mersenneforum.org Minimal set of the strings for primes with at least two digits
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2021-11-28, 17:20   #221
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

5×7×89 Posts

These are all minimal primes (start with b+1) in base b=25 up to 2^32

Base 25 is a very hard base (however, of course, bases > 25 which are coprime to 6 are harder than it), we can imagine an alien force, vastly more powerful than us, landing on Earth and demanding the set of minimal primes (start with b+1) in base b=17 (or 19, 21, 22, 23, 28, 30, 36) (including primality proving of the probable primes in this set) or they will destroy our planet. In that case, I claim, we should marshal all our computers and all our mathematicians and attempt to find the set and to prove the primality of all numbers in this set. But suppose, instead, that they ask for the set of minimal primes (start with b+1) in base b=25 (or 26, 27, 29, 31, 32, 33, 34, 35). In that case, I believe, we should attempt to destroy the aliens.
Attached Files
 base 25.txt (458.2 KB, 11 views)

Last fiddled with by sweety439 on 2021-11-28 at 18:46

2021-11-28, 17:23   #222
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

5·7·89 Posts

Quote:
 Originally Posted by sweety439 Upload text files, searched up to length 5000 For 11{0}1 (the dual of 1{0}11), see https://www.rieselprime.de/ziki/Williams_prime_PP_least For 10{z} (the dual of {z}yz), see https://www.rieselprime.de/ziki/Williams_prime_PM_least
This is the text file for {z0}z1 (i.e. generalized Wagstaff primes, see A084742, but exclude p=3, and use the length of the primes (i.e. use p-1 instead of p)), also searched to length 5000

very important note: they are minimal primes (start with b'+1) in base b'=b^2 instead of base b'=b
Attached Files
 z0z0z1.txt (8.5 KB, 12 views)

Last fiddled with by sweety439 on 2021-11-28 at 17:25

2021-11-30, 19:59   #223
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

1100001010112 Posts

more datas
Attached Files
 base 25.txt (551.0 KB, 9 views) base 20.txt (19.9 KB, 9 views) base 24.txt (19.2 KB, 9 views)

2021-11-30, 20:41   #224
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

5·7·89 Posts

Quote:
 Originally Posted by sweety439 In base 10, the set such strings are not simply to write, however, if "primes > base" is not needed, then such strings are any strings n such that A039995(n) = 0 (not A062115, since A062115 is for substring instead of subsequence, i.e. A062115 is the numbers n such that A039997(n) = 0 instead of the numbers n such that A039995(n) = 0) with any number (including 0) of leading zeros. Such strings are called primefree strings in this post.
In base 10, a sting is primefree string if and only if:

The string is of the form {0}S{0,2,4,5,6,8} and S is of one of these forms:

* empty string
* "gcd of its digits" <> 1 (note: gcd(0,n) = n for all n, including n=0)
* X{0}Y with X+Y divisible by 3 (this includes: 2{0}1, 2{0}7, 5{0}1, 5{0}7, 8{0}1, 8{0}7)
* 28{0}7
* 4{6}9
* 221
* 2021
* 2201
* 22001
* 220001
* 2200001
* (5^n)1 with n<11
* 581
* 5(0^n)27 with n<28
* 5207
* 52007
* 520007
* 649
* 6649
* 66649
* 6049
* 60049
* 600049
* 6000049
* 66049
* 660049
* 6600049
* 666049
* 6660049
* 8(5^n)1 with n<11
* 8051
* 80551
* 805551
* 8055551
* 91
* 901
* 921
* 951
* 981
* 9021
* 9051
* 9081
* 9201
* 9501
* 9581
* 9801
* 90581
* 95081
* 949
* 9469
* 94669

reference: https://math.stackexchange.com/quest...e-number-in-it

Last fiddled with by sweety439 on 2021-11-30 at 20:47

2021-11-30, 20:52   #225
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

5·7·89 Posts

Update data for minimal primes, see https://sites.google.com/view/data-of-minimal-primes
Attached Files
 data for minimal primes.txt (5.9 KB, 9 views)

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