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Old 2021-11-28, 17:20   #221
sweety439
 
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"99(4^34019)99 palind"
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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 all 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.
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File Type: txt base 25.txt (458.2 KB, 40 views)

Last fiddled with by sweety439 on 2021-12-11 at 18:48
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Old 2021-11-28, 17:23   #222
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Quote:
Originally Posted by sweety439 View Post
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
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File Type: txt z0z0z1.txt (8.5 KB, 38 views)

Last fiddled with by sweety439 on 2021-11-28 at 17:25
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Old 2021-11-30, 19:59   #223
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"99(4^34019)99 palind"
Nov 2016
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more datas
Attached Files
File Type: txt base 25.txt (551.0 KB, 33 views)
File Type: txt base 20.txt (19.9 KB, 33 views)
File Type: txt base 24.txt (19.2 KB, 35 views)
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Old 2021-11-30, 20:41   #224
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Quote:
Originally Posted by sweety439 View Post
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
* single-digit number
* "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
* 949
* 9469
* 94669

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

Last fiddled with by sweety439 on 2021-12-20 at 21:44 Reason: remove 95081 from the list since 5081 is prime
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Old 2021-11-30, 20:52   #225
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Update data for minimal primes, see https://sites.google.com/view/data-of-minimal-primes
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Old 2021-12-06, 10:14   #226
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Quote:
Originally Posted by sweety439 View Post
Related search for minimal primes (generalized form: (a*b^n+c)/d) in PRP top:

b^n+c

b^n-c

a*b^n+c

a*b^n-c

(b^n+c)/d

(b^n-c)/d

(a*b^n+c)/d

(a*b^n-c)/d

Also for the special case c = +-1 and d = 1, they are proven primes, the search page in top 5000 primes:

https://primes.utm.edu/primes/search_proth.php
Forms in https://docs.google.com/spreadsheets...RwmKME/pubhtml:

1{0}2 b^n+2: http://www.primenumbers.net/prptop/s...&action=Search

1{0}3 b^n+3: http://www.primenumbers.net/prptop/s...&action=Search

1{0}4 b^n+4: http://www.primenumbers.net/prptop/s...&action=Search

{z}w b^n-4: http://www.primenumbers.net/prptop/s...&action=Search

{z}x b^n-3: http://www.primenumbers.net/prptop/s...&action=Search

{z}y b^n-2: http://www.primenumbers.net/prptop/s...&action=Search
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Old 2021-12-06, 13:17   #227
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Apprciation is felt for your effort @sweety439 and all
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Old 2021-12-07, 14:15   #228
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Quote:
Originally Posted by sweety439 View Post
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 (Note: {62}:63 for b = 125 can be ruled out as contain no primes > base, by sum-of-cubes factorization, thus the smallest prime of the form 12:{62}:63 for b = 125 (if exists) must be minimal prime (start with b+1) in base b = 125)
* 16:{0}:1 for b = 128
* 36:{0}:1 for b = 216
* 24:{171}:172 for b = 343 (Note: {171}:172 for b = 343 can be ruled out as contain no primes > base, by sum-of-cubes factorization, thus the smallest prime of the form 24:{171}:172 for b = 343 (if exists) must be minimal prime (start with b+1) in base 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
For GFN families in bases b <= 1024 which are odd powers (i.e. in https://oeis.org/A070265) (other bases already have information in http://jeppesn.dk/generalized-fermat.html and http://www.noprimeleftbehind.net/crus/GFN-primes.htm and http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt, which are 1:{0}:1 for even bases and {((b-1)/2)}:((b+1)/2) for odd bases):

b=8:

1:{0}:1, ruled out as only contain composite numbers
2:{0}:1, prime at length 2
4:{0}:1, prime at length 3

b=27:

{13}:14, ruled out as only contain composite numbers
1:{13}:14, prime at length 2, also 1:{13} has prime at length 3
4:{13}:14, prime at length 11, also 4:{13} has prime at length 24

b=32:

1:{0}:1, ruled out as only contain composite numbers
2:{0}:1, prime at length 4
4:{0}:1, unsolved family
8:{0}:1, prime at length 2
16:{0}:1, unsolved family

b=64:

1:{0}:1, ruled out as only contain composite numbers
4:{0}:1, prime at length 2
16:{0}:1, prime at length 3

b=125:

{62}:63, ruled out as only contain composite numbers
2:{62}:63, prime at length 2
12:{62}:63, unsolved family

b=128:

1:{0}:1, ruled out as only contain composite numbers
2:{0}:1, prime at length 2
4:{0}:1, prime at length 3
8:{0}:1, ruled out as only contain composite numbers
16:{0}:1, unsolved family
32:{0}:1, ruled out as only contain composite numbers
64:{0}:1, ruled out as only contain composite numbers

b=216:

1:{0}:1, ruled out as only contain composite numbers
6:{0}:1, prime at length 2
36:{0}:1, unsolved family

b=243:

{121}:122, ruled out as only contain composite numbers
1:{121}:122, prime at length 4, also 1:{121} has prime at length 15
4:{121}:122, prime at length 7, also 4:{121} has prime at length 2
13:{121}:122, although no known prime or PRP in this family, but 13:{121} has prime at length 3, thus 13:{121}:122 is still not unsolved family since the smallest prime (if exists) in this family will not be minimal prime (start with b+1)
40:{121}:122, prime at length 13, however 40:{121} has no known prime or PRP, and 40:{121} is unsolved family (however, 40:{121} is GRU family instead of GFN family, for reference of this family, see https://oeis.org/A028491, there are no single known number in https://oeis.org/A028491 which is == 4 mod 5)

b=343:

{171}:172, ruled out as only contain composite numbers
3:{171}:172, prime at length 2
24:{171}:172, unsolved family

b=512:

1:{0}:1, ruled out as only contain composite numbers
2:{0}:1, unsolved family
4:{0}:1, unsolved family
8:{0}:1, ruled out as only contain composite numbers
16:{0}:1, unsolved family
32:{0}:1, unsolved family
64:{0}:1, ruled out as only contain composite numbers
128:{0}:1, prime at length 2
256:{0}:1, unsolved family

b=729:

{364}:365, ruled out as only contain composite numbers
4:{364}:365, prime at length 6
40:{364}:365, prime at length 3

b=1000:

1:{0}:1, ruled out as only contain composite numbers
10:{0}:1, unsolved family
100:{0}:1, unsolved family

b=1024:

1:{0}:1, ruled out as only contain composite numbers
4:{0}:1, unsolved family
16:{0}:1, unsolved family
64:{0}:1, prime at length 2
256:{0}:1, unsolved family

Last fiddled with by sweety439 on 2021-12-11 at 15:34
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Old 2021-12-08, 08:21   #229
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Quote:
Originally Posted by sweety439 View 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
* single-digit number
* "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
* 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
* 949
* 9469
* 94669
Like minimal element of prime numbers > b in base b, we can find the maximal element of primefree strings:

{0}S{0,2,4,5,6,8} with these sets S:

* {0,3,6,9}
* {0,7}
* 2{0}1
* 5{0}1
* 5{0}7
* 8{0}1
* 28{0}7
* 4{6}9
* 2021
* 2200001
* (5^10)1
* 581
* 5(0^27)27
* 520007
* 6000049
* 6600049
* 6660049
* 8(5^10)1
* 8055551
* 9021
* 9201
* 9801
* 90581
* 94669

(note: {0,2,4,6,8} and {0,5} are already included in the digits after S, i.e. {0,2,4,5,6,8})
(note: 2{0}7 and 8{0}7 are already sub-families of 28{0}7)
(note: 66649 is already subsequence of 6660049)
(note: 581, 9051, 9081, 9501, 9581 are already subsequences of 90581 or/and 95081)

Last fiddled with by sweety439 on 2021-12-20 at 21:45 Reason: remove 95081 from the list since 5081 is prime
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Old 2021-12-08, 08:29   #230
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and we have these "maximal primefree strings"

b=2:

* {0}1{0}

b=3:

* {0}1{0}1{0}
* {0,2}

b=4:

* {0}2{0}1{0,2}
* {0}{0,3}{0,2}

b=10:

* {0}{0,3,6,9}{0,2,4,5,6,8}
* {0}{0,7}{0,2,4,5,6,8}
* {0}2{0}1{0,2,4,5,6,8}
* {0}5{0}1{0,2,4,5,6,8}
* {0}5{0}7{0,2,4,5,6,8}
* {0}8{0}1{0,2,4,5,6,8}
* {0}28{0}7{0,2,4,5,6,8}
* {0}4{6}9{0,2,4,5,6,8}
* {0}2021{0,2,4,5,6,8}
* {0}2200001{0,2,4,5,6,8}
* {0}55555555551{0,2,4,5,6,8}
* {0}500000000000000000000000000027{0,2,4,5,6,8}
* {0}520007{0,2,4,5,6,8}
* {0}6000049{0,2,4,5,6,8}
* {0}6600049{0,2,4,5,6,8}
* {0}6660049{0,2,4,5,6,8}
* {0}855555555551{0,2,4,5,6,8}
* {0}8055551{0,2,4,5,6,8}
* {0}9021{0,2,4,5,6,8}
* {0}9201{0,2,4,5,6,8}
* {0}9801{0,2,4,5,6,8}
* {0}90581{0,2,4,5,6,8}
* {0}95081{0,2,4,5,6,8}
* {0}94669{0,2,4,5,6,8}

Last fiddled with by sweety439 on 2021-12-08 at 14:20
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Old 2021-12-10, 16:59   #231
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These factor pattern can show that such families contain no primes > base:

Reference: the divisibility rule for base b:

* For prime p dividing b, the number is divisible by p if and only if the last digit of this number is divisible by p.
* For prime p dividing b-1, the number is divisible by p if and only if the sum of the digits of this number is divisible by p.
* For prime p dividing b+1, the number is divisible by p if and only if the alternating sum of the digits of this number is divisible by p. (this can also show that all palindromic primes in any base b have an odd number of digits, the only possible exception is "11" in base b)

(in these examples, only list the numbers > base)

Example 1: base 10, family 4{6}9

Code:
49 = 7 * 7
469 = 7 * 67
4669 = 7 * 667
46669 = 7 * 6667
466669 = 7 * 66667
4666669 = 7 * 666667
Example 2: base 10, family 28{0}7

Code:
287 = 7 * 41
2807 = 7 * 401
28007 = 7 * 4001
280007 = 7 * 40001
2800007 = 7 * 400001
28000007 = 7 * 4000001
Example 3: base 9, family {1}

Code:
11 = 2 * 5
111 = 7 * 14
1111 = 22 * 45
11111 = 67 * 144
111111 = 222 * 445
1111111 = 667 * 1444
11111111 = 2222 * 4445
111111111 = 6667 * 14444
1111111111 = 22222 * 44445
11111111111 = 66667 * 144444
111111111111 = 222222 * 444445
1111111111111 = 666667 * 1444444
Example 4: base 9, family 3{8}

Code:
38 = 5 * 7
388 = 18 * 21
3888 = 58 * 61
38888 = 188 * 201
388888 = 588 * 601
3888888 = 1888 * 2001
38888888 = 5888 * 6001
388888888 = 18888 * 20001
3888888888 = 58888 * 60001
38888888888 = 188888 * 200001
388888888888 = 588888 * 600001
Example 5: base 8, family 1{0}1

Code:
11 = 3 * 3
101 = 5 * 15
1001 = 11 * 71
10001 = 21 * 361
100001 = 41 * 1741
1000001 = 101 * 7701
10000001 = 201 * 37601
100000001 = 401 * 177401
1000000001 = 1001 * 777001
10000000001 = 2001 * 3776001
100000000001 = 4001 * 17774001
1000000000001 = 10001 * 77770001
Example 6: base 11, family 2{5}

Code:
25 = 3 * 9
255 = 2 * 128
2555 = 3 * 919
25555 = 2 * 12828
255555 = 3 * 91919
2555555 = 2 * 1282828
25555555 = 3 * 9191919
255555555 = 2 * 128282828
2555555555 = 3 * 919191919
25555555555 = 2 * 12828282828
Example 7: base 12, family {B}9B

Code:
9B = 7 * 15
B9B = 11 * AB
BB9B = B7 * 105
BBB9B = 11 * B0AB
BBBB9B = BB7 * 1005
BBBBB9B = 11 * B0B0AB
BBBBBB9B = BBB7 * 10005
BBBBBBB9B = 11 * B0B0B0AB
BBBBBBBB9B = BBBB7 * 100005
BBBBBBBBB9B = 11 * B0B0B0B0AB
Example 8: base 14, family B{0}1

Code:
B1 = 5 * 23
B01 = 3 * 395
B001 = 5 * 22B3
B0001 = 3 * 39495
B00001 = 5 * 22B2B3
B000001 = 3 * 3949495
B0000001 = 5 * 22B2B2B3
B00000001 = 3 * 394949495
B000000001 = 5 * 22B2B2B2B3
B0000000001 = 3 * 39494949495
Example 9: base 13, family 3{0}95

Code:
395 = 14 * 2B
3095 = 7 * 58A
30095 = 5 * 7A71
300095 = 7 * 5758A
3000095 = 14 * 23A92B
30000095 = 7 * 575758A
300000095 = 5 * 7A527A71
3000000095 = 7 * 57575758A
30000000095 = 14 * 23A923A92B
300000000095 = 7 * 5757575758A
3000000000095 = 5 * 7A527A527A71
Example 10: base 16, family {4}D

Code:
4D = 7 * B
44D = 3 * 16F
444D = D * 541
4444D = 7 * 9C0B
44444D = 3 * 16C16F
444444D = D * 540541
4444444D = 7 * 9C09C0B
44444444D = 3 * 16C16C16F
444444444D = D * 540540541
4444444444D = 7 * 9C09C09C0B
44444444444D = 3 * 16C16C16C16F
444444444444D = D * 540540540541
Example 11: base 17, family 1{9}

Code:
19 = 2 * D
199 = B * 27
1999 = 2 * D4D
19999 = AB * 287
199999 = 2 * D4D4D
1999999 = AAB * 2887
19999999 = 2 * D4D4D4D
199999999 = AAAB * 28887
1999999999 = 2 * D4D4D4D4D
19999999999 = AAAAB * 288887
199999999999 = 2 * D4D4D4D4D4D
1999999999999 = AAAAAB * 2888887
Example 12: base 36, family O{Z}

Code:
OZ = T * V
OZZ = 4Z * 51
OZZZ = TZ * U1
OZZZZ = 4ZZ * 501
OZZZZZ = TZZ * U01
OZZZZZZ = 4ZZZ * 5001
OZZZZZZZ = TZZZ * U001
OZZZZZZZZ = 4ZZZZ * 50001
OZZZZZZZZZ = TZZZZ * U0001
OZZZZZZZZZZ = 4ZZZZZ * 500001
OZZZZZZZZZZZ = TZZZZZ * U00001
OZZZZZZZZZZZZ = 4ZZZZZZ * 5000001
Some references of this, see:

http://www.worldofnumbers.com/wing.htm for:

{1}0{1} (base 10):

Code:
101 = 1 * 101 (the only possible prime case)
11011 = 11 * 1001
1110111 = 111 * 10001
111101111 = 1111 * 100001
11111011111 = 11111 * 1000001
1111110111111 = 111111 * 10000001
{1}2{1} (base 10):

Code:
121 = 11 * 11
11211 = 101 * 111
1112111 = 1001 * 1111
111121111 = 10001 * 11111
11111211111 = 100001 * 111111
1111112111111 = 1000001 * 1111111
{3}2{3} (base 10):

Code:
323 = 17 * 19
33233 = 167 * 199
3332333 = 1667 * 1999
333323333 = 16667 * 19999
33333233333 = 166667 * 199999
3333332333333 = 1666667 * 1999999
{3}4{3} (base 10):

Code:
343 = 7 * 49
33433 = 67 * 499
3334333 = 667 * 4999
333343333 = 6667 * 49999
33333433333 = 66667 * 499999
3333334333333 = 666667 * 4999999
http://www.worldofnumbers.com/deplat.htm for:

1{2}1 (base 10):

Code:
11 = 11 * 1 (the only possible prime case)
121 = 11 * 11
1221 = 11 * 111
12221 = 11 * 1111
122221 = 11 * 11111
1222221 = 11 * 111111
12222221 = 11 * 1111111
7{3}7 (base 10):

Code:
77 = 11 * 7
737 = 11 * 67
7337 = 11 * 667
73337 = 11 * 6667
733337 = 11 * 66667
7333337 = 11 * 666667
73333337 = 11 * 6666667
9{7}9 (base 10):

Code:
99 = 11 * 9
979 = 11 * 89
9779 = 11 * 889
97779 = 11 * 8889
977779 = 11 * 88889
9777779 = 11 * 888889
97777779 = 11 * 8888889
9{4}9 (base 10):

Code:
99 = 11 * 9
949 = 13 * 73
9449 = 11 * 859
94449 = 3 * 31483
944449 = 11 * 85859
9444449 = 7 * 1349207
94444449 = 11 * 8585859
944444449 = 13 * 72649573
9444444449 = 11 * 858585859
94444444449 = 3 * 31481481483
944444444449 = 11 * 85858585859
9444444444449 = 7 * 1349206349207
94444444444449 = 11 * 8585858585859
http://www.worldofnumbers.com/Append...s%20to%20n.txt for:

37{1} (base 10):

Code:
37 = 37 * 1 (the only possible prime case)
371 = 7 * 53
3711 = 3 * 1237
37111 = 37 * 1003
371111 = 13 * 28547
3711111 = 3 * 1237037
37111111 = 37 * 1003003
371111111 = 7 * 53015873
3711111111 = 3 * 1237037037
37111111111 = 37 * 1003003003
371111111111 = 13 * 28547008547
3711111111111 = 3 * 1237037037037
37111111111111 = 37 * 1003003003003
38{1} (base 10):

Code:
38 = 2 * 19
381 = 3 * 127
3811 = 37 * 103
38111 = 23 * 1657
381111 = 3 * 127037
3811111 = 37 * 103003
38111111 = 233 * 163567
381111111 = 3 * 127037037
3811111111 = 37 * 103003003
38111111111 = 2333 * 16335667
381111111111 = 3 * 127037037037
3811111111111 = 37 * 103003003003
38111111111111 = 23333 * 1633356667

Last fiddled with by sweety439 on 2021-12-19 at 23:20
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