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Old 2021-10-05, 12:07   #188
sweety439
 
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"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

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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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"99(4^34019)99 palind"
Nov 2016
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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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"99(4^34019)99 palind"
Nov 2016
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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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"99(4^34019)99 palind"
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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, ... 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.
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Old 2021-10-16, 13:54   #192
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"99(4^34019)99 palind"
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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

Last fiddled with by sweety439 on 2021-10-17 at 12:55
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Old 2021-10-17, 17:00   #193
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"99(4^34019)99 palind"
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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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