)). Once the numbers with small divisors had been removed, it remained to test the remaining numbers using a probable primality test. For this we used the software LLR by Jean Penne. Although undocumented, it is possible to run this program on numbers of the form (a*b^n+c)/gcd(a+c,b-1) when gcd(a+c,b-1)>1, so this program required no modifications (also, LLR can prove the primality for numbers of the form a*b^n+-1 (i.e. the special case c=+-1 and gcd(a+c,b-1)=1) with b^n>a, the case c=1 and gcd(a+c,b-1)=1 is corresponding to families *{0}1, and the case c=-1 and gcd(a+c,b-1)=1 is corresponding to families *{z}). A script was also written which allowed one to run srsieve while LLR was testing the remaining candidates, so that when a divisor was found by srsieve on a number which had not yet been tested by LLR it would be removed from the list of candidates. In the cases where the elements of M(Lb) could be proven prime rigorously, we employed Primo by Marcel Martin, an elliptic curve primality proving implementation. Our algorithm then proceeds as follows: 1. Let M := {minimal primes in base b of length ≤ 3} L := where x ≠ 0 and Y is the set of digits y such that xyz has no subword in M. 2. While L contains non-simple families: (a) Explore each family of L, and update L. (b) Examine each family of L: i. Let w be the shortest string in the family. If w has a subword in M, then remove the family from L. If w represents a prime, then add w to M and remove the family from L. ii. If possible, simplify the family. iii. Check if the family can be proven to contain no primes > base, and if so then remove the family from L. (c) As much as possible and update L; after each split examine the new families as in (b). Links of the programs to solve the problem for this project: Sieving programs for the simple families (families of the form *{?}*, where * represents any strings of digits (may be empty string), ? represents any digit) (the numbers in these families are of the form (a*b^n+c)/gcd(a+c,b-1) for fixed integers a>=1, b>=2 (b is exactly the base), c != 0, gcd(a,c) = 1, gcd(b,c) = 1) (we use srsieve to sieve the sequence a*b^n+c with primes not dividing gcd(a+c,b-1), and delete the n such that (a*b^n+c)/gcd(a+c,b-1) is not coprime to gcd(a+c,b-1)): srsieve (broken link, new link: srsieve, sr1sieve, sr2sieve, PFGW, LLR and srbsieve, also the BOINC Confederation for srsieve, sr1sieve, sr2sieve, srbsieve) mtsieve Primality testing programs: PFGW LLR primo (except in the special case c = +-1 and gcd(a+c,b-1) = 1, when n is large the known primality tests for such a number are too inefficient to run. In this case one must resort to a probable primality test such as a Miller–Rabin test or a Baillie–PSW test, unless a divisor of the number can be found (trial division)) Download these programs: srsieve, sr1sieve, sr2sieve, PFGW, LLR mtsieve PFGW LLR (completed source for LLR) primo Currently, only bases 2, 3, 4, 5, 6, 7, 8, 10, 12 are completely solved, the complete list of the minimal primes (start with b+1) in these bases are Code:
base 2: 11 base 3: 12 21 111 base 4: 11 13 23 31 221 base 5: 12 21 23 32 34 43 104 111 131 133 313 401 414 3101 10103 14444 30301 33001 33331 44441 300031 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013 base 6: 11 15 21 25 31 35 45 51 4401 4441 40041 base 7: 14 16 23 25 32 41 43 52 56 61 65 113 115 131 133 155 212 221 304 313 335 344 346 364 445 515 533 535 544 551 553 1022 1051 1112 1202 1211 1222 2111 3031 3055 3334 3503 3505 3545 4504 4555 5011 5455 5545 5554 6034 6634 11111 11201 30011 30101 31001 31111 33001 33311 35555 40054 100121 150001 300053 351101 531101 1100021 33333301 5100000001 33333333333333331 base 8: 13 15 21 23 27 35 37 45 51 53 57 65 73 75 107 111 117 141 147 161 177 225 255 301 343 361 401 407 417 431 433 463 467 471 631 643 661 667 701 711 717 747 767 3331 3411 4043 4443 4611 5205 6007 6101 6441 6477 6707 6777 7461 7641 47777 60171 60411 60741 444641 500025 505525 3344441 4444477 5500525 5550525 55555025 444444441 744444441 77774444441 7777777777771 555555555555525 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447 base 10: 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 227 251 257 277 281 349 409 449 499 521 557 577 587 727 757 787 821 827 857 877 881 887 991 2087 2221 5051 5081 5501 5581 5801 5851 6469 6949 8501 9001 9049 9221 9551 9649 9851 9949 20021 20201 50207 60649 80051 666649 946669 5200007 22000001 60000049 66000049 66600049 80555551 555555555551 5000000000000000000000000000027 base 12: 11 15 17 1B 25 27 31 35 37 3B 45 4B 51 57 5B 61 67 6B 75 81 85 87 8B 91 95 A7 AB B5 B7 221 241 2A1 2B1 2BB 401 421 447 471 497 565 655 665 701 70B 721 747 771 77B 797 7A1 7BB 907 90B 9BB A41 B21 B2B 2001 200B 202B 222B 229B 292B 299B 4441 4707 4777 6A05 6AA5 729B 7441 7B41 929B 9777 992B 9947 997B 9997 A0A1 A201 A605 A6A5 AA65 B001 B0B1 BB01 BB41 600A5 7999B 9999B AAAA1 B04A1 B0B9B BAA01 BAAA1 BB09B BBBB1 44AAA1 A00065 BBBAA1 AAA0001 B00099B AA000001 BBBBBB99B B0000000000000000000000000009B 400000000000000000000000000000000000000077 Code:
b number of minimal primes base b base-b form of largest known minimal prime base b length of largest known minimal prime base b algebraic ((a×bn+c)/d) form of largest known minimal prime base b 2 1 11 2 3 3 3 111 3 13 4 5 221 3 41 5 22 1(0^93)13 96 5^95+8 6 11 40041 5 5209 7 71 (3^16)1 17 (7^17−5)/2 8 75 (4^220)7 221 (4*8^221+17)/7 10 77 5(0^28)27 31 5*10^30+27 12 106 4(0^39)77 42 4*12^41+91 Code:
1{0}1
b == 1 mod 2: Finite covering set {2}
b = m^r with odd r>1: Sum-of-rth-powers factorization
1{0}2
b == 0 mod 2: Finite covering set {2}
b == 1 mod 3: Finite covering set {3}
1{0}3
b == 1 mod 2: Finite covering set {2}
b == 0 mod 3: Finite covering set {3}
1{0}4
b == 0 mod 2: Finite covering set {2}
b == 1 mod 5: Finite covering set {5}
b == 14 mod 15: Finite covering set {3, 5}
b = m^4: Aurifeuillian factorization of x^4+4y^4
1{0}z
(none)
{1}
b = m^r with r>1: Difference-of-rth-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2))
1{2}
b == 0 mod 2: Finite covering set {2}
b such that b and 2(b+1) are both squares: Difference-of-squares factorization (such bases are 49)
1{3}
b == 0 mod 3: Finite covering set {3}
b such that b and 3(b+2) are both squares: Difference-of-squares factorization (such bases are 25, 361)
b == 1 mod 2 such that 3(b+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865)
1{4}
b == 0 mod 2: Finite covering set {2}
b such that b and 4(b+3) are both squares: Difference-of-squares factorization
1{z}
(none)
2{0}1
b == 1 mod 3: Finite covering set {3}
2{0}3
b == 0 mod 3: Finite covering set {3}
b == 1 mod 5: Finite covering set {5}
{2}1
b such that b and 2(b+1) are both squares: Difference-of-squares factorization (such bases are 49)
2{z}
b == 1 mod 2: Finite covering set {2}
3{0}1
b == 1 mod 2: Finite covering set {2}
3{0}2
b == 0 mod 2: Finite covering set {2}
b == 1 mod 5: Finite covering set {5}
3{0}4
b == 0 mod 2: Finite covering set {2}
b == 1 mod 7: Finite covering set {7}
{3}1
b such that b and 3(2b+1) are both squares: Difference-of-squares factorization (such bases are 121)
3{z}
b == 1 mod 3: Finite covering set {3}
b == 14 mod 15: Finite covering set {3, 5}
b = m^2: Difference-of-squares factorization
b == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)
4{0}1
b == 1 mod 5: Finite covering set {5}
b == 14 mod 15: Finite covering set {3, 5}
b = m^4: Aurifeuillian factorization of x^4+4y^4
4{0}3
b == 0 mod 3: Finite covering set {3}
b == 1 mod 7: Finite covering set {7}
{4}1
b such that b and 4(3b+1) are both squares: Difference-of-squares factorization (such bases are 16, 225)
4{z}
b == 1 mod 2: Finite covering set {2}
5{0}1
b == 1 mod 2: Finite covering set {2}
b == 1 mod 3: Finite covering set {3}
5{z}
b == 1 mod 5: Finite covering set {5}
b == 34 mod 35: Finite covering set {5, 7}
b = 6m^2 with m == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014)
6{0}1
b == 1 mod 7: Finite covering set {7}
b == 34 mod 35: Finite covering set {5, 7}
6{z}
b == 1 mod 2: Finite covering set {2}
b == 1 mod 3: Finite covering set {3}
7{0}1
b == 1 mod 2: Finite covering set {2}
7{z}
b == 1 mod 7: Finite covering set {7}
b == 20 mod 21: Finite covering set {3, 7}
b == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993)
b = m^3: Difference-of-cubes factorization
8{0}1
b == 1 mod 3: Finite covering set {3}
b == 20 mod 21: Finite covering set {3, 7}
b == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022)
b = 467: Finite covering set {3, 5, 7, 19, 37}
b = 722: Finite covering set {3, 5, 13, 73, 109}
b = m^3: Sum-of-cubes factorization
b = 128: Cannot have primes since 7n+3 cannot be power of 2
8{z}
b == 1 mod 2: Finite covering set {2}
b = m^2: Difference-of-squares factorization
b == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd)
9{0}1
b == 1 mod 2: Finite covering set {2}
b == 1 mod 5: Finite covering set {5}
9{z}
b == 1 mod 3: Finite covering set {3}
b == 32 mod 33: Finite covering set {3, 11}
A{0}1
b == 1 mod 11: Finite covering set {11}
b == 32 mod 33: Finite covering set {3, 11}
A{z}
b == 1 mod 2: Finite covering set {2}
b == 1 mod 5: Finite covering set {5}
b == 14 mod 15: Finite covering set {3, 5}
B{0}1
b == 1 mod 2: Finite covering set {2}
b == 1 mod 3: Finite covering set {3}
b == 14 mod 15: Finite covering set {3, 5}
B{z}
b == 1 mod 11: Finite covering set {11}
b == 142 mod 143: Finite covering set {11, 13}
b = 307: Finite covering set {5, 11, 29}
b = 901: Finite covering set {7, 11, 13, 19}
C{0}1
b == 1 mod 13: Finite covering set {13}
b == 142 mod 143: Finite covering set {11, 13}
b = 296, 901: Finite covering set {7, 11, 13, 19}
b = 562, 828, 900: Finite covering set {7, 13, 19}
b = 563: Finite covering set {5, 7, 13, 19, 29}
b = 597: Finite covering set {5, 13, 29}
y{z}
(none)
{y}z
(none)
z{0}1
(none)
{z}1
(none)
{z}y
b == 0 mod 2: Finite covering set {2}
Code:
1{0}1
(none)
1{0}2
(none)
1{0}3
(none)
1{0}4
53 (13403)
113 (10647)
1{0}z
113 (20089)
123 (64371)
{1}
152 (270217)
184 (16703)
200 (17807)
311 (36497)
326 (26713)
331 (25033)
371 (15527)
485 (99523)
629 (32233)
649 (43987)
670 (18617)
684 (22573)
691 (62903)
693 (41189)
731 (15427)
752 (32833)
872 (10093)
932 (20431)
1{z}
107 (21911)
170 (166429)
278 (43909)
303 (40175)
383 (20957)
515 (58467)
522 (62289)
578 (129469)
590 (15527)
647 (21577)
662 (16591)
698 (127559)
704 (62035)
845 (39407)
938 (40423)
969 (24097)
989 (26869)
2{0}1
101 (192276)
206 (46206)
218 (333926)
236 (161230)
257 (12184)
305 (16808)
467 (126776)
578 (44166)
626 (174204)
695 (94626)
752 (26164)
788 (72918)
869 (49150)
887 (27772)
899 (15732)
932 (13644)
2{z}
432 (16003)
3{0}1
(none)
3{z}
72 (1119850)
212 (34414)
218 (23050)
270 (89662)
303 (198358)
312 (51566)
422 (21738)
480 (93610)
513 (38032)
527 (46074)
566 (23874)
650 (498102)
686 (16584)
758 (15574)
783 (12508)
800 (33838)
921 (98668)
947 (10056)
4{0}1
107 (32587)
227 (13347)
257 (160423)
355 (10990)
410 (144079)
440 (56087)
452 (14155)
482 (30691)
542 (15983)
579 (67776)
608 (20707)
635 (11723)
650 (96223)
679 (69450)
737 (269303)
740 (58043)
789 (149140)
797 (468703)
920 (103687)
934 (101404)
962 (84235)
4{z}
14 (19699)
68 (13575)
254 (15451)
800 (20509)
5{0}1
326 (400786)
350 (20392)
554 (10630)
662 (13390)
926 (40036)
5{z}
258 (212135)
272 (148427)
299 (64898)
307 (26263)
354 (25566)
433 (283919)
635 (36163)
678 (40859)
692 (45447)
719 (20552)
768 (70214)
857 (23083)
867 (61411)
972 (36703)
6{0}1
108 (16318)
129 (16797)
409 (369833)
522 (52604)
587 (24120)
643 (164916)
762 (11152)
789 (27297)
986 (21634)
6{z}
68 (25396)
332 (15222)
338 (42868)
362 (146342)
488 (33164)
566 (164828)
980 (50878)
986 (12506)
1016 (23336)
7{0}1
398 (17473)
1004 (54849)
7{z}
97 (192336)
170 (15423)
194 (38361)
202 (155772)
282 (21413)
283 (164769)
332 (13205)
412 (29792)
560 (19905)
639 (10668)
655 (53009)
811 (31784)
814 (17366)
866 (108591)
908 (61797)
962 (31841)
992 (10605)
997 (15815)
8{0}1
23 (119216)
53 (227184)
158 (123476)
254 (67716)
320 (52004)
410 (279992)
425 (94662)
513 (19076)
518 (11768)
596 (148446)
641 (87702)
684 (23387)
695 (39626)
788 (11408)
893 (86772)
920 (107822)
962 (47222)
998 (81240)
1013 (43872)
8{z}
138 (35686)
412 (12154)
788 (11326)
990 (23032)
9{0}1
248 (39511)
592 (96870)
9{z}
431 (43574)
446 (152028)
458 (126262)
599 (11776)
846 (12781)
A{0}1
173 (264235)
198 (47665)
311 (314807)
341 (106009)
449 (18507)
492 (42843)
605 (12395)
708 (17563)
710 (31039)
743 (285479)
786 (68169)
800 (15105)
802 (149320)
879 (25004)
929 (13065)
977 (125873)
986 (48279)
1004 (10645)
A{z}
368 (10867)
488 (10231)
534 (80328)
662 (13307)
978 (14066)
B{0}1
710 (15272)
740 (33520)
878 (227482)
B{z}
153 (21660)
186 (112718)
439 (18752)
593 (16064)
602 (36518)
707 (10573)
717 (67707)
C{0}1
68 (656922)
219 (29231)
230 (94751)
312 (21163)
334 (83334)
353 (20262)
359 (61295)
457 (10024)
481 (45941)
501 (20140)
593 (42779)
600 (11242)
604 (17371)
641 (26422)
700 (91953)
887 (13961)
919 (45359)
923 (64365)
992 (10300)
y{z}
38 (136212)
83 (21496)
113 (286644)
188 (13508)
401 (103670)
417 (21003)
458 (46900)
494 (21580)
518 (129372)
527 (65822)
602 (17644)
608 (36228)
638 (74528)
663 (47557)
723 (24536)
758 (50564)
833 (12220)
904 (13430)
938 (50008)
950 (16248)
z{0}1
202 (46774)
251 (102979)
272 (16681)
297 (14314)
298 (60671)
326 (64757)
347 (69661)
363 (142877)
452 (71941)
543 (10042)
564 (38065)
634 (84823)
788 (13541)
869 (12289)
890 (37377)
953 (60995)
1004 (29685)
{z}1
(none)
{z}y
317 (13896)
Code:
1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥223)
1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000)
1{0}3: 646, 718, 998 (length limit: 2000)
1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000)
1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000)
{1}: 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 (length limit: ≥100000)
1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000)
1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995 (length limit: 2000)
1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000)
1{z}: 581, 992, 1019 (length limit: ≥100000)
2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000)
2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000)
{2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000)
2{z}: 588, 972 (length limit: ≥100000)
3{0}1: 718, 912 (length limit: ≥100000)
3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000)
3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000)
{3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000)
3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000)
4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000)
4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000)
{4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000)
4{z}: 338, 998 (length limit: ≥100000)
5{0}1: 308, 512, 824 (length limit: ≥100000)
5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000)
6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000)
6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000)
7{0}1: (none)
7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000)
8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 785, 824, 828, 866, 908, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000)
8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000)
9{0}1: 724, 884 (length limit: ≥100000)
9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000)
A{0}1: 185, 338, 417, 432, 614, 668, 744, 773, 863, 935, 1000 (length limit: ≥100000)
A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000)
B{0}1: 560, 770, 968 (length limit: ≥100000)
B{z}: 263, 615, 912, 978 (length limit: ≥100000)
C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000)
{y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000)
y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000)
z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000)
{z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000)
{z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000)
{z}x: (none)
{z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000)
List of the length of the minimal primes (start with b+1) in given family for bases 2<=b<=1024 (only list families which must be minimal primes (start with b+1)): https://docs.google.com/spreadsheets...RwmKME/pubhtml ("RC" means this family can be ruled out as only contain composite numbers (only count numbers > base), "NB" means this family is not interpretable in this base (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime > base) in this base), "unknown" means this family is unsolved family) More information of minimal primes (start with b+1) in given family for bases 2<=b<=1024 (only list families which must be minimal primes (start with b+1)): https://en.wikipedia.org/w/index.php...did=1017467222 Last fiddled with by sweety439 on 2021-07-09 at 04:42 |
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2021-02-17, 14:32
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#141 |
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"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
5×7×83 Posts |
I tried to write a PARI/GP code that can print all minimal primes (start with b+1) up to length 1000 in given base b in <10 minutes, but not success, since the code for updating L (when L contains non-simple families) by "let w be the shortest sting in this family, if w has a subword in M, then remove the family from L, if w represents a prime, then add w to M, if the family can be proven to only contain composites, then remove the family from L" (see page of https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf) is very complex.
Thus, I only have the program that looks the small primes one-by-one, and I only checked the simple families of the from x{y}, x{y}, and x{0}y (where x,y are digits) to find the smallest primes in these simple families (or to prove that these simple families only contain composites). The more difficult case is: Non-simple families that can be proven to only contain composites, if the gcd (greatest common divisor) of the digits in these families is >1, then these families clearly only contain composites (note: we only count the numbers > base), but there exist many non-simple families with gcd of the digits = 1 and can be proven to only contain composites (and all subsequences of all numbers in these families represent composites, when we only count the numbers > base), e.g. {1}6{1} in base 9 {3}{0}5 in base 9 {3}{6}8 in base 9 (base 9 is the first base which has such families) Interestingly, base 9 is also the first base with some simple families x{y} or {x}y which are ruled out as only contain composites by 2-cover (i.e. full numerical covering set with only two prime factors), since base b has such families x{y} or {x}y if and only if b+1 is not prime or prime power, and gcd(repeating digit, b+1) = 1, the first base b such that b+1 is not prime or prime power is 5, but for base 5, the only such families are 3{1}, 4{1}, {1}3, {1}4, but the smallest prime in the family whose repeating digit is 1 may not be minimal prime (start with b+1), unless base b has no repunit primes (the first such bases b are 9, 25, 32, 49, 64, ...), and base 5 has repunit prime 111 (=31 in base 10), thus base 5 has no simple families x{y} or {x}y which are ruled out as only contain composites by 2-cover (i.e. full numerical covering set with only two prime factors), and next base b such that b+1 is not prime or prime power is 9, and base 9 has these simple families: 2{7}, 5{1}, 5{7}, 6{1}, {7}2, {1}5, {3}5, {7}5, {3}8, which are ruled out as only contain composites by covering set {2,5} (also the families 5{3}, 8{3}, {1}6, but they are already ruled out as only contain composites by trivial 1-cover set {3}) Since in any base b, for a repdigit (a number whose all digits are all same) to be prime (only count numbers > base), it must be a repunit and have a prime number of digits in its base (b), and for the simple families x{y} and {x}y in base b, the only chance of their smallest primes (if exist) are not minimal primes (start with b+1) in base b is the base b repdigit is prime, thus the repeating digit in these families must be 1, and since in bases 9, 25, 32, 49, 64, ... there are no repunit primes, thus in these bases, the smallest primes (if exist) in all simple families x{y} and {x}y are always minimal primes (start with b+1) in base b Last fiddled with by sweety439 on 2021-02-19 at 19:54 |
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2021-02-17, 14:36
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#142 |
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"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
5·7·83 Posts |
Except the families *{0}1 and *{z} (where * represents any string of digits), when the corresponding prime is large, the known primality tests for such a number are too inefficient to run (*{0}1 can be proven prime by N-1 primality test, *{z} can be proven prime by N+1 primality test). In this case one must resort to a probable primality test such as Miller–Rabin primality test and Baillie–PSW primality test, unless a divisor of the number can be found, since we are testing many numbers in an exponential sequence, it is possible to use a sieving process (such as srsieve software) to find divisors rather than using trial division.
There are three levels for these large minimal (probable) primes (start with b+1) base b: (let the large minimal (probable) primes (start with b+1) base b be N, and assume N > 10^3000, since all (probable) primes < 10^3000 can be easily proven prime by Primo) 1. either N-1 or N+1 can be trivially 100% factored (i.e. the primes in families *{0}1 and *{z} (where * represents any string of digits) in any base) (e.g. 2B3(0^15197)1 base 13, 4(D^19698) base 14, A(0^1355)1 base 17, F1(0^18523)1 base 19, 5D(0^19848)1 base 21, 4(0^341)1 base 23, 8(0^119214)1 base 23, C(0^1022)1 base 30, O(T^34205) base 30, 13(0^23614)1 base 33, Q(X^3086) base 34, 1B(0^56061)1 base 35, 2(0^2728)1 base 38, V(0^1527)1 base 38, L(b^1579) base 38, a(b^136211) base 38, 2(f^2523) base 42, N(i^153355) base 45, O(0^18521)1 base 45, 3(k^1555) base 47, T(0^133041)1 base 48, 7(0^515)1 base 50, g(0^4821)1 base 52, 8(0^227182)1 base 53, E(0^14954)1 base 57, L(0^1030)1 base 58), in this case we can use either Pocklington N-1 method or Morrison N+1 method to prove the primility of this minimal prime (start with b+1) base b. 2. neither N-1 nor N+1 can be trivially 100% factored, but either N-1 or N+1 can be trivially factored to product to a small number and a large base b repunit number (e.g. the generalized repunit primes (the primes in family {1}) (since {1} - 1 = {1}0 and {1}0 = 10 (small number) * {1} (repunit number)), the generalized half Fermat primes (the primes in family {#}$, for odd base b, # = (b-1)/2, $ = (b+1)/2) (since {#}$ - 1 = {#} and {#} = # (small number) * {1} (repunit number)), and the primes in families 1{2} (since 1{2} - 1 = 1{2}1 and 1{2}1 = 11 (small number) * {1} (repunit number)), 1{3} (since 1{3} - 1 = 1{3}2 and 1{3}2 = 12 (small number) * {1} (repunit number)), 1{4} (since 1{4} - 1 = 1{4}3 and 1{4}3 = 13 (small number) * {1} (repunit number)), {2}1 (since {2}1 - 1 = {2}0 and {2}0 = 20 (small number) * {1} (repunit number), also {2}1 + 1 = {2} and {2} = 2 (small number) * {1} (repunit number)), {3}1 (since {3}1 - 1 = {3}0 and {3}0 = 30 (small number) * {1} (repunit number)), {4}1 (since {4}1 - 1 = {4}0 and {4}0 = 40 (small number) * {1} (repunit number)), {2}3 (since {2}3 - 1 = {2} and {2} = 2 (small number) * {1} (repunit number)), {3}2 (since {3}2 + 1 = {3} and {3} = 3 (small number) * {1} (repunit number)), {3}4 (since {3}4 - 1 = {3} and {3} = 3 (small number) * {1} (repunit number)), {4}3 (since {4}3 + 1 = {4} and {4} = 4 (small number) * {1} (repunit number)), in any base) (e.g. 1(B^576) base 13, (7^1504)1 base 13, (9^308)1 base 13, (B^563)C base 13, (9^292)1 base 17, (G^2034)1 base 19, (3^1063)2 base 21, (7^230)1 base 21, (F^1091)G base 23, (H^1020)1 base 23, (K^3761)L base 23, (B^305)C base 25, (8^354)1 base 26, 1(H^4272) base 27, (2^1986)1 base 31, (3^4260)1 base 31, (P^1025)Q base 31, (V^251)W base 33, (1^313) base 35, (1^349) base 39, (1^4229) base 51) or can be factored to product to a small number and b^n+1 with large n (e.g. 9(0^3542)91 base 16, F(0^293)1 base 19, B(0^3529)C base 25, C(0^544)D base 29), in this case we require the factored part at least 33.333% for the (base b) repunit number, and the base b repunit number with length n has algebra factors: Phi_d(b) (where Phi is cyclotomic polynomial) for all d>1 dividing n), thus these numbers can be proven prime if these Phi_d(b) can be factored to make N-1 or N+1 over 33.333% factored, and this is equivalent to factor the Cunningham numbers b^n+-1 (references for factoring Cunningham numbers: b<=12 13<=b<=99 b=10 any b), if this base b repunit number at least 33.333% factored part, then we can prove the primility for this minimal prime (start with b+1) base b, otherwise we can only use probable primality test (since the known primality tests for such a number are too inefficient to run) such as Miller–Rabin primality test and Baillie–PSW primality test to show that this number is probable prime, and the possibility of this number is in fact composite is less than 10^(-679) if this minimal prime (start with b+1) base b is larger than 10^5000, reference: https://primes.utm.edu/notes/prp_prob.html 3. neither N-1 nor N+1 can be trivially 100% factored or trivially factored to product to a small number and a large base b repunit number, in this case we can only use probable primality test (since the known primality tests for such a number are too inefficient to run) such as Miller–Rabin primality test and Baillie–PSW primality test to show that this number is probable prime, and the possibility of this number is in fact composite is less than 10^(-679) if this minimal prime (start with b+1) base b is larger than 10^5000, reference: https://primes.utm.edu/notes/prp_prob.html however, in some primes which are case 2 or case 3, N-1 or N+1 still has algebra factors (like that some generalized Cullen/Woodall numbers have algebra factors) to make it over 33.333% factored, such as difference-of-squares factorization or difference-of-cubes factorization, e.g. 8(0^298)B base 18, N+1 = 8*18^299+12 = (18^2)*(8*18^297)+12 = 12*27*(8*18^297)+12 = 12*(27*(8*18^297)+1) = 12*(3*(2*18^99)+1)*(9*(4*18^198)-3*(2*18^99)+1) has sum-of-cubes factorization, to make it over 33.333% factored and thus this number can be proven prime with N+1 method, a non-example is 2(0^313)7 base 24, N+1 = 2*24^314+8 = 2^943*3^314+8 = 8*(2^940*3^314+1), N-1 = 2*24^314+6 = 2*(24^314+3) = 2*(2^942*3^314+3) = 6*(2^941*3^313+1), neither of them has algebra factorization, thus we can do nothing but using Primo to prove its primality. examples of prove the primility for the generalized repunit primes by factoring Phi_d(b) for d dividing n, click the link of the numbers in "Prime for Exponent" column Last fiddled with by sweety439 on 2021-07-02 at 18:58 |
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2021-02-17, 14:42
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#143 |
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"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36
5×7×83 Posts |
If the Sierpinski/Riesel CK for base b is <b (see http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm for the list of the CK for bases 2<=b<=1030), then the "minimal primes (start with b+1) base b problem" covers the Sierpinski base b problem and the Riesel base b problem, since all primes for the Sierpinski base b problem and the Riesel base b problem are minimal primes (start with b+1) base b
Last fiddled with by sweety439 on 2021-02-17 at 14:47 |
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