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Palindromic Wing Primes (PWP's)
(near-repdigit palindromes)
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313717919727929
131737141747949
151353757959161
767171373979181
383787989191797


Palindromic Wing Primes

Palindromic Wing Primes (or PWP's for short) are numbers that
are primes, palindromic in base 10, and consisting of one central digit
surrounded by two wings having an equal amount of identical digits and
different from the central one. E.g.

101
99999199999
333333313333333
7777777777772777777777777
11111111111111111111111111111111411111111111111111111111111111111

While setting up this collection of palprimes I realised that perhaps
this kind of integers was already considered under another description.
And so it turned out!
Near-repunit palindromes
Near-repdigit palindromes
are the names used in the sources where I found more PWP's ¬
The Complete List of the Largest Known Primes by Chris Caldwell
The Top Ten Prime Numbers by Rudolf Ondrejka
Palindrome prime number patterns by Harvey Heinz
Liczby pierwsze o szczególnym rozmieszczeniu cyfr by Andrzej Nowicki
    Translated in Dutch "Priemgetallen met een speciale rangschikking van cijfers"
    Translated in English "Prime numbers with a special arrangement of digits"
In case one should discover more sources I will be most happy
to add them to the list. Just let me know.

PWP's sorted by length

[ October 3, 2002 ]
Harvey Dubner (email) informed me that he co-authored an article :

" I think your idea to collect PWP's is a great and worthwhile project.
Most of the work I did on this subject was included in a paper that Chris Caldwell
and I wrote that was published in the
“Journal of Recreational Mathematics”, Volume 28, No. 1, 1996-97, pp. 1-9.
With the new hardware and software that is now available it would be easy to extend
these results. You really do good work. Please keep it up."

[ October 4, 2002 ]
Daniel Heuer (email) started to search at the beginning of 2001 :

" You have a very good initiative. I have studied numbers of the form
p(k,n) = 10^(2n+1)–k*10^n–1 from n = 1 to 38500 and 0 < k < 10.
if k = 3, 6 or 9 p(k,n)%3 = = 0
result for other value of k is:
k = 1, n = 26, 378, 1246, 1798, 2917, 23034
k = 2, n = 118
k = 4, n = 88, 112, 198, 622, 4228, 10052
k = 5, n = 14, 22, 36, 104, 1136, 17864, 25448
k = 7, n = 1, 8, 9, 352, 530, 697, 1315, 1918, 2874, 5876, 6768
k = 8, n = 1, 5, 13, 43, 169, 181, 1579, 18077, 22652
and I am still continuing..."

[ December 28, 2002 ]
Jeff Heleen prime proved a record PWP with Primo.
His prime (104769–1)/3 – 2*102384 consists of 4769 digits.

" Here is the prime cert for (3)2384_1_(3)2384.
It finished on Christmas day.
I send in 3 parts: Primo116A, Primo116B and Primo116C.
Happy Holidays.
jeff "
Jeff positioned himself at Rank 4 in Marcel Martin's
'primo top-20' with this feat !

[ January 2, 2003 ]
Daniel Heuer made a new PWP world record for the new year...

"1095019–1047509–1 is prime.
Have a good year.
Daniel "

[ January 14, 2003 ]
Daniel Heuer just finished scanning the provable PWP's
of the form 102n+1–k*10n–1 up to n = 50,000 (100,001 digits).
And he is continuing...
Ranges of this form are
(9)w1(9)w
(9)w2(9)w
(9)w4(9)w
(9)w5(9)w
(9)w7(9)w
(9)w8(9)w

[ January 27, 2003 ]
Daniel Heuer found a new PWP.

(9)521408(9)52140    or    (10104281-1) - 1052140
" 10104281 – 1052140 –1
is the first PWP with over 100,000 digits."

Some nontrivial combinations can never produce primes since
these generate infinite patterns of products of two factors.

(1)w0(1)w = (10w+1 + 1)(10w - 1)/9
101 x 1 = 101 ( is the only possible prime case when w = 1 )
1001 x 11 = 11011
10001 x 111 = 1110111
...
general formula 1(0)k1 x 1(1)k-21 ; ( k ⩾ 2 )
(1)w2(1)w = (10w+1 - 1)(10w + 1)/9
11 x 11 = 121
101 x 111 = 11211
1001 x 1111 = 1112111
...
general formula 1(0)k1 x 1(1)k1
(3)w2(3)w = (5.10w + 1)(2.10w - 1)/3
17 x 19 = 323
167 x 199 = 33233
1667 x 1999 = 3332333
...
general formula 1(6)k7 x 1(9)k+1
(3)w4(3)w = (5.10w - 1)(2.10w + 1)/3
7 x 49 = 343
67 x 499 = 33433
667 x 4999 = 3334333
...
general formula (6)k7 x 4(9)k+1


PWP Factorization Projects

( n = 2.w + 1 )

(3)w1(3)w = (10n–1)/3 – 2.10w Factorizations of 33...33133...33 (M. Kamada)
(7)w1(7)w = (7.10n–1)/9 – 6.10w Factorizations of 77...77177...77 (M. Kamada)
(9)w1(9)w = (10n–1) – 8.10w Factorizations of 99...99199...99 (M. Kamada)
(7)w2(7)w = (7.10n–1)/9 – 5.10w Factorizations of 77...77277...77 (M. Kamada)
(9)w2(9)w = (10n–1) – 7.10w Factorizations of 99...99299...99 (M. Kamada)
(1)w3(1)w = (10n–1)/9 + 2.10w Factorizations of 11...11311...11 (M. Kamada)
(7)w3(7)w = (7.10n–1)/9 – 4.10w Factorizations of 77...77377...77 (M. Kamada)
(1)w4(1)w = (10n–1)/9 + 3.10w Factorizations of 11...11411...11 (M. Kamada)
(7)w4(7)w = (7.10n–1)/9 – 3.10w Factorizations of 77...77477...77 (M. Kamada)
(9)w4(9)w = (10n–1) – 5.10w Factorizations of 99...99499...99 (M. Kamada)
(1)w5(1)w = (10n–1)/9 + 4.10w Factorizations of 11...11511...11 (M. Kamada)
(3)w5(3)w = (10n–1)/3 + 2.10w Factorizations of 33...33533...33 (M. Kamada)
(7)w5(7)w = (7.10n–1)/9 – 2.10w Factorizations of 77...77577...77 (M. Kamada)
(9)w5(9)w = (10n–1) – 4.10w Factorizations of 99...99599...99 (M. Kamada)
(1)w6(1)w = (10n–1)/9 + 5.10w Factorizations of 11...11611...11 (M. Kamada)
(7)w6(7)w = (7.10n–1)/9 – 10w Factorizations of 77...77677...77 (M. Kamada)
(1)w7(1)w = (10n–1)/9 + 6.10w Factorizations of 11...11711...11 (M. Kamada)
(3)w7(3)w = (10n–1)/3 + 4.10w Factorizations of 33...33733...33 (M. Kamada)
(9)w7(9)w = (10n–1) – 2.10w Factorizations of 99...99799...99 (M. Kamada)
(1)w8(1)w = (10n–1)/9 + 7.10w Factorizations of 11...11811...11 (M. Kamada)
(3)w8(3)w = (10n–1)/3 + 5.10w Factorizations of 33...33833...33 (M. Kamada)
(7)w8(7)w = (7.10n–1)/9 + 10w Factorizations of 77...77877...77 (M. Kamada)
(9)w8(9)w = (10n–1) – 10w Factorizations of 99...99899...99 (M. Kamada)
(1)w9(1)w = (10n–1)/9 + 8.10w Factorizations of 11...11911...11 (M. Kamada)
(7)w9(7)w = (7.10n–1)/9 + 2.10w Factorizations of 77...77977...77 (M. Kamada)
Complete list of the factorizations of all possible “Palindromic Wing Numbers” can be found here Factorization of AA...AABAA...AA (M. Kamada)
with the exception of 11...11011...11, 11...11211...11, 33...33233...33 and 33...33433...33. They are not explicitly in this Makoto Kamada's page since
11...11011...11 = [ 111...111 ] * [ 1000...0001 ] and general formula Rn * 1(0)n1 [1^^n] * [1][0^^n][1]
11...11211...11 = [ 111...111 ] * [ 1000...0001 ] and general formula Rn * 1(0)n-21 [1^^n] * [1][0^^(n-2)][1]
33...33233...33 = [ 1666...6667 ] * [ 1999...999 ] and general formula 1(6)n7 * 1(9)n+1 [1][6^^n][7] * [1][9^^(n+1)]
33...33433...33 = [ 666...6667 ] * [ 4999...999 ] and general formula (6)n7 * 4(9)n+1 [6^^n][7] * [4][9^^(n+1)]
and their factorizations [ 111...111 (Repunit) ], [ 1000...0001 ], [ 1666...6667 ], [ 1999...999 ], [ 666...6667 ], [ 4999...999 ] are already to be found in Kamada's pages.

(Note: '^^' is symbol for concatenation )


The “PWP” Table


The reference table for
Palindromic Wing Primes
This collection is complete for
probable primes up to 40001
digits (by DH) and for proven
primes up to  3000  digits.
DB = Darren Bedwell
DH = Daniel Heuer
HD = Harvey Dubner
JH = Jeff Heleen
P&B = Propper & Batalov
PDG = Patrick De Geest
RP = Robert Price
PWPFormula
Blue exp = # of digits
Accolades = prime exp		WhoWhenStatusOutput
Logs
 ¬ 
(1)10(1)1 (10{3}–1)/9 – 101
IMPORTANT NOTE		 PDG Sep 23 2002 PRIME View
A077775 ¬
A183174 ¬		 
   [ n ⩾ 200000 ]
(3)11(3)1 (10{3}–1)/3 – 2*101 PDGSep 23 2002PRIME View
(3)31(3)3 (10{7}–1)/3 – 2*103 PDGSep 23 2002PRIME View
(3)71(3)7 (1015–1)/3 – 2*107 PDGSep 23 2002PRIME View
(3)611(3)61 (10123–1)/3 – 2*1061 PDGSep 23 2002PRIME View
(3)901(3)90 (10{181}–1)/3 – 2*1090 PDGSep 23 2002PRIME View
(3)921(3)92 (10185–1)/3 – 2*1092 PDGSep 23 2002PRIME View
(3)2691(3)269 (10539–1)/3 – 2*10269 PDGSep 23 2002PRIME View
(3)2981(3)298 (10597–1)/3 – 2*10298 PDGSep 23 2002PRIME View
(3)3211(3)321 (10{643}–1)/3 – 2*10321 PDGSep 23 2002PRIME View
(3)3711(3)371 (10{743}–1)/3 – 2*10371 PDGSep 23 2002PRIME View
(3)7761(3)776 (10{1553}–1)/3 – 2*10776 JHSep 28 2002PRIME View
(3)15671(3)1567 (103135–1)/3 – 2*101567 JHOct 25 2002PRIME View
(3)23841(3)2384 (104769–1)/3 – 2*102384 JHDec 25 2002PRIME View
(3)25661(3)2566 (105133–1)/3 – 2*102566 PDGSep 26 2002PRIME View
(3)30881(3)3088 (106177–1)/3 – 2*103088 PDGSep 26 2002PRIME View
(3)58661(3)5866 (1011733–1)/3 – 2*105866 DHOct 31 2002PROBABLE
PRIME		 View
(3)80511(3)8051 (10{16103}–1)/3 – 2*108051 DHOct 31 2002PROBABLE
PRIME		 View
(3)94981(3)9498 (1018997–1)/3 – 2*109498 DHNov 04 2002PROBABLE
PRIME		 View
(3)126351(3)12635 (1025271–1)/3 – 2*1012635 DHNov 07 2002PROBABLE
PRIME		 View
(3)245121(3)24512 (1049025–1)/3 – 2*1024512 PDGJul 05 2005PROBABLE
PRIME		 View
(3)325211(3)32521 (1065043–1)/3 – 2*1032521 RPJan 29 2016PROBABLE
PRIME		 View
(3)439821(3)43982 (1087965–1)/3 – 2*1043982 RPJan 29 2016RECORD
PROBABLE
PRIME		 View
 ¬ 
   [ n ⩾ 200000 ] searched from july 2005 till january 2006 !
(7)1161(7)116 7*(10{233}–1)/9 – 6*10116 PDGSep 23 2002PRIME View
A077776 ¬
A183184 ¬		 
   [ n ⩾ 100000 ]
(9)11(9)1 (10{3}–1) – 8*101 PDGSep 23 2002PRIME View
(9)51(9)5 (10{11}–1) – 8*105 PDGSep 23 2002PRIME View
(9)131(9)13 (1027–1) – 8*1013 PDGSep 23 2002PRIME View
(9)431(9)43 (1087–1) – 8*1043 PDGSep 23 2002PRIME View
(9)1691(9)169 (10339–1) – 8*10169 PDGSep 23 2002PRIME View
(9)1811(9)181 (10363–1) – 8*10181 PDGSep 23 2002PRIME View
(9)15791(9)1579 (103159–1) – 8*101579 PDGSep 23 2002PRIME View
(9)180771(9)18077 (1036155–1) – 8*1018077 DHJul 21 2001PRIME View
(9)226521(9)22652 (1045305–1) – 8*1022652 DHSep 18 2001PRIME View
(9)1573631(9)157363 (10314727–1) – 8*10157363 DBJan 8 2013PRIME View
A077777 ¬
A183178 ¬		 
(7)12(7)1 7*(10{3}–1)/9 – 5*101 PDGSep 23 2002PRIME View
(7)32(7)3 7*(10{7}–1)/9 – 5*103 PDGSep 23 2002PRIME View
(7)72(7)7 7*(1015–1)/9 – 5*107 PDGSep 23 2002PRIME View
(7)102(7)10 7*(1021–1)/9 – 5*1010 PDGSep 23 2002PRIME View
(7)122(7)12 7*(1025–1)/9 – 5*1012 PDGSep 23 2002PRIME View
(7)4802(7)480 7*(10961–1)/9 – 5*10480 PDGSep 23 2002PRIME View
(7)9492(7)949 7*(101899–1)/9 – 5*10949 JHSep 30 2002PRIME View
(7)19452(7)1945 7*(103891–1)/9 – 5*101945 PDGSep 23 2002PRIME View RC
(7)75482(7)7548 7*(1015097–1)/9 – 5*107548 DHOct 31 2002PROBABLE
PRIME		 View
(7)89232(7)8923 7*(1017847–1)/9 – 5*108923 DHOct 31 2002PROBABLE
PRIME		 View
A077778 ¬
A115073 ¬		 
(9)12(9)1 (10{3}–1) – 7*101 PDGSep 23 2002PRIME View
(9)82(9)8 (10{17}–1) – 7*108 PDGSep 23 2002PRIME View
(9)92(9)9 (10{19}–1) – 7*109 PDGSep 23 2002PRIME View
(9)3522(9)352 (10705–1) – 7*10352 PDGSep 23 2002PRIME View
(9)5302(9)530 (10{1061}–1) – 7*10530 JHSep 28 2002PRIME View
(9)6972(9)697 (101395–1) – 7*10697 JHSep 28 2002PRIME View
(9)13152(9)1315 (102631–1) – 7*101315 HD––– –– 1989PRIME View
(9)19182(9)1918 (103837–1) – 7*101918 HD––– –– 1999PRIME View
(9)28742(9)2874 (10{5749}–1) – 7*102874 HD––– –– 1999PRIME View
(9)58762(9)5876 (1011753–1) – 7*105876 HD––– –– 1999PRIME View
(9)67682(9)6768 (10{13537}–1) – 7*106768 HD––– –– 1999PRIME View
(9)629382(9)62938 (10125877–1) – 7*1062938 DBOct 31 2010PRIME View
(9)1347392(9)134739 (10269479–1) – 7*10134739 DBFeb 29 2012PRIME View
A077779 ¬
A107123 ¬		 
(1)13(1)1 (10{3}–1)/9 + 2*101 PDGSep 23 2002PRIME View
(1)23(1)2 (10{5}–1)/9 + 2*102 PDGSep 23 2002PRIME View
(1)193(1)19 (1039–1)/9 + 2*1019 PDGSep 23 2002PRIME View
(1)973(1)97 (10195–1)/9 + 2*1097 PDGSep 23 2002PRIME View
(1)98183(1)9818 (1019637–1)/9 + 2*109818 DHNov 04 2002PROBABLE
PRIME		 View
 ¬ 
(7)23(7)2 7*(10{5}–1)/9 – 4*102 PDGSep 23 2002PRIME View
A077780 ¬
A107124 ¬		 
(1)24(1)2 (10{5}–1)/9 + 3*102 PDGSep 23 2002PRIME View
(1)34(1)3 (10{7}–1)/9 + 3*103 PDGSep 23 2002PRIME View
(1)324(1)32 (1065–1)/9 + 3*1032 PDGSep 23 2002PRIME View
(1)454(1)45 (1091–1)/9 + 3*1045 PDGSep 23 2002PRIME View
(1)15444(1)1544 (10{3089}–1)/9 + 3*101544 PDGSep 23 2002PRIME View DB
A077781 ¬
A183179 ¬		 
   [ n ⩾ 200000 ]
(7)24(7)2 7*(10{5}–1)/9 – 3*102 PDGSep 23 2002PRIME View
(7)34(7)3 7*(10{7}–1)/9 – 3*103 PDGSep 23 2002PRIME View
(7)64(7)6 7*(10{13}–1)/9 – 3*106 PDGSep 23 2002PRIME View
(7)234(7)23 7*(10{47}–1)/9 – 3*1023 PDGSep 23 2002PRIME View
(7)364(7)36 7*(10{73}–1)/9 – 3*1036 PDGSep 23 2002PRIME View
(7)694(7)69 7*(10{139}–1)/9 – 3*1069 PDGSep 23 2002PRIME View
(7)5614(7)561 7*(10{1123}–1)/9 – 3*10561 JHSep 28 2002PRIME View
(7)7234(7)723 7*(10{1447}–1)/9 – 3*10723 JHOct 01 2002PRIME View
(7)34384(7)3438 7*(106877–1)/9 – 3*103438 PDGOct 10 2002PRIME View
(7)41044(7)4104 7*(10{8209}–1)/9 – 3*104104 PDGOct 10 2002PROBABLE
PRIME		 View
(7)90204(7)9020 7*(10{18041}–1)/9 – 3*109020 DHNov 04 2002PROBABLE
PRIME		 View
(7)139774(7)13977 7*(1027955–1)/9 – 3*1013977 DHNov 13 2002PROBABLE
PRIME		 View
(7)196554(7)19655 7*(1039311–1)/9 – 3*1019655 DHNov 25 2002PROBABLE
PRIME		 View
(7)324004(7)32400 7*(1064801–1)/9 – 3*1032400 RPNov 23 2015PROBABLE
PRIME		 View
A077782 ¬
A183185 ¬		 
(9)144(9)14 (10{29}–1) – 5*1014 PDGSep 23 2002PRIME View
(9)224(9)22 (1045–1) – 5*1022 PDGSep 23 2002PRIME View
(9)364(9)36 (10{73}–1) – 5*1036 PDGSep 23 2002PRIME View
(9)1044(9)104 (10209–1) – 5*10104 PDGSep 23 2002PRIME View
(9)11364(9)1136 (10{2273}–1) – 5*101136 JHOct 13 2002PRIME View
(9)178644(9)17864 (10{35729}–1) – 5*1017864 DHJul 02 2001PRIME View
(9)254484(9)25448 (1050897–1) – 5*1025448 DHOct 29 2001PRIME View
A077783 ¬
A107125 ¬		 
(1)15(1)1 (10{3}–1)/9 + 4*101 PDGSep 23 2002PRIME View
(1)75(1)7 (1015–1)/9 + 4*107 PDGSep 23 2002PRIME View
(1)455(1)45 (1091–1)/9 + 4*1045 PDGSep 23 2002PRIME View
(1)1155(1)115 (10231–1)/9 + 4*10115 PDGSep 23 2002PRIME View
(1)6815(1)681 (101363–1)/9 + 4*10681 JHSep 29 2002PRIME View
(1)12485(1)1248 (102497–1)/9 + 4*101248 JHOct 09 2002PRIME View
(1)24815(1)2481 (104963–1)/9 + 4*102481 PDGSep 23 2002PRIME View
(1)26895(1)2689 (105379–1)/9 + 4*102689 PDGOct 11 2002PRIME View
(1)61985(1)6198 (1012397–1)/9 + 4*106198 DHOct 31 2002PROBABLE
PRIME		 View
(1)131975(1)13197 (1026395–1)/9 + 4*1013197 DHNov 06 2002PROBABLE
PRIME		 View
A077784 ¬
A183175 ¬		 
(3)15(3)1 (10{3}–1)/3 + 2*101 PDGSep 23 2002PRIME View
(3)25(3)2 (10{5}–1)/3 + 2*102 PDGSep 23 2002PRIME View
(3)175(3)17 (1035–1)/3 + 2*1017 PDGSep 23 2002PRIME View
(3)795(3)79 (10159–1)/3 + 2*1079 PDGSep 23 2002PRIME View
(3)1185(3)118 (10237–1)/3 + 2*10118 PDGSep 23 2002PRIME View
(3)1625(3)162 (10325–1)/3 + 2*10162 PDGSep 23 2002PRIME View
(3)1775(3)177 (10355–1)/3 + 2*10177 PDGSep 23 2002PRIME View
(3)1855(3)185 (10371–1)/3 + 2*10185 PDGSep 23 2002PRIME View
(3)2405(3)240 (10481–1)/3 + 2*10240 PDGSep 23 2002PRIME View
(3)8245(3)824 (101649–1)/3 + 2*10824 JHSep 29 2002PRIME View
(3)18205(3)1820 (103641–1)/3 + 2*101820 PDGSep 23 2002PRIME View DB
(3)23545(3)2354 (104709–1)/3 + 2*102354 PDGSep 23 2002PRIME View RC
A077785 ¬
A183180 ¬		 
(7)15(7)1 7*(10{3}–1)/9 – 2*101 PDGSep 23 2002PRIME View
(7)75(7)7 7*(1015–1)/9 – 2*107 PDGSep 23 2002PRIME View
(7)135(7)13 7*(1027–1)/9 – 2*1013 PDGSep 23 2002PRIME View
(7)585(7)58 7*(10117–1)/9 – 2*1058 PDGSep 23 2002PRIME View
(7)1295(7)129 7*(10259–1)/9 – 2*10129 PDGSep 23 2002PRIME View
(7)2535(7)253 7*(10507–1)/9 – 2*10253 PDGSep 23 2002PRIME View
(7)16575(7)1657 7*(103315–1)/9 – 2*101657 PDGSep 23 2002PRIME View DB
(7)22445(7)2244 7*(104489–1)/9 – 2*102244 PDGSep 23 2002PRIME View DB
(7)24375(7)2437 7*(104875–1)/9 – 2*102437 PDGSep 23 2002PRIME View
(7)79245(7)7924 7*(1015849–1)/9 – 2*107924 DHOct 31 2002PROBABLE
PRIME		 View
(7)99035(7)9903 7*(1019807–1)/9 – 2*109903 DHNov 04 2002PROBABLE
PRIME		 View
(7)118995(7)11899 7*(1023799–1)/9 – 2*1011899 DHNov 05 2002PROBABLE
PRIME		 View
(7)181575(7)18157 7*(1036315–1)/9 – 2*1018157 DHNov 18 2002PROBABLE
PRIME		 View
(7)189575(7)18957 7*(1037915–1)/9 – 2*1018957 DHNov 20 2002PROBABLE
PRIME		 View
(7)236655(7)23665 7*(1047331–1)/9 – 2*1023665 RPJun 25 2017PROBABLE
PRIME		 View
A077786 ¬
A183186 ¬		 
(9)885(9)88 (10177–1) – 4*1088 PDGSep 23 2002PRIME View
(9)1125(9)112 (10225–1) – 4*10112 PDGSep 23 2002PRIME View
(9)1985(9)198 (10{397}–1) – 4*10198 PDGSep 23 2002PRIME View
(9)6225(9)622 (101245–1) – 4*10622 JHOct 02 2002PRIME View
(9)42285(9)4228 (108457–1) – 4*104228 PDGOct 04 2002PRIME View
(9)100525(9)10052 (1020105–1) – 4*1010052 DHMar 28 2001PRIME View
(9)558625(9)55862 (10111725–1) – 4*1055862 DBSep 19 2010PRIME View
A077787 ¬
A107126 ¬		 
(1)106(1)10 (1021–1)/9 + 5*1010 PDGSep 23 2002PRIME View
(1)146(1)14 (10{29}–1)/9 + 5*1014 PDGSep 23 2002PRIME View
(1)406(1)40 (1081–1)/9 + 5*1040 PDGSep 23 2002PRIME View
(1)596(1)59 (10119–1)/9 + 5*1059 PDGSep 23 2002PRIME View
(1)1606(1)160 (10321–1)/9 + 5*10160 PDGSep 23 2002PRIME View
(1)4126(1)412 (10825–1)/9 + 5*10412 PDGSep 23 2002PRIME View
(1)5606(1)560 (101121–1)/9 + 5*10560 JHOct 02 2002PRIME View
(1)12896(1)1289 (10{2579}–1)/9 + 5*101289 JHOct 07 2002PRIME View
(1)18466(1)1846 (103693–1)/9 + 5*101846 PDGSep 23 2002PRIME View DB
A077788 ¬
A183181 ¬		 
(7)46(7)4 7*(109–1)/9 – 104 PDGSep 23 2002PRIME View
(7)56(7)5 7*(10{11}–1)/9 – 105 PDGSep 23 2002PRIME View
(7)86(7)8 7*(10{17}–1)/9 – 108 PDGSep 23 2002PRIME View
(7)116(7)11 7*(10{23}–1)/9 – 1011 PDGSep 23 2002PRIME View
(7)12446(7)1244 7*(102489–1)/9 – 101244 JHOct 16 2002PRIME View
(7)16856(7)1685 7*(10{3371}–1)/9 – 101685 PDGSep 23 2002PRIME View DB
(7)20096(7)2009 7*(10{4019}–1)/9 – 102009 PDGSep 23 2002PRIME View DB
(7)146576(7)14657 7*(1029315–1)/9 – 1014657 DHNov 13 2002PROBABLE
PRIME		 View
(7)151186(7)15118 7*(1030237–1)/9 – 1015118 DHNov 13 2002PROBABLE
PRIME		 View
A077789 ¬
A107127 ¬		 
   [ n ⩾ 100000 ]
(1)37(1)3 (10{7}–1)/9 + 6*103 PDGSep 23 2002PRIME View
(1)337(1)33 (10{67}–1)/9 + 6*1033 PDGSep 23 2002PRIME View
(1)3117(1)311 (10623–1)/9 + 6*10311 PDGSep 23 2002PRIME View
(1)29337(1)2933 (10{5867}–1)/9 + 6*102933 JHJun 21 2003PRIME View
(1)222357(1)22235 (1044471–1)/9 + 6*1022235 RPApr 30 2017PROBABLE
PRIME		 View
(1)391657(1)39165 (1078331–1)/9 + 6*1039165 RPApr 30 2017PROBABLE
PRIME		 View
(1)415857(1)41585 (1083171–1)/9 + 6*1041585 RPApr 30 2017PROBABLE
PRIME		 View
A077790 ¬
A183176 ¬		 
(3)17(3)1 (10{3}–1)/3 + 4*101 PDGSep 23 2002PRIME View
(3)37(3)3 (10{7}–1)/3 + 4*103 PDGSep 23 2002PRIME View
(3)77(3)7 (1015–1)/3 + 4*107 PDGSep 23 2002PRIME View
(3)117(3)11 (10{23}–1)/3 + 4*1011 PDGSep 23 2002PRIME View
(3)137(3)13 (1027–1)/3 + 4*1013 PDGSep 23 2002PRIME View
(3)177(3)17 (1035–1)/3 + 4*1017 PDGSep 23 2002PRIME View
(3)297(3)29 (10{59}–1)/3 + 4*1029 PDGSep 23 2002PRIME View
(3)317(3)31 (1063–1)/3 + 4*1031 PDGSep 23 2002PRIME View
(3)337(3)33 (10{67}–1)/3 + 4*1033 PDGSep 23 2002PRIME View
(3)777(3)77 (10155–1)/3 + 4*1077 PDGSep 23 2002PRIME View
(3)9337(3)933 (10{1867}–1)/3 + 4*10933 JHOct 01 2002PRIME View
(3)15557(3)1555 (103111–1)/3 + 4*101555 PDGSep 23 2002PRIME View DB
(3)117587(3)11758 (1023517–1)/3 + 4*1011758 DHNov 13 2002PROBABLE
PRIME		 View
 ¬ 
(9)1187(9)118 (10237–1) – 2*10118 PDGSep 23 2002PRIME View
(9)1451267(9)145126 (10290253–1) – 2*10145126 DBApr 11 2012PRIME View
A077791 ¬
A107648 ¬		 
(1)18(1)1 (10{3}–1)/9 + 7*101 PDGSep 23 2002PRIME View
(1)48(1)4 (109–1)/9 + 7*104 PDGSep 23 2002PRIME View
(1)68(1)6 (10{13}–1)/9 + 7*106 PDGSep 23 2002PRIME View
(1)78(1)7 (1015–1)/9 + 7*107 PDGSep 23 2002PRIME View
(1)3848(1)384 (10{769}–1)/9 + 7*10384 PDGSep 23 2002PRIME View
(1)6668(1)666 (101333–1)/9 + 7*10666 JHOct 02 2002PRIME View
(1)6758(1)675 (101351–1)/9 + 7*10675 JHOct 02 2002PRIME View
(1)31658(1)3165 (106331–1)/9 + 7*103165 DHOct 31 2002PRIME View
A077792 ¬
A183177 ¬		 
   [ n ⩾ 100000 ]
(3)18(3)1 (10{3}–1)/3 + 5*101 PDGSep 23 2002PRIME View
(3)78(3)7 (1015–1)/3 + 5*107 PDGSep 23 2002PRIME View
(3)858(3)85 (10171–1)/3 + 5*1085 PDGSep 23 2002PRIME View
(3)948(3)94 (10189–1)/3 + 5*1094 PDGSep 23 2002PRIME View
(3)2738(3)273 (10{547}–1)/3 + 5*10273 PDGSep 23 2002PRIME View
(3)3568(3)356 (10713–1)/3 + 5*10356 PDGSep 23 2002PRIME View
(3)10778(3)1077 (102155–1)/3 + 5*101077 JHOct 11 2002PRIME View
(3)17978(3)1797 (103595–1)/3 + 5*101797 PDGSep 23 2002PRIME View DB
(3)67588(3)6758 (1013517–1)/3 + 5*106758 DHOct 31 2002PROBABLE
PRIME		 View
(3)302328(3)30232 (1060465–1)/3 + 5*1030232 RPApr 21 2016PROBABLE
PRIME		 View
A077793 ¬
A183182 ¬		 
(7)18(7)1 7*(10{3}–1)/9 + 101 PDGSep 23 2002PRIME View
(7)38(7)3 7*(10{7}–1)/9 + 103 PDGSep 23 2002PRIME View
(7)398(7)39 7*(10{79}–1)/9 + 1039 PDGSep 23 2002PRIME View
(7)548(7)54 7*(10{109}–1)/9 + 1054 PDGSep 23 2002PRIME View
(7)1688(7)168 7*(10{337}–1)/9 + 10168 PDGSep 23 2002PRIME View
(7)2408(7)240 7*(10481–1)/9 + 10240 PDGSep 23 2002PRIME View
(7)53288(7)5328 7*(10{10657}–1)/9 + 105328 DHOct 31 2002PROBABLE
PRIME		 View
(7)61598(7)6159 7*(1012319–1)/9 + 106159 DHOct 31 2002PROBABLE
PRIME		 View
A077794 ¬
A183187 ¬		 
(9)268(9)26 (10{53}–1) – 1026 PDGSep 23 2002PRIME View
(9)3788(9)378 (10{757}–1) – 10378 PDGSep 23 2002PRIME View
(9)12468(9)1246 (102493–1) – 101246 HD––– –– 1989PRIME View
(9)17988(9)1798 (103597–1) – 101798 PDGSep 23 2002PRIME View
(9)29178(9)2917 (105835–1) – 102917 PDGOct 04 2002PRIME View
(9)230348(9)23034 (1046069–1) – 1023034 DHSep 22 2001PRIME View
(9)475098(9)47509 (1095019–1) – 1047509 DHJan 02 2003PRIME View
(9)521408(9)52140 (10{104281}–1) – 1052140 DHJan 27 2003PRIME View
(9)674048(9)67404 (10134809–1) – 1067404 DBNov 24 2010PRIME View
(9)9442648(9)944264 (101888529–1) – 10944264 P&BOct 18 2021 RECORD
PROVEN
PRIME		 View
A077795 ¬
A107649 ¬		 
(1)19(1)1 (10{3}–1)/9 + 8*101 PDGSep 23 2002PRIME View
(1)49(1)4 (109–1)/9 + 8*104 PDGSep 23 2002PRIME View
(1)269(1)26 (10{53}–1)/9 + 8*1026 PDGSep 23 2002PRIME View
(1)1879(1)187 (10375–1)/9 + 8*10187 PDGSep 23 2002PRIME View
(1)2269(1)226 (10453–1)/9 + 8*10226 PDGSep 23 2002PRIME View
(1)8749(1)874 (101749–1)/9 + 8*10874 JHOct 03 2002PRIME View
(1)133099(1)13309 (1026619–1)/9 + 8*1013309 DHNov 13 2002PROBABLE
PRIME		 View
A077796 ¬
A183183 ¬		 
(7)19(7)1 7*(10{3}–1)/9 + 2*101 PDGSep 23 2002PRIME View
(7)29(7)2 7*(10{5}–1)/9 + 2*102 PDGSep 23 2002PRIME View
(7)89(7)8 7*(10{17}–1)/9 + 2*108 PDGSep 23 2002PRIME View
(7)199(7)19 7*(1039–1)/9 + 2*1019 PDGSep 23 2002PRIME View
(7)209(7)20 7*(10{41}–1)/9 + 2*1020 PDGSep 23 2002PRIME View
(7)2129(7)212 7*(10425–1)/9 + 2*10212 PDGSep 23 2002PRIME View
(7)2809(7)280 7*(10561–1)/9 + 2*10280 PDGSep 23 2002PRIME View
(7)8879(7)887 7*(101775–1)/9 + 2*10887 JHOct 03 2002PRIME View
(7)10219(7)1021 7*(102043–1)/9 + 2*101021 JHOct 04 2002PRIME View
(7)55159(7)5515 7*(1011031–1)/9 + 2*105515 DHOct 31 2002PROBABLE
PRIME		 View
(7)81169(7)8116 7*(1016233–1)/9 + 2*108116 DHOct 31 2002PROBABLE
PRIME		 View
(7)118529(7)11852 7*(1023705–1)/9 + 2*1011852 DHNov 05 2002PROBABLE
PRIME		 View


Sources Revealed


Neil Sloane's "Integer Sequences" Encyclopedia can be consulted online :
Neil Sloane's Integer Sequences
Various numbers, primes and palindromic primes are categorised as follows :
%N Wing numbers. Start is identical to sequence A046075
%N Palindromic wing primes. under A077798
%N Palindromic wing primes exist for digitlengths a(n). under A077797
Wing numbers otherwise ordered by its wing digit and not its central digit by Amarnath Murthy
%N a(1) = 11; for n > 1, palindromic primes in which a single digit is sandwiched between strings of '1's. A088281
%N a(1) = 11; for n > 1, palindromic primes in which a single digit is sandwiched between strings of '3's. A088282
%N a(1) = 11; for n > 1, palindromic primes in which a single digit is sandwiched between strings of '7's. A088283
%N a(1) = 11; for n > 1, palindromic primes in which a single digit is sandwiched between strings of '9's. A088284
Click here to view some of the author's [P. De Geest] entries to the table.
Click here to view some entries to the table about palindromes.


Prime Curios! - site maintained by G. L. Honaker Jr. and Chris Caldwell
101
131
151
181
191
313
353
373
383
727
757
787
797
919
929
11311
1114111
1115111
111181111
777767777
77777677777
99999199999
1111118111111
11111...5...11111 (91-digits)
77777...8...77777 (109-digits)
77777...2...77777 (961-digits)
99999...2...99999 (1061-digits)
99999...8...99999 (2493-digits)
99999...2...99999 (2631-digits)
99999...2...99999 (5749-digits)

All of Daniel Heuer's probable primes above 10000 digits are also
submitted to the PRP TOP records table maintained by Henri & Renaud Lifchitz.
See : http://www.primenumbers.net/prptop/prptop.php

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( © All rights reserved ) - Last modified : September 27, 2022.
Patrick De Geest - Belgium flag - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com