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 2020-10-31, 03:36 #2 sweety439     "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 23·389 Posts Link 1: * Smallest prime of the form k*b^n+-1 with n>=1 for bases 2<=b<=1030 and k=b-1 for bases 3<=b<=10000 * Primes of the form n*b^n+1 with n>=1 for bases 101<=b<=10000 * Primes of the form x^y*y^x+1 * Primes of the form x^y*y^x-1 * Primes of the form (n+-1)*b^n+-1 with n>=1 for bases 2<=b<=128 * Primes of the form (x^y+-1)*y^+-1 with x=2 or y=2 * Primes of the form b*(b+1)^n-1 with n>=1 and bases 2<=b<=2048 * Primes of the form (b^n+-1)^2-2 with n>=1 and bases b = 2, 6, 10, 14, 22, 204 * Numbers n such that k*2^n+1 and k*2^(n+1)+1 are both primes Link 3: * Primes of the form n*b^n+1 with n>=1 for bases 3<=b<=100 Link 4: * Primes of the form (b^p-1)/(b-1) with prime p for 2<=b<=160 * Primes of the form (b^p+1)/(b+1) with odd prime p for 2<=b<=160 * Primes of the form (b+1)^p-b^p with prime p for 1<=b<=160 * Primes p such that n^p-1 == 1 mod p^2 for 2<=n<=10125 Link 5: * Numbers n such that k*2^n+-1 are both primes * Numbers n such that k*2^n-1 and k*2^(n+1)-1 are both primes Link 6: * Smallest prime of the form k*2^n+-1 with even/odd n>=1 for k
 2020-10-31, 03:45 #3 sweety439     "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 C2816 Posts GitHub Pages for the smallest n>=1 such that (k*b^n+-1)/gcd(k+-1,b-1) created by me: k
2020-10-31, 23:14   #4
sweety439

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

23·389 Posts

Links for the dual Sierpinski problem:

* https://oeis.org/A076336/a076336c.html
* http://sierpinski.insider.com/dual (broken link and excluded from wayback machine: from archive today cached copy)
* http://www.mit.edu/~kenta/three/prim...rp-excerpt.txt
* https://www.rechenkraft.net/wiki/Five_or_Bust
* https://oeis.org/A067760
* https://oeis.org/A033919

Links for the dual Riesel problem:

* https://oeis.org/A252168
* https://oeis.org/A096502
* https://oeis.org/A276417

The "mixed Sierpinski conjecture base 2" is proven.

The k remaining for the original Sierpinski conjecture base 2 are: {21181, 22699, 24737, 55459, 67607} (references: http://www.prothsearch.com/sierp.html, http://www.primegrid.com/forum_thread.php?id=1647, http://www.noprimeleftbehind.net/cru...es-powers2.htm, http://www.primegrid.com/stats_sob_llr.php)

and for the dual primes (2^n+k):

Code:
k   n
21181   28
22699   26
24737   17
55459   14
67607   16389
The "mixed Sierpinski conjecture base 5" is proven. (in the weaker sense of allowing probable primes, the only unproven probable prime is 5^50669+31712 (for k = 31712))

The k remaining for the original Sierpinski conjecture base 5 are: {6436, 7528, 10918, 26798, 29914, 31712, 36412, 41738, 44348, 44738, 45748, 51208, 58642, 60394, 62698, 64258, 67612, 67748, 71492, 74632, 76724, 83936, 84284, 90056, 92906, 93484, 105464, 126134, 139196, 152588} (references: http://www.primegrid.com/forum_thread.php?id=5087, http://www.noprimeleftbehind.net/cru...e5-reserve.htm, http://primegrid.com/stats_sr5_llr.php)

and for the dual primes (5^n+k):

Code:
k   n
6436   24
7528   36
10918   144
26798   1505
29914   4
31712   50669
36412   458
41738   3
44348   9
44738   485
45748   12
51208   12
58642   46
60394   12
62698   2
64258   2
67612   10
67748   41
71492   13
74632   74
76724   7
83936   3
84284   21
90056   181
92906   23
93484   4
105464   11
126134   11
139196   1
152588   15
Primality certificates of proven primes >=300 digits: 2^16389+67607 5^1505+26798 5^458+36412 (5^485+44738 is proven prime by N+1 method, this is the primality certificate for the large prime factor of N+1)

Some pdf files:

1. http://www.kurims.kyoto-u.ac.jp/EMIS...rs/i61/i61.pdf

2. https://scholar.rose-hulman.edu/cgi/...&context=rhumj

3. https://www.utm.edu/staff/caldwell/preprints/2to100.pdf

4. https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf

5. https://www.ams.org/journals/mcom/19...-0427213-2.pdf

6. http://chesswanks.com/num/LTPs/

7. https://oeis.org/A028491/a028491.pdf

8. http://emis.impa.br/EMIS/journals/JI...NER/dubner.pdf

Sierpinski conjectures in bases 2<=b<=128 and b = 256, 512, 1024

Riesel conjectures in bases 2<=b<=128 and b = 256, 512, 1024

Sierpinski conjectures in bases 2<=b<=200 and b = 256, 512, 1024

Riesel conjectures in bases 2<=b<=200 and b = 256, 512, 1024

First 4 Riesel conjectures in selected bases 2<=b<=64 and b = 100, 128, 256, 512, 1024

(the zip file is for the archive today's archive for website which are excluded from wayback machine)
Attached Files
 broken link for dual sierpinski problem.zip (19.8 KB, 41 views)

Last fiddled with by sweety439 on 2021-09-13 at 07:18

 2020-11-04, 01:10 #5 sweety439     "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 C2816 Posts Last fiddled with by sweety439 on 2021-06-11 at 18:24
 2020-11-06, 12:07 #6 sweety439     "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 23×389 Posts A link of generalized repunit primes (achieved): Index (broken link: from wayback machine cached copy) List of generalized repunit proven primes >= 1000 decimal digits (broken link: from wayback machine cached copy) List of generalized repunit (probable) primes bases 2 to 999 (broken link: from wayback machine cached copy) List of generalized repunit proven primes >= 1000 decimal digits (broken link: from wayback machine cached copy) List of generalized repunit probable primes >= 1000 decimal digits (broken link: from wayback machine cached copy) Last fiddled with by sweety439 on 2021-08-10 at 13:41
2020-11-08, 11:18   #7
sweety439

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

23×389 Posts
Base-dependent types of primes (in various bases)

Links for left truncatable primes:

http://primerecords.dk/left-truncatable.txt
http://rosettacode.org/wiki/Truncatable_primes
http://chesswanks.com/num/LTPs/
http://www.primerecords.dk/left-truncatable.htm
http://www.worldofnumbers.com/truncat.htm
http://www.lifl.fr/~jdelahay/dnalor/ChasseursNbPr.pdf
https://rosettacode.org/wiki/Find_la...n_a_given_base
https://www.ams.org/journals/mcom/19...-0427213-2.pdf
http://www.wschnei.de/digit-related-...ar-primes.html (broken link: from wayback machine cached copy)
https://www.primepuzzles.net/puzzles/puzz_002.htm

Links for right truncatable primes:

http://primerecords.dk/right-truncatable.txt
http://fatphil.org/maths/rtp/rtp.html
http://www.worldofnumbers.com/truncat.htm
http://www.lifl.fr/~jdelahay/dnalor/ChasseursNbPr.pdf
https://codegolf.meta.stackexchange....es/17229#17229
https://hlma.math.cuhk.edu.hk/wp-con...ea1b2408fa.pdf
https://www.ams.org/journals/mcom/19...-0427213-2.pdf
http://www.wschnei.de/digit-related-...ar-primes.html (broken link: from wayback machine cached copy)
https://www.primepuzzles.net/puzzles/puzz_002.htm

Links for deletable primes:

http://www.wschnei.de/digit-related-...ar-primes.html (broken link: from wayback machine cached copy)
https://www.primepuzzles.net/puzzles/puzz_002.htm

Links for minimal primes:

http://www.wiskundemeisjes.nl/wp-con...02/minimal.pdf
https://www.pourlascience.fr/sd/math...idaux-4744.php
https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf
https://cs.uwaterloo.ca/~cbright/tal...mal-slides.pdf
https://doi.org/10.1080/10586458.2015.1064048 (need access, pdf file attached below)
http://recursed.blogspot.com/2006/12/prime-game.html
https://github.com/curtisbright/mepn-data
https://github.com/RaymondDevillers/primes
https://www.primepuzzles.net/puzzles/puzz_178.htm

Links for weakly primes:

https://arxiv.org/pdf/1510.03401
https://arxiv.org/pdf/2101.08898
https://arxiv.org/pdf/0802.3361
https://people.math.sc.edu/filaseta/...Primes2021.pdf
https://doi.org/10.1017%2FS1446788712000043
https://www.quantamagazine.org/mathe...rimes-20210330
http://people.missouristate.edu/lesreid/Soln12.html
https://www.primepuzzles.net/puzzles/puzz_017.htm

Links for permutable primes:

https://arxiv.org/pdf/1811.08613.pdf
https://www.jstor.org/stable/2689862
https://www.jstor.org/stable/2689222
https://www.tandfonline.com/doi/abs/....1974.11976408
https://oeis.org/A003459/a003459.pdf
http://www.lifl.fr/~jdelahay/dnalor/ChasseursNbPr.pdf
https://zbmath.org/?format=complete&q=an:0054.02305
http://www.lacim.uqam.ca/~plouffe/OE...ute_Primes.pdf (broken link: from wayback machine, pdf file attached below)
http://www.wschnei.de/digit-related-...ar-primes.html (broken link: from wayback machine cached copy)

Links for circular primes:

http://www.worldofnumbers.com/circular.htm
https://tangente-mag.com/maths_etonnantes.php?id=3139
http://www.lifl.fr/~jdelahay/dnalor/ChasseursNbPr.pdf
http://www.wschnei.de/digit-related-...ar-primes.html (broken link: from wayback machine cached copy)

Links for repunit primes:

https://stdkmd.net/nrr/repunit/
http://www.kurtbeschorner.de/
http://www.elektrosoft.it/matematica...it/repunit.htm
https://gmplib.org/~tege/repunit.html
http://repunit:1031@repunits.skoberne.net/list/ (broken link: from wayback machine cached copy)
http://www.fermatquotient.com/PrimSerien/GenRepu.txt
https://listserv.nodak.edu/cgi-bin/w...;417ab0d6.0906 (archive today cannot automatically return the archive page, if you use archive today, click https://archive.is/WCvbi)
http://www.users.globalnet.co.uk/~aads/primes.html (broken link: from wayback machine cached copy)

Links for palindromic primes:

http://www.worldofnumbers.com/palpri.htm
https://www.mathpages.com/home/kmath359.htm (excluded from wayback machine: cached copy)
https://arxiv.org/pdf/math/0405056
http://www.maa.org/mathland/mathtrek_5_10_99.html (broken link: from wayback machine cached copy)
https://primes.utm.edu/top20/page.php?id=53

Links for reversible primes:

https://www.primepuzzles.net/puzzles/puzz_020.htm
https://www.primepuzzles.net/puzzles/puzz_973.htm

Links for Smarandache primes:

http://sprott.physics.wisc.edu/picko...ianglegod.html
http://www.worldofnumbers.com/factorlist.htm
https://vixra.org/pdf/1005.0104v2.pdf
http://fs.unm.edu/micha.txt
http://www.asahi-net.or.jp/~KC2H-MSM...tha1/micha.txt
http://chesswanks.com/pxp/smfactors.html
http://99.121.249.54:1200/ (broken link and both wayback machine and archive today did not achieve)
https://www.primepuzzles.net/puzzles/puzz_008.htm

Links for Smarandache-Wellin primes:

https://www.primepuzzles.net/puzzles/puzz_008.htm
http://www.asahi-net.or.jp/~KC2H-MSM...1/sm_prime.htm
http://fs.unm.edu/SmConPri.txt

My GitHub page for left truncatable primes, right truncatable primes, minimal primes, two-sided primes (both left truncatable and right truncatable) (all in bases 2 to 160):

https://github.com/xayahrainie4793/m...catable-primes

My GitHub pages for all these types of primes:

https://raw.githubusercontent.com/xa...%20to%2012.txt
https://raw.githubusercontent.com/xa...%20to%2036.txt

My sites for all these types of primes:

Prime Glossary pages: (no article for "Smarandache prime")

left truncatable prime
right truncatable prime
deletable prime
minimal prime
weakly prime
permutable prime
circular prime
repunit
palindromic prime
reversible prime
Smarandache-Wellin prime

truncatable prime
deletable prime
minimal prime
weakly prime
permutable prime
circular prime
repunit
palindromic prime
reversible prime
Smarandache prime
Smarandache-Wellin prime

Mathworld pages: (no article for "minimal prime")

truncatable prime
deletable prime
weakly prime
permutable prime
circular prime
repunit
palindromic prime
reversible prime
Smarandache prime
Smarandache-Wellin prime

Prime Curios! pages for the largest/smallest such primes:

left truncatable prime (b=10)
right truncatable prime (b=10)
two-sided prime (b=10)
minimal prime (b=5)
minimal prime (b=6)
minimal prime (b=7)
minimal prime (b=8)
minimal prime (b=9)
minimal prime (b=10)
weakly prime (b=10)
permutable prime (b=10)
circular prime (b=10)
repunit prime (b=10)

Bitman pages: (b is the base) (no "data" for "two-sided prime" and "permutable prime" and "Smarandache prime" and "Smarandache-Wellin prime", for the data see corresponding "article", also there is an article about the conjecture about "Smarandache prime")

truncatable prime (article)
left truncatable prime (data for 2<=b<=20)
right truncatable prime (data for 2<=b<=20)
deletable prime (article)
deletable prime (data for 2<=b<=20)
minimal prime (article)
minimal prime (data for 2<=b<=16)
minimal prime (data for b=17)
minimal prime (data for b=18)
minimal prime (data for b=19)
minimal prime (data for b=20)
weakly prime (article)
weakly prime (data for 2<=b<=20)
permutable prime (article)
circular prime (article)
circular prime (data for 2<=b<=20)
repunit prime (article)
repunit prime (data for 2<=b<=100)
palindromic prime (article)
palindromic prime (data for 2<=b<=10)
palindromic prime (data for 11<=b<=20)
reversible prime (article)
reversible prime (data for 2<=b<=20)
Smarandache prime, Smarandache-Wellin prime (article)

OEIS sequences:

#1 = left truncatable primes
#2 = right truncatable primes
#3 = two-sided primes (both left truncatable and right truncatable)
#4 = deletable primes
#5 = minimal primes
#6 = weakly primes
#7 = permutable primes (for sequences B, C, D and "sequences of such primes in various bases b", repunit primes are excluded)
#8 = circular primes (for sequences B, C, D and "sequences of such primes in various bases b", repunit primes are excluded)
#9 = repunit primes
#10 = palindromic primes
#11 = reversible primes
#12 = Smarandache primes
#13 = Smarandache-Wellin primes (for sequence B and C, the prime 2 is excluded, since 2 is Smarandache-Wellin prime in every base, make it trivial and uninteresting)

A = Such primes in base 10
B = Largest (or smallest, in case of weakly prime and repunit prime and Smarandache prime and Smarandache-Wellin prime) such prime in base n (written in base 10)
C = Length of largest (or smallest, in case of weakly prime and repunit prime and Smarandache prime and Smarandache-Wellin prime) such prime in base n
D = Number of such primes in base n

Code:
         A         B         C         D
#1    A024785   A103443   A103463   A076623
#2    A024770   A023107   A103483   A076586
#3    A020994   A323137   A??????   A323390
#4    A305352   (no sequences B, C, D since such primes are conjectured to exist infinitely many in every base)
#5    A071062   A326609   A330049   A330048
#6    A050249   A186995   A??????   (no sequences D since such primes are conjectured to exist infinitely many in every base)
#7    A003459   A317689   A??????   A??????
#8    A068652   A293142   A??????   A??????
#9    A004022   A084738   A084740   (no sequences D since such primes are conjectured to exist infinitely many in every base which is not perfect power, but there is OEIS sequence A085104 for the primes which is nontrivial such primes in some base)
#10   A002385   (no sequences B, C, D since such primes are conjectured to exist infinitely many in every base, but there is OEIS sequence A016038 for the numbers which is not nontrivial such primes in any base, all such numbers >6 are primes)
#11   A006567   (no sequences B, C, D since such primes are conjectured to exist infinitely many in every base)
#12   A??????   A??????   A??????   (no sequences D since such primes are conjectured to exist infinitely many in every base)
#13   A069151   A??????   A??????   (no sequences D since such primes are conjectured to exist infinitely many in every base)
sequences of such primes in various bases b:

Code:
b        2         3         4         5         6         7         8         9         10        11        12
#1    A000000   A??????   A129940   A129941   A129942   A129943   A129944   A129945   A024785   A??????   A??????
#2    A000000   A129669   A129670   A129671   A129672   A129673   A129692   A129693   A024770   A??????   A??????
#3    A000000   A??????   A??????   A??????   A??????   A??????   A??????   A??????   A020994   A??????   A??????
#4    A000000   A319596   A321657   A321700   A322173   A321910   A322443   A322471   A305352   A322475   A322477
#5    A??????   A??????   A??????   A??????   A??????   A??????   A??????   A??????   A071062   A??????   A110600
#6    A137985   A??????   A??????   A??????   A??????   A??????   A??????   A??????   A050249   A??????   A??????
#7    A000000   A??????   A??????   A??????   A??????   A??????   A??????   A??????   A317688   A??????   A??????
#8    A000000   A??????   A293657   A293658   A293659   A293660   A293661   A293662   A293663   A??????   A??????
#9    A000668   A076481   A??????   A086122   A165210   A102170   A??????   A000000   A004022   A??????   A??????
#10   A016041   A029971   A029972   A029973   A029974   A029975   A029976   A029977   A002385   A029978   A029979
#11   A080790   A??????   A??????   A??????   A??????   A??????   A??????   A??????   A006567   A??????   A??????
#12   A??????   A??????   A??????   A??????   A??????   A??????   A??????   A??????   A??????   A??????   A??????
#13   A??????   A??????   A??????   A??????   A??????   A??????   A??????   A??????   A069151   A??????   A??????
Note: A000000 is the empty sequence, and for the sequence which has too few terms to be included in OEIS: ("#7,b=3" and "#8,b=3 are only conjectured to be complete, others are proven to be complete)

Code:
#1,b=3      2, 5, 23
#3,b=3      2, 23
#3,b=4      2, 3, 11
#5,b=2      2, 3
#5,b=3      2, 3, 13
#5,b=4      2, 3, 5
#7,b=3      2, 5, 7
#8,b=3      2, 5, 7
#9,b=4      5
#9,b=8      73
By definition, repunit primes also counted as permutable primes and circular primes, but palindromic primes (and hence repunit primes) are not counted as reversible primes.

Conjectured cardinality of the set of these 13 types of primes in given base:

#1 "left truncatable primes": Conjectured finite in every base (reference: https://www.ams.org/journals/mcom/19...-0427213-2.pdf https://rosettacode.org/wiki/Find_la...n_a_given_base http://chesswanks.com/num/LTPs/)
#2 "right truncatable primes": Conjectured finite in every base (reference: https://www.ams.org/journals/mcom/19...-0427213-2.pdf http://fatphil.org/maths/rtp/rtp.html)
#3 "two-sided primes": Conjectured finite in every base (since all two-sided primes are also left truncatable primes and right truncatable primes)
#4 "deletable primes": Conjectured infinite in every base (reference: https://primes.utm.edu/glossary/xpag...ablePrime.html)
#5 "minimal primes": Proven finite in every base (reference: http://www.wiskundemeisjes.nl/wp-con...02/minimal.pdf https://cs.uwaterloo.ca/~cbright/tal...mal-slides.pdf http://recursed.blogspot.com/2006/12/prime-game.html
#6 "weakly primes": Proven infinite in every base (reference: https://arxiv.org/pdf/0802.3361.pdf https://www.cambridge.org/core/journ...5432734E9A87FD)
#7 "permutable primes": Conjectured finite in every base if repunit primes are not counted (reference: https://www.jstor.org/stable/2689222 https://www.tandfonline.com/doi/abs/....1974.11976408 https://oeis.org/A317689)
#8 "circular primes" Conjectured finite in every base if repunit primes are not counted (reference: https://oeis.org/A293142 https://oeis.org/A327835)
#9 "repunit primes": Proven finite (only 0 or 1 such primes) in every perfect-power base and conjecture infinite in every non-perfect-power base (reference: https://listserv.nodak.edu/cgi-bin/w...;417ab0d6.0906 https://web.archive.org/web/20021111...ds/primes.html http://www.fermatquotient.com/PrimSerien/GenRepu.txt)
#10 "palindromic primes": Conjectured infinite in every base (reference: http://www.worldofnumbers.com/nobase10.htm http://web.archive.org/web/201212291...k_5_10_99.html https://arxiv.org/pdf/math/0405056.pdf)
#11 "reversible primes": Conjectured infinite in every base (reference: https://en.wikipedia.org/wiki/Emirp)
#12 "Smarandache primes": Conjectured infinite in every base (reference: https://mersenneforum.org/showthread.php?t=20527 https://mersenneforum.org/showthread.php?t=20535)
#13 "Smarandache-Wellin primes": Conjectured infinite in every base (reference: https://en.wikipedia.org/wiki/Talk:S...Wellin_primes?)

All single-digit primes (i.e. primes < base) are trivial these 13 types of primes (except "weakly prime", "repunit prime", "reversible prime", "Smarandache primes", "Smarandache-Wellin primes") in the same base.

All repunit primes are trivial permutable primes, circular primes, and palindromic primes in the same base, besides, the smallest repunit prime (if exists) is always minimal prime in the same base, and this prime is the only repunit prime which is also minimal prime in the same base, also, a repunit prime cannot be left truncatable prime, right truncatable prime, or deletable prime in the same base, since 1 is not prime.

All left truncatable primes and all right truncatable primes are deletable primes in the same base.

A minimal prime cannot be left truncatable prime, right truncatable prime, or deletable prime in the same base, unless it is single-digit prime.

The attached text file does not include "deletable prime", "palindromic prime", and "reversible prime", since it is conjectured there are infinitely many such primes in every base, and unlikely weakly prime and repunit prime (it is also conjectured that there are infinitely many weakly primes in every base, and it is also conjectured that there are infinitely many repunit primes in every base which is not perfect power; for perfect power bases, all repunit numbers are not primes; the only possible exception is when the base b is m^(p^r) with p prime and m>=2, r>=1, only the repunit with length p may be prime; if the base b is m^r with m>=2 and r is not prime or prime power, then all repunit numbers are not primes), the smallest such primes are only 2

Edit: There seems to be a proof that there are infinitely many weakly primes in every base, but I do not know that there is also a proof for deletable primes, palindromic primes, and reversible primes (I only know that there is no proof for repunit primes).

All base b given type in these 13 types of primes (except "weakly prime", "repunit prime", "palindromic prime", "reversible prime", "Smarandache primes", "Smarandache-Wellin primes") are also the same type of primes in base b^n for every n>=1

By the theorem that there are no infinite antichains for the subsequence ordering, there are only finitely minimal primes in every base, but no proof is known for left truncatable primes, right truncatable primes, and two-sided primes, however, it is likely that there are only finitely many left truncatable primes and finitely many right truncatable primes (thus only finitely many two-sided primes, since all two-sided primes are also left truncatable primes and right truncatable primes) in every base, even no proof is known for non-repunit permutable primes and non-repunit circular primes, but it is very likely that there are only finitely many non-repunit circular primes (thus only finitely many non-repunit permutable primes, since all permutable primes are also circular primes) in every base, e.g. in base 10, there are no non-repunit permutable primes >991 and <10^(6*10^175), reference: https://zbmath.org/?format=complete&q=an:0054.02305

The expected largest base b left truncatable prime is (see http://chesswanks.com/num/LTPs/), and the expected largest base b right truncatable prime is b^(e^sqrt(b)), and for the expected largest base b minimal prime, we should use the sense of https://mersenneforum.org/showpost.p...65&postcount=7 and https://mersenneforum.org/showpost.p...86&postcount=3 and https://mersenneforum.org/showpost.p...3&postcount=18 (finding the Nash weight (or difficulty)) to every unsolved families (example: base 25 and base 34), and the expected smallest base b weakly prime is b^(e^sqrt(eulerphi(b))), and the expected number of base b repunit primes <=n digits is (see http://www.fermatquotient.com/PrimSerien/GenRepu.txt and https://listserv.nodak.edu/cgi-bin/w...;417ab0d6.0906), and the expected number of base b palindromic primes <=n digits is 1/log(b^n)*Sum {k=1..n} (b-1)*b^floor[(k-1)/2], and the expected number of base b Smarandache primes <=n digits is (see https://mersenneforum.org/showpost.p...73&postcount=1)

For the conjectured cardinality of the set of given types of primes in given base, and the excepted (number of such primes, the largest/smallest such primes) of given types of primes in given base, see https://oeis.org/wiki/User:Charles_R...special_primes for more information.
Attached Files
 permutable primes.pdf (138.5 KB, 139 views) minimal primes.pdf (585.3 KB, 126 views) data for base dependent types of primes for bases up to 36.txt (14.8 KB, 28 views) data for base dependent types of primes for bases up to 12.zip (1.28 MB, 24 views)

Last fiddled with by sweety439 on 2021-10-20 at 10:02

 2020-11-08, 13:45 #8 sweety439     "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 311210 Posts Last fiddled with by sweety439 on 2021-08-10 at 13:45
 2020-11-08, 13:50 #9 sweety439     "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 23×389 Posts Last fiddled with by sweety439 on 2021-06-11 at 18:27
 2020-11-08, 14:02 #10 sweety439     "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 23×389 Posts GIMPS main page (factorization of Mersenne numbers): https://www.mersenne.org/ For all known Mersenne primes (total 51), see https://www.mersenne.org/primes/ Last fiddled with by sweety439 on 2021-06-11 at 18:12
 2020-11-09, 16:35 #11 sweety439     "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 23·389 Posts

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