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Forum: sweety439 2021-11-30, 20:52
Replies: 224
Views: 23,478
Posted By sweety439
Update data for minimal primes, see...

Update data for minimal primes, see https://sites.google.com/view/data-of-minimal-primes
Forum: sweety439 2021-11-30, 20:41
Replies: 224
Views: 23,478
Posted By sweety439
In base 10, a sting is primefree string if and...

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) =...
Forum: sweety439 2021-11-30, 19:59
Replies: 224
Views: 23,478
Posted By sweety439
more datas

more datas
Forum: sweety439 2021-11-28, 19:42
Replies: 136
Views: 19,474
Posted By sweety439
For the minimal set of "noncomposites" (i.e. 0,...

For the minimal set of "noncomposites" (i.e. 0, 1, primes), all bases <=24 are completely solved:


b largest minimal noncomposite
2 1
3 2
4 3
5 3
6 5
7 5
Forum: sweety439 2021-11-28, 17:23
Replies: 224
Views: 23,478
Posted By sweety439
This is the text file for {z0}z1 (i.e....

This is the text file for {z0}z1 (i.e. generalized Wagstaff primes, see A084742 (https://oeis.org/A084742), but exclude p=3, and use the length of the primes (i.e. use p-1 instead of p)), also...
Forum: sweety439 2021-11-28, 17:20
Replies: 224
Views: 23,478
Posted By sweety439
These are all minimal primes (start with b+1) in...

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...
Forum: sweety439 2021-11-28, 15:12
Replies: 136
Views: 19,474
Posted By sweety439
If 1 is regarded as prime...

If 1 is regarded as prime (https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.pdf), then all bases <=20 are solved, in fact, all bases <=24 except the base 21 family G{0}FK is solved.


b ...
Forum: sweety439 2021-11-26, 22:14
Replies: 224
Views: 23,478
Posted By sweety439
For families X{Y}Zb = (a*b^n+c)/gcd(a+c,b-1)...

For families X{Y}Zb = (a*b^n+c)/gcd(a+c,b-1) (a>=1, b>=2, c != 0, gcd(a,c) = 1, gcd(b,c) = 1), which can be ruled out as only containing composite, which are by covering congruence, and which are by...
Forum: sweety439 2021-11-26, 17:44
Replies: 224
Views: 23,478
Posted By sweety439
Upload text files, searched up to length 5000 ...

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...
Forum: sweety439 2021-11-26, 14:56
Replies: 224
Views: 23,478
Posted By sweety439
e.g. (list all sub-families of given unsolved...

e.g. (list all sub-families of given unsolved family, in the same base)

base 11 unsolved family 5{7}:

{7}: divisible by 7

base 13 unsolved family A{3}A:

A{3}: covering set {2,7}
{3}A:...
Forum: sweety439 2021-11-25, 11:48
Replies: 224
Views: 23,478
Posted By sweety439
Now I consider to add these families to the list...

Now I consider to add these families to the list (https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml), although...
Forum: sweety439 2021-11-24, 18:59
Replies: 224
Views: 23,478
Posted By sweety439
Since triangular number * 9 + 1 are always...

Since triangular number * 9 + 1 are always triangular numbers, and no triangular numbers > 3 are primes, thus these families in base 9 contain no primes (only count numbers > base), since they can be...
Forum: sweety439 2021-11-24, 18:45
Replies: 224
Views: 23,478
Posted By sweety439
Our result about minimal primes (start with b+1)...

Our result about minimal primes (start with b+1) assume that a number which has passed Miller–Rabin tests (https://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test) to the first 9 prime...
Forum: sweety439 2021-11-24, 18:02
Replies: 163
Views: 22,079
Posted By sweety439
OEIS sequence for Williams primes: 1st:...

OEIS sequence for Williams primes:

1st: (b-1)*b^n-1
2nd: (b-1)*b^n+1
3rd: (b+1)*b^n-1
4th: (b+1)*b^n+1
1st dual: b^n-(b-1)
2nd dual: b^n+(b-1)
3rd dual: b^n-(b+1)
4th dual: b^n+(b+1)
Forum: sweety439 2021-11-23, 18:13
Replies: 224
Views: 23,478
Posted By sweety439
See https://stdkmd.net/nrr/prime/primesize.txt,...

See https://stdkmd.net/nrr/prime/primesize.txt, sequences of the form k*10^n+1 includes Generalized Cullen prime numbers, and sequences of the form k*10^n-1 includes Generalized Woodall prime...
Forum: sweety439 2021-11-23, 15:45
Replies: 224
Views: 23,478
Posted By sweety439
Every base b family is the dual (for the...

Every base b family is the dual (for the definition, see http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf and https://oeis.org/A076336/a076336c.html and...
Forum: sweety439 2021-11-23, 14:29
Replies: 395
Views: 28,458
Posted By sweety439
Since the multiplicative order...

Since the multiplicative order (https://en.wikipedia.org/wiki/Multiplicative_order) of b mod prime p (i.e. znorder(Mod(b,p))) always divides p-1, thus, p must divide Phi(n,b) (where Phi is cyclotomic...
Forum: sweety439 2021-11-21, 15:27
Replies: 224
Views: 23,478
Posted By sweety439
For the forms in...

For the forms in https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml, it is conjectured that every base has...
Forum: sweety439 2021-11-20, 10:57
Replies: 224
Views: 23,478
Posted By sweety439
This problem is better than the original minimal...

This problem is better than the original minimal prime problem since this problem is regardless whether 1 is considered as prime or not (https://primes.utm.edu/notes/faq/one.html), i.e. no matter 1...
Forum: sweety439 2021-11-20, 10:39
Replies: 224
Views: 23,478
Posted By sweety439
Although {z0}z1 almost cannot be minimal prime...

Although {z0}z1 almost cannot be minimal prime (start with b+1), but always minimal prime (start with b'+1) in base b'=b^2, its formula is (b^n+1)/(b+1), like the generalized repunit {1} =...
Forum: And now for something completely different 2021-11-18, 12:30
Replies: 71
Views: 8,556
Posted By sweety439
but A004023 (https://oeis.org/A004023) already...

but A004023 (https://oeis.org/A004023) already shown that "8177207" as a(11) ....
Forum: sweety439 2021-11-18, 12:22
Replies: 395
Views: 28,458
Posted By sweety439
n is a quadratic nonresidue modulo all odd primes...

n is a quadratic nonresidue modulo all odd primes p <= sqrt(n) which do not divide n, for these n:

{1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 14, 15, 17, 18, 20, 21, 23, 24, 27, 30, 32, 33, 35, 38, 42,...
Forum: sweety439 2021-11-18, 06:09
Replies: 9
Views: 437
Posted By sweety439
Also: Conjecture 1: (A) A143539...

Also:

Conjecture 1:

(A) A143539 (https://oeis.org/A143539) = 0
(AB) A046923 (https://oeis.org/A046923) = 0
(BAA+BBA+CA) A064272 (https://oeis.org/A064272) = 0
(BBA+CA) A002471...
Forum: sweety439 2021-11-18, 05:41
Replies: 9
Views: 437
Posted By sweety439
The counterexamples listed as sequences: (A)...

The counterexamples listed as sequences:

(A) Odd counterexamples
(AA) Odd prime counterexamples
(AB) Odd composite counterexamples
(AC) Odd unit (only for "1") counterexamples
[note: Odd...
Forum: sweety439 2021-11-18, 04:56
Replies: 9
Views: 437
Posted By sweety439
For these counterexamples for conjecture 1, all...

For these counterexamples for conjecture 1, all numbers >10 except 1542 are == 2 mod 3 (reference: https://oeis.org/A060003, this says that these counterexamples for conjecture 1, all odd numbers >3...
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