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Old 2021-10-05, 12:07   #188
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Quote:
Originally Posted by sweety439 View Post
Some known unsolved families for bases b<=64 not in the list for bases 2 to 30 or list for bases 28 to 50:

Base 11: 5{7} (found by me)
Base 13: 9{5} (found by me)
Base 13: A{3}A (found by me)
Base 17: 15{0}D (found by me)
Base 17: 1F{0}7 (found by me)
Base 18: C{0}C5 (found by me)
Base 25: F{2} (found by extended generalized Riesel conjecture base 25 with k > CK)
Base 31: 2{F} (found by extended generalized Riesel conjecture base 31)
Base 31: 3{5} (found by extended generalized Riesel conjecture base 31)
Base 32: 8{0}V (see https://oeis.org/A247952)
Base 32: S{V} (found by CRUS generalized Riesel conjecture base 1024)
Base 37: 2K{0}1 (found by CRUS generalized Sierpinski conjecture base 37)
Base 37: {I}J (see http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt)
Base 38: 1{0}V (see https://math.stackexchange.com/quest...the-form-38n31)
Base 43: 2{7} (found by extended generalized Riesel conjecture base 43)
Base 43: 3b{0}1 (found by CRUS generalized Sierpinski conjecture base 43)
Base 53: 19{0}1 (found by CRUS generalized Sierpinski conjecture base 53)
Base 53: 4{0}1 (found by CRUS generalized Sierpinski conjecture base 53)
Base 55: a{0}1 (found by CRUS generalized Sierpinski conjecture base 55)
Base 55: {R}S (see http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt)
Base 60: Z{x} (see CRUS generalized Riesel conjecture base 60)
Base 62: 1{0}1 (see http://jeppesn.dk/generalized-fermat.html)
Base 63: {V}W (see http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt)
No, base 32 family 8{0}V has covering set {3, 5, 41}, thus can be ruled out as only contain composite numbers.
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Old 2021-10-06, 23:52   #189
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New minimal prime (start with b+1) in base b is found for b=908: 8(0^243438)1, see post https://mersenneforum.org/showpost.p...&postcount=992

File https://docs.google.com/spreadsheets...RwmKME/pubhtml updated.
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Old 2021-10-16, 13:38   #190
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Conjecture: If sequence (a*b^n+c)/gcd(a+c,b-1) (a>=1 is integer, b>=2 is integer, c is (positive or negative) integer, |c|>=1, gcd(a,c) = 1, gcd(b,c) = 1) does not have covering set (full numerical covering set, full algebraic covering set, or partial algebraic/partial numerical covering set), then the sum of the reciprocals of the positive integers n such that (a*b^n+c)/gcd(a+c,b-1) is prime is converge (i.e. not infinity) and transcendental number. (of course, this conjecture will imply that there are infinitely many such n)

For the examples of (a,b,c) triples (a>=1 is integer, b>=2 is integer, c is (positive or negative) integer, |c|>=1, gcd(a,c) = 1, gcd(b,c) = 1) such that (a*b^n+c)/gcd(a+c,b-1) have covering set (full numerical covering set, full algebraic covering set, or partial algebraic/partial numerical covering set), see post https://mersenneforum.org/showpost.p...&postcount=678
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Old 2021-10-16, 13:45   #191
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Another conjecture (seems to already be proven, but I am not sure that): If all but finitely many primes p divide (a*b^n+c)/gcd(a+c,b-1) (a>=1 is integer, b>=2 is integer, c is (positive or negative) integer, |c|>=1, gcd(a,c) = 1, gcd(b,c) = 1) for some n>=1, then a=1 and c=-1, i.e. (a*b^n+c)/gcd(a+c,b-1) is generalized repunit number (b^n-1)/(b-1)

The factor tables have many examples for the special case that b=10, e.g. {2}1 in base 10 is (a,b,c) = (2,10,-11), the section Prime factors that appear periodically lists the primes that divide (a*b^n+c)/gcd(a+c,b-1) = (2*10^n-11)/9 for some n, and we note that the primes 2, 5, 11, 31, 37, 41, 43, 53, 71, 73, 79, 83, 101, 103, 107, 127, 137, 157, 173, 191, 199, 227, 239, 241, 251, 271, 281, 283, 307, 311, 317, 331, 347, 349, 353, 397, 409, 449, 523, 547, 563, 569, 599, 601, 613, 617, 631, 641, 643, 653, 661, 673, 691, 719, 733, 739, 751, 757, 761, 769, 773, 787, 797, 809, 827, 829, 839, 853, 859, 907, 911, 967, 991, 997, ..., divides no numbers of the form (a*b^n+c)/gcd(a+c,b-1) = (2*10^n-11)/9, and this sequence of primes seems to be infinite, another example is 1{3} in base 10 is (a,b,c) = (4,10,-1), the section Prime factors that appear periodically lists the primes that divide (a*b^n+c)/gcd(a+c,b-1) = (4*10^n-1)/3 for some n, and we note that the primes 2, 3, 5, 11, 37, 41, 53, 73, 79, 101, 103, 137, 139, 173, 211, 239, 241, 271, 277, 281, 317, 331, 349, 353, 397, 421, 449, 463, 521, 547, 607, 613, 617, 661, 673, 733, 751, 757, 773, 797, 829, 853, 859, 907, 967, ..., divides no numbers of the form (a*b^n+c)/gcd(a+c,b-1) = (4*10^n-1)/3, and this sequence of primes seems to be infinite.

Of course this conjecture also include for bases other than 10, i.e. for every (a*b^n+c)/gcd(a+c,b-1) (a>=1 is integer, b>=2 is integer, c is (positive or negative) integer, |c|>=1, gcd(a,c) = 1, gcd(b,c) = 1) family other than generalized repunit family (b^n-1)/(b-1) (i.e. a=1 and c=-1), there are infinitely many primes not dividing any number of the form (a*b^n+c)/gcd(a+c,b-1), e.g. for the base 11 unsolved family 5{7} = (57*11^n-7)/10, the primes dividing no numbers of the form (a*b^n+c)/gcd(a+c,b-1) = (57*11^n-7)/10 are 3, 7, 11, 19, 37, 43, 61, 83, 89, 107, 131, 137, 157, 191, 193, 199, 211, 229, 241, 257, 269, 307, 311, 313, 317, 379, 389, 397, 421, 431, 439, 449, 457, 479, 503, 509, 521, 523, 541, 547, 571, 577, 607, 617, 631, 641, 653, 659, 661, 691, 727, 739, 743, 751, 757, 773, 787, 797, 811, 827, 829, 907, 911, 919, 967, ..., and for the base 17 unsolved family F1{9} = (4105*17^n-9)/16, the primes dividing no numbers of the form (a*b^n+c)/gcd(a+c,b-1) = (4105*17^n-9)/16 are 3, 5, 17, 29, 43, 59, 67, 71, 79, 101, 103, 137, 151, 157, 163, 179, 181, 191, 199, 223, 229, 239, 241, 257, 263, 281, 293, 307, 331, 337, 353, 359, 373, 383, 389, 409, 433, 443, 457, 461, 463, 491, 509, 541, 563, 587, 601, 619, 631, 647, 659, 661, 727, 733, 739, 757, 761, 769, 773, 797, 811, 821, 829, 859, 863, 877, 883, 919, 937, 947, 953, 967, 977, 991, ... (it is surprising that many primes do not divide (4105*17^n-9)/16 for any n, since (4105*17^n-9)/16 is a low-weight form, i.e. (4105*17^n-9)/16 is divisible by a small prime for most n, note that (4105*17^n-9)/16 has no algebraic factorization for any n, since 4105 is not perfect power)

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Old 2021-10-16, 13:54   #192
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The smallest generalized near-repdigit primes (i.e. of the form x{y} or {x}y) base b is always minimal primes (start with b+1) in base b unless the repeating digit (i.e. y for x{y}, or x for {x}y) is 1, since the generalized repunit numbers base b may be prime unless b is 9, 25, 32, 49, 64, 81, 121, 125, 144, ... (A096059) bases without any generalized repunit primes, and for a repdigit to be prime, it must be a repunit (i.e. the repeating digit is 1) and have a prime number of digits in its base (except trivial single-digit numbers), since, for example, the repdigit 77777 is divisible by 7, in any base > 7, thus (i.e. of the form x{y} or {x}y) base b is always minimal primes (start with b+1) in base b if the repeating digit (i.e. y for x{y}, or x for {x}y) is not 1, thus, the families A{1} in base 22 and 8{1} in base 33 and 4{1} in base 40 are not unsolved families in this problem (i.e. finding all minimal primes (start with b+1) in base b) although all they are near-repdigit families and all they have no known primes or PRPs and none of them can be ruled out as only contain composites (only count numbers > base), since their repeating digit are 1, and the prime F(1^957) in base 24 (its value is (346*24^957-1)/23) is not minimal prime (start with b+1) in base b=24, since its repeating digit is 1

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Old 2021-10-17, 17:00   #193
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The algebra form ((a*b^n+c)/d) of the unsolved families are:

Code:
base   unsolved family   algebra form
11   5(7^n)   (57*11^n-7)/10
13   9(5^n)   (113*13^n-5)/12
13   A(3^n)A   (41*13^(n+1)+27)/4
16   (3^n)AF   (16^(n+2)+619)/5
16   (4^n)DD   (4*16^(n+2)+2291)/15
See https://stdkmd.net/nrr/exprgen.htm for the algebra form calculator (only for base 10 families), also see page 16 of https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf
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Old 2021-10-18, 18:04   #194
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Families which can be ruled out as contain no primes (only count numbers > base) by reasons other than trivial 1-cover are:

Code:
Base 5:
{1}3 (covering set {2,3}) (not produce minimal primes (start with b+1) since 111 is prime)
{1}4 (covering set {2,3}) (not produce minimal primes (start with b+1) since 111 is prime)
3{1} (covering set {2,3}) (not produce minimal primes (start with b+1) since 111 is prime)
4{1} (covering set {2,3}) (not produce minimal primes (start with b+1) since 111 is prime)
Base 8:
1{0}1 (sum of cubes)
6{4}7 (covering set {3,5,13}) (not produce minimal primes (start with b+1) since 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447 is prime)
Base 9:
{1} (difference of squares)
{1}5 (covering set {2,5})
2{7} (covering set {2,5})
3{1} (difference of squares)
{3}5 (covering set {2,5})
3{8} (difference of squares)
{3}8 (covering set {2,5})
5{1} (covering set {2,5})
5{7} (covering set {2,5})
6{1} (covering set {2,5})
{7}2 (covering set {2,5})
{7}5 (covering set {2,5})
{8}5 (difference of squares)
Base 11:
{1}3 (covering set {2,3}) (not produce minimal primes (start with b+1) since 11111111111111111 is prime)
{1}4 (covering set {2,3}) (not produce minimal primes (start with b+1) since 11111111111111111 is prime)
{1}9 (covering set {2,3}) (not produce minimal primes (start with b+1) since 11111111111111111 is prime)
{1}A (covering set {2,3}) (not produce minimal primes (start with b+1) since 11111111111111111 is prime)
2{5} (covering set {2,3})
3{1} (covering set {2,3}) (not produce minimal primes (start with b+1) since 11111111111111111 is prime)
3{5} (covering set {2,3})
3{7} (covering set {2,3})
4{1} (covering set {2,3}) (not produce minimal primes (start with b+1) since 11111111111111111 is prime)
4{7} (covering set {2,3})
{5}2 (covering set {2,3})
{5}3 (covering set {2,3})
{5}8 (covering set {2,3})
{5}9 (covering set {2,3})
{7}4 (covering set {2,3})
{7}9 (covering set {2,3})
{7}A (covering set {2,3})
8{5} (covering set {2,3})
9{1} (covering set {2,3}) (not produce minimal primes (start with b+1) since 11111111111111111 is prime)
9{5} (covering set {2,3})
9{7} (covering set {2,3})
A{1} (covering set {2,3}) (not produce minimal primes (start with b+1) since 11111111111111111 is prime)
A{7} (covering set {2,3})
Base 12:
{B}9B (combined with covering set {13} and difference of squares)

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Old 2021-10-19, 11:04   #195
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Like http://www.wiskundemeisjes.nl/wp-con...02/primes2.pdf, if you write down a prime > 10, then I can always strike out 0 or more digits to get a prime on this set: {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}

In fact, if a primes which do not contain any of the 1st to the 76th primes in the set, then this prime must be of the form 5{0}27 = 5*10^(n+2)+27, and if a primes which do not contain any of the 1st to the 75th primes in the set, then the prime must be of one of these forms: 5{0}27 = 5*10^(n+2)+27, {5}1 = (5*10^(n+1)-41)/9, 8{5}1 = (77*10^(n+1)-41)/9, by our theorem, such primes must contain either the 76th prime or the 77th prime as subsequence, however any such prime cannot contain both the 76th prime and the 77th prime as subsequences, since the 76th prime contain "1" and the 77th prime contain "7", any prime (in fact, any number, need not to be prime) containing both the 76th prime and the 77th prime as subsequences will contain both "1" and "7", and hence contain either "17" or "71" (or both) as subsequence, but both 17 and 71 are primes > 10

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Old 2021-10-19, 13:04   #196
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A minimal prime (start with b+1) base b is called infinite-minimal prime iff there are infinitely many primes which have this minimal prime (start with b+1) as subsequence (when written as base b string) but not have any other minimal prime (start with b+1) base b as subsequence (when written as base b string).

e.g. the largest minimal prime (start with b+1) 5000000000000000000000000000027 is likely infinite-minimal prime in base 10, since there are likely infinitely many primes of the form 5{0}27 in base 10 (reference: https://stdkmd.net/nrr/prime/primecount.txt, search: “50w27”), so is the second-largest minimal prime (start with b+1) 555555555551, since there are likely infinitely many primes of the form {5}1 (search “5w1” in the reference page) or 8{5}1 (search “85w1” in the reference page), but the third-largest minimal prime (start with b+1) and the fourth-largest minimal prime (start with b+1), 80555551 and 66600049 are not infinite-minimal prime in base 10, since the only primes which have no minimal prime (start with b+1) base b=10 but 80555551 as subsequence are 80555551 and 8055555551, and the only prime which have no minimal prime (start with b+1) base b=10 but 66600049 as subsequence is 66600049.

In base 2, of course, 11 is infinite-minimal prime, since all odd primes have 11 as subsequence (at least, the first digit and the last digit of all odd primes are both 1), and there are infinitely many such primes (reference: Euclid's Proof), and in base 3, all the three minimal primes (start with b+1) are very likely infinite-minimal primes, since for 12, all primes of the form 1{2} (A003307) or 1{0}2 (A051783) have no subsequence 21 or 111, and for 21, all primes of the form {2}1 (A014224) or 2{0}1 (A003306) have no subsequence 12 or 111, and for 111, all primes of the form {1} (A028491) or 1{0}11 (A058958) or 11{0}1 (A005537) have no subsequence 12 or 21, and all these seven sequences are conjectured to be infinite.

Problem: In base 10, how many of the 77 minimal primes (start with b+1) are infinite-minimal primes? (assume the conjecture in this post)

In base 10, 1235607889460606009419 the smallest prime containing all 26 original minimal primes (i.e. p>b not needed) as subsequences (see https://www.primepuzzles.net/puzzles/puzz_178.htm and https://primes.utm.edu/curios/page.p...606009419.html), problem: Find the smallest prime containing all 77 minimal primes (start with b+1) in base b=10, also this problem can be generalized to other bases, in base 2, this prime is clearly 11, and in base 3, this prime is 1121 (since this number is the smallest number containing both 21 and 111 as subsequences, and indeed this number is prime and containing 12 as subsequence).

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Old 2021-10-23, 20:29   #197
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Base b=72 has largest known minimal prime (start with b+1) for all bases 2<=b<=1024: 3(71^1119849), its value is 4*72^1119849-1, it has 2079933 digits when written in decimal (this number is proven prime, for unproven probable prime, the largest known minimal prime (start with b+1) for all bases 2<=b<=1024 is base b=23, the PRP 9(14^800873), its value is (106*23^800873-7)/11, it has 1090573 digits when written in decimal)
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Old 2021-10-23, 20:39   #198
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Quote:
Originally Posted by sweety439 View Post
New minimal prime (start with b+1) in base b is found for b=650: 3:{649}^(498101), see https://mersenneforum.org/showpost.p...&postcount=931

Added it to excel file https://docs.google.com/spreadsheets...RwmKME/pubhtml

Base 108 is an interesting base since ....

* For the family {1}, length 2 is prime, but the next prime is large (length 449)
* For the family 1{0}1, length 2 is prime, the next prime is not known
* For the family y{z}, first prime is large (length 411)
* For the family 11{0}1, first prime is large (length 400)
* For the family {y}z, first prime is large (length 492) (note that length 1 is also prime, but length 1 is not allowed in this project)
* For the family 6{0}1, first prime is large (length 16318)
* For the family #{z} (# = (base/2)-1)), first prime is large (length 7638)

This situation is not common in bases with many divisors, but although 108 has many divisors, this situation occurs in this base, this is why this base is interesting :))

Also base 282 ....

* For the family A{0}1, first prime is large (length 1474)
* For the family C{0}1, first prime is large (length 2957)
* For the family z{0}1, first prime is large (length 277)

http://www.noprimeleftbehind.net/cru...82-reserve.htm only tells you that all these three families have a prime with length <= 100001 ....

* For the family 7{z}, first prime is large (length 21413)
* For the family 10{z}, first prime is large (length 780)

http://www.noprimeleftbehind.net/cru...82-reserve.htm only tells you that the farmer family has a prime with length <= 100001, and the letter family has a prime with length <= 100002
Another interesting base is b=23, this base is SOLVED when single-digit primes (i.e. the primes p<23) and the prime "10" (i.e. the prime p=23) are included, and this base is the solved base with the largest minimal prime ((106*23^800873-7)/11, or 9(E^800873), which has 1090573 digits when written in decimal) and the largest number of minimal primes (6021 primes or PRPs, while the second-largest number (base 42) has 4551 proven primes), this base seems be easy and high weight, but in fact this base is low-weight base and this base has large primes for special forms listed in https://docs.google.com/spreadsheets...RwmKME/pubhtml ....

* 1{0}2 (b^n+2): 12 digits [a record value for bases b] (minimal prime in my project, but not minimal prime in original project (i.e. the primes p<=base are also included), since this prime has "10" and "2" as subsequences)
* 4{0}1 (4*b^n+1): 343 digits [a record value for bases b]
* {4}1: 13 digits [not a record value for bases b, since base b=11 requires 45 digits]
* 8{0}1 (8*b^n+1): 119216 digits [a record value for bases b]
* {K}L (extended Sierpinski problem with k>base (k=10)): 3762 digits [a record value for base b for extended Sierpinski problem with k=10 (the prime (10*23^3762+1)/11, exponent is 3762), the previous record is b=17, the prime 10*17^1356+1, exponent is 1356]
* y{z} ((b-1)*b^n-1, Williams prime of the 1st kind): 56 digits [a record value for bases b]
* z{0}1 ((b-1)*b^n+1, Williams prime of the 2nd kind): 15 digits [not a record value for bases b, since base b=19 requires 30 digits, also the same as base b=20, both require 15 digits]
* {z}1 (b^n-(b-1), dual Williams prime of the 1st kind): 17 digits [a record value for bases b, the same as base b=20, both require 17 digits]
* {z}y (b^n-2): 24 digits [a record value for bases b]

Last fiddled with by sweety439 on 2021-10-26 at 22:08
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