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Plateau and Depression Primes
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101 131141151
161171181191313
323343353373383
717727747757767
787797919929989


Palindromic Plateau and Depression Primes

Plateau and Depression Primes (or PDP's for short) are numbers that
are primes, palindromic in base 10, and consisting of a repdigital interior
bordered by two identical single digits D different from the repdigit R.
D_RRR...RRR_D or D(R)nD
We have Plateau Primes when D < R
We have Depression Primes when D > R
E.g.
101
3222223
74444444447
79999999999999999999999999997


Sources were I found some PDP's ¬
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.


PDP's sorted by length

PDP's after division by 2 and 5


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

1(2)w1 = divisible by 11
11 x 11 = 121
111 x 11 = 1221
1111 x 11 = 12221
11111 x 11 = 122221
111111 x 11 = 1222221
...
general formula (1)k x 11 ; ( k ⩾ 2 )

7(3)w7 = divisible by 11
67 x 11 = 737
667 x 11 = 7337
6667 x 11 = 73337
66667 x 11 = 733337
666667 x 11 = 7333337
...
general formula (6)k7 x 11 ; ( k ⩾ 1 )

9(7)w9 = divisible by 11
89 x 11 = 979
889 x 11 = 9779
8889 x 11 = 97779
88889 x 11 = 977779
888889 x 11 = 9777779
...
general formula (8)k9 x 11 ; ( k ⩾ 1 )

9(4)w9 = always composite because
if w = even 11 is a divisor
if w@3 = 0 3 is a divisor
if w = odd and w@3 = 1 13 is a divisor
if w = odd and w@3 = 2 7 is a divisor

9(5)w9 = always composite because
if w = even 11 is a divisor
if w@3 = 0 3 is a divisor
if w = odd and w@3 = 1 7 is a divisor
if w = odd and w@3 = 2 13 is a divisor

7(1)w7 = is composite in the following general cases (J. C. Rosa)
if w = even 11 is a divisor
if (w–1)@6 = 0 3 is a divisor
if (w+1)@6 = 0 13 is a divisor
The only interesting cases to search for possible primes are when w = 6m + 3, for m ⩾ 0
E.g.: w = 10905 m = 1817 (J. K. Andersen)

Julien Peter Benney (email) adds to that [ May 12, 2004 ] :
if w = 18m + 3, for m ⩾ 0, then 19 is a divisor, as with 71117.
Thus, the statement should say :
The only interesting cases to search for possible primes are when w = 18m + 9 or 18m + 15, for m ⩾ 0

1(0)w1 = (C. Rivera & J. C. Rosa)
if w = even 11 is a divisor
Case for (w–1)@8 = 0 101 is a divisor, except for w=1 then 101 is prime.
Case for (w–3)@8 = 0 10001 is a divisor
Case for (w–5)@8 = 0 101 is a divisor
So only for (w+1)@8 = 0 this formula has some possibilities of being prime.
In fact only for (w+1)@(2^n) = 0 this formula has some possibilities of being prime.

This asks for some explanation (thanks JCR) :

1(0)w1 = 10(w+1)+1
1°) if w is even :
one has : 10 = –1 mod 11
hence 10^(w+1) = (–1)^(w+1) = –1 mod 11
and thus 10^(w+1)+1 = 0 mod 11

2°) if w is odd :

Suppose there exists an odd p, prime,
such that : 10^(w+1)+1 = 0 mod p
hence 10^(w+1) = –1 mod p
and (10^(w+1))^k = (–1)^k mod p

but (10^(w+1))^k = 10^(k*(w+1))
hence 10^(k*(w+1)) = (–1)^k mod p
So if k odd : 10^(k*(w+1)) = –1 mod p

Conclusion : If 10^(w+1)+1 is divisible by p,
then 10^(k*(w+1))+1, with k odd, is also divisible by p.

Examples
a) 10^2+1 = 101 prime hence 10^6+1, 10^10+1, 10^14+1, ...
are divisible by 101.
b) 10^4+1 = 0 mod 73 hence 10^12+1, 10^20+1, 10^28+1, ...
are divisible by 73.
c) 10^8+1 = 0 mod 17 hence 10^24+1, 10^40+1, 10^56+1, ...
are divisible by 17.
And so on...
Final explanatory note (thanks CR) :

There are no primes for 10x+1 if x is not of the form 2n
Here are some sources to back up the above statement:
http://yves.gallot.pagesperso-orange.fr/primes/math.html (theorem)
http://yves.gallot.pagesperso-orange.fr/primes/stat.html (finiteness)
http://mathworld.wolfram.com/GeneralizedFermatNumber.html
http://primes.utm.edu/glossary/page.php?sort=GeneralizedFermatPrime
See also: p. 359 of the Ribenboim's well known book
“The New Book of Prime Number Records”
See also: p. 426-427 of Riesel's well known book
“Prime numbers and computer Methods for factorization”


Messages

[ January 23, 2003 ]
David Broadhurst announced a new PDP record formerly at
http://groups.yahoo.com/group/primenumbers/message/11084
4*(102898-1)/3-1
He focused on three patterns that have a nice N^2-1 for PFGW :

My method can handle a(b)a only when
b = 2*a +/- 1 .
Since we must restrict a to {1,3,7,9},
I am limited to 1(3)1, 3(5)3, 3(7)3.
In addition to 1(3)_{2897}1
I have proven two smaller titanic primes:
1(3)_{1469}1
3(5)_{1973}3
both of which were in the Ondrejka tables.

I uploaded the helper files for the three PFGW proofs.
To complete the 3 proofs, one should prove that
every factor in these files is prime, but that doesn't take long.

David also proved the smallest titanic plateau and depression primes:
1(7)_{1001}1
9(1)_{1139}9
Primo certificates are available.


[ May 28, 2003 ]
Message from KAMADA Makoto

" We completed factorization of the sequence (8)w9 up to 150-digits.
(8)w9 is factor of plateau and depression number 9(7)w9.

My factorization project page is here.
Factorization of near-repdigit numbers
http://stdkmd.net/nrr/

Contributions of factorization are welcome.

Cheers,
(email)
http://stdkmd.net


[ August 2, 2003 ]
Message from Patrick De Geest
29*(103036+7)/9

" The largest PDP is now (29*10^3036+7)/9 or
2*(103037-1)/9 + (103036+1) or 3(2)30353 having a prime length of 3037 digits.
It was proved prime with 'Primo 2.1.1' using a 3000 MHz Pentium 4 cpu.
Certificate Primo-B29190474C134-01.out available by simple email request (945 KB).
Total timing = 170h 38mn 53s (around ~7,11 days) "


[ March 2, 2006 ]
Message from Greg Childers
(34*1015768–43)/9 the largest proven PDP to this date

" Patrick,

I have a new palprime with prime digits for your page at
http://www.worldofnumbers.com/won150.htm.
The proof of the 15769-digit prime (34*10^15768-43)/9 is located
at http://www.pa.uky.edu/~childers/certs/P15769.zip (zip file not available).
The zip file contained a readme.txt detailing the method of proof and
the certificates.
Available zip file (by courtesy of Chen Xinyao) at https://stdkmd.net/nrr/cert/3/#CERT_37773_15768

Thanks,
Greg "


[ March 8, 2009 ]
Message from Serge Batalov
(13*1067038-31)/9 the largest PDP to this date

" Dear Patrick,
I have found a rather big PRP last November, but I guess I never wrote about it to you.
I've reported all other quasi-rep-digit PRPs to M.Kamada. So, here goes :

http://stdkmd.net/nrr/1/14441.htm

(13*10^67038-31)/9 = 1(4)670371 <67039> is PRP. (Serge Batalov / PFGW / Nov 2, 2008)

It is also submitted in the Lifchitz PRP site, because it wasn't there yet,
so I decided that I may have discovered it, really. I realize that there may be
a chance that it is found not for the first time, but anyway, finally decided
to report it to you as well.

This is the only PDP number in my collection, all others are ABBBB or ABBBC-type.

Cheers,

Serge Batalov "


[ May 2009 ]
Messages from Serge Batalov (email)

" After a long desert in my PRP mining, I have hit another gem -- (5*10^66394-17)/3

(5*10^66394-17)/3 is 3-PRP! (217.2589s+0.0029s)
(5*10^66394-17)/3 is 23-PRP! (286.7449s+0.0033s)

It is a PD 166...661
and apparently I haven't beaten my own previous one. (13*10^67038-31)/9 "144...441"

This one is out of sequence -- it is a part of the "hopeless" quasi-rep-unit twin prime project
(which runs for more than half a year on 1 cpu, previously on 3; I've pre-sieved all possible pairs
and now PRP-ing slowly... then I'll need a bit of cleanup and after a month or so I will have removed
any possibility of any additional quasi-rep-unit twin primes up to 100000 digits)

P.S. No, it doesn't have a twin prime 166..663. :-)


Because of this number, I will now do this whole 16661 series in order. (For 14441, I've done that.)
I am sieving it now, and then will do 50000 ⩽ n ⩽ 100000
(the trivial test shows that only n=0 and 4 (mod 6) exponents are good)

Maybe I'll continue with all remaining 1xxx1 numbers, maybe not.
My computational resources are now quite limited...


Well... What do you know, here's another one --
(16*10^56082-61)/9 is 3-PRP! (199.1310s+0.0043s)
that's a 17771.

Serge Batalov "


Here's a PRP out of sequence. It's a 76667
(and I have started a run to make the 76667 in sequence to fill the gaps)

(23*10^95326+1)/3 is 3-PRP! (455.4071s+0.0046s)
(23*10^95326+1)/3 is 7-PRP! (562.4393s+4.2830s)

Brillhart-Lehmer-Selfridge test is running now.
Also, 15551 and 17771 were fully tested to n⩽100,000.

Serge


[ June 2009 ]
Messages from Serge Batalov (email)

" By filling the gaps in 76667 found yet another, in sequence
(23*10^81214+1)/3 is 3-PRP! (327.7524s+0.0038s)
It is now tested up to n⩽98,300. These are now the two PRPs, nothing else.

Serge


[ June 10, 2022 ]
Message from Xinyao Chen (email)

Concerning the search limit for the Plateau and Depression Primes ( '^^' is symbol for concatenation )

The current search limit for 1(0^^(n-1))1 = 10^n+1 is n=2^31-1,
since the next possible prime of the form 10^n+1 after 101 is 10^(2^31)+1,
10^n+1 is composite for all 2<n<2^31,
see http://www.prothsearch.com/GFN10.html










PDP Factorization Projects

( n = w + 1 )


Members can be prime.
1(0)w1 = 10n+1 Factorization of 100...001 (M. Kamada)
The Cunningham Project (search for '10^n+1')
1(3)w1 = (4.10n–7)/3 Factorization of 133...331 (P. De Geest) For w ⩽ 100
Factorization of 133...331 (M. Kamada)
1(4)w1 = (13.10n–31)/9 Factorization of 144...441 (M. Kamada)
1(5)w1 = (14.10n–41)/9 Factorization of 155...551 (M. Kamada)
1(6)w1 = (5.10n–17)/3 Factorization of 166...661 (M. Kamada)
1(7)w1 = (16.10n–61)/9 Factorization of 177...771 (M. Kamada)
1(8)w1 = (17.10n–71)/9 Factorization of 188...881 (M. Kamada)
1(9)w1 = 2.10n–9 Factorization of 199...991 (M. Kamada)
3(1)w3 = (28.10n+17)/9 Factorization of 311...113 (J.C. Rosa) For w ⩽ 100
Factorization of 311...113 (M. Kamada)
3(2)w3 = (29.10n+7)/9 Factorization of 322...223 (M. Kamada)
3(4)w3 = (31.10n–13)/9 Factorization of 344...443 (M. Kamada)
3(5)w3 = (32.10n–23)/9 Factorization of 355...553 (M. Kamada)
3(7)w3 = (34.10n–43)/9 Factorization of 377...773 (M. Kamada)
3(8)w3 = (35.10n–53)/9 Factorization of 388...883 (M. Kamada)
7(1)w7 = (64.10n+53)/9 Factorization of 711...117 (M. Kamada)
7(2)w7 = (65.10n+43)/9 Factorization of 722...227 (M. Kamada)
7(4)w7 = (67.10n+23)/9 Factorization of 744...447 (M. Kamada)
7(5)w7 = (68.10n+13)/9 Factorization of 755...557 (M. Kamada)
7(6)w7 = (23.10n+1)/3 Factorization of 766...667 (M. Kamada)
7(8)w7 = (71.10n–17)/9 Factorization of 788...887 (M. Kamada)
7(9)w7 = 8.10n–3 Factorization of 799...997 (M. Kamada)
9(1)w9 = (82.10n+71)/9 Factorization of 911...119 (M. Kamada)
9(2)w9 = (83.10n+61)/9 Factorization of 922...229 (M. Kamada)
9(8)w9 = (83.10n+61)/9 Factorization of 988...889 (M. Kamada)
Members (n>1) are always composite.
1(2)w1 = 11.(10n–1)/9 Factorization of 122...221 / 11 (M. Kamada)
The Cunningham Project (search for 10^n-1 or 10^n+1)
2(1)w2 = (19.10n+8)/9 Factorization of 211...112 (M. Kamada)
2(3)w2 = (7.10n–4)/3 Factorization of 233...332 (M. Kamada)
2(5)w2 = (23.10n–32)/9 Factorization of 255...552 (M. Kamada)
2(7)w2 = (25.10n–52)/9 Factorization of 277...772 (M. Kamada)
2(9)w2 = 3.10n–8 Factorization of 299...992 (M. Kamada)
4(1)w4 = (37.10n+26)/9 Factorization of 411...114 (M. Kamada)
4(3)w4 = (13.10n+2)/3 Factorization of 433...334 (M. Kamada)
4(5)w4 = (41.10n–14)/9 Factorization of 455...554 (M. Kamada)
4(7)w4 = (43.10n–34)/9 Factorization of 477...774 (M. Kamada)
4(9)w4 = 5.10n–6 Factorization of 499...994 (M. Kamada)
5(1)w5 = (46.10n+35)/9 Factorization of 511...115 (M. Kamada)
5(2)w5 = (47.10n+25)/9 Factorization of 522...225 (M. Kamada)
5(3)w5 = (16.10n+5)/3 Factorization of 533...335 (Patrick De Geest)

      Free to factor    70 remaining  
5(4)w5 = (49.10n+5)/9 Factorization of 544...445 (M. Kamada)
5(6)w5 = (17.10n–5)/3 Factorization of 566...665 (M. Kamada)
5(7)w5 = (52.10n–25)/9 Factorization of 577...775 (M. Kamada)
5(8)w5 = (53.10n–35)/9 Factorization of 588...885 (M. Kamada)
5(9)w5 = 6.10n–5 Factorization of 599...995 (M. Kamada)
6(1)w6 = 11.(5.10n+4)/9 Factorization of 611...116 / 11 * 18 (M. Kamada)
6(5)w6 = (59.10n+4)/9 Factorization of 655...556 (M. Kamada)
6(7)w6 = (61.10n–16)/9 Factorization of 677...776 (M. Kamada)
7(3)w7 = 11.(2.10n+1)/3 Factorization of 733...337 / 11 (M. Kamada)
8(1)w8 = (73.10n+62)/9 Factorization of 811...118 (M. Kamada)
8(3)w8 = (25.10n+14)/3 Factorization of 833...338 (M. Kamada)
8(5)w8 = 11.(7.10n+2)/9 Factorization of 855...558 / 2 / 11 (M. Kamada)
8(7)w8 = (79.10n+2)/9 Factorization of 877...778 (M. Kamada)
8(9)w8 = 9.10n–2 Factorization of 899...998 (M. Kamada)
9(4)w9 = (85.10n+41)/9 Factorization of 944...449 (M. Kamada)
9(5)w9 = (86.10n+31)/9 Factorization of 955...559 (M. Kamada)
9(7)w9 = 11.(8.10n+1)/9 Factorization of 977...779 / 11 (M. Kamada)


[ August 31, 2022 ] Message from Chen Xinyao

Complete list of the factorization of all possible “Palindromic Depression and Plateau Numbers” can be found here Factorization of ABB...BBA (M. Kamada)
with an exception of 5333...3335, which is on hold for Kamada's page because the script does not support the longer algebraic factor
(i.e. 16*10^(4*n/5) 8*10^(3*n/5) + 4*10^(2*n/5) 2*10^(n/5) + 1).
Link https://stdkmd.net/nrr/news2022.htm#NEWS_20220424

Sum of 5th powers, 5(3^^n)5 ('^^' is symbol for concatenation) is (16*10^(n+1)+5)/3, which has sum-of-5th-power factorization if n = 5*m.

Following condition must be imposed that gcd(A,B) = 1, i.e. A and B are coprime, since if A and B have a common factor > 1, then we can divide this factor from the number,
e.g. factor 6999...9996 is equivalent to factor 2333...3332.

Factoring Calculator from 'Number Empire' gives with input (16*10^(5*n+1)+5)/3 :

(5*(5^n*2^(n+1)+1)*(5^(4*n)*2^(4*n+4)–5^(3*n)*2^(3*n+3)+5^(2*n)*2^(2*n+2)–5^n*2^(n+1)+1))/3


Same exercise in an alternative calculator (link added by PDG).

Factoring Calculator from 'EMath' gives with input (16*10^(5n+1)+5)/3 its answer at the end of the step by step procedure:

5 * (2*10^(n/5)+1) * [ 16*10^(4*n/5) – 8*10^(3*n/5) + 4*10^(2*n/5) – 2*10^(n/5) + 1 ] / 3

if n is divisible by 5.


Well, allow me to make that missing file facpdp535.htm myself.



The Table


The reference table for
Plateau and Depression Primes
This collection is complete for
probable primes up to 100,000 (ref. RC)
digits and for proven
primes up to  7363  digits.
DB = David Broadhurst
GC = Greg Childers
JCR = Jean Claude Rosa
JKA = Jens Kruse Andersen
PDG = Patrick De Geest
RC = Ray Chandler
RPSB = Ryan Propper & Serge Batalov
SB = Serge Batalov
TB = Tyler Busby
PDPFormula
Blue exp = # of digits
Accolades = prime exp
WhoWhenStatusOutput
Logs
 ¬ 
  10n+1   [ n = (# of digits) – 1]
[n >= 2^31 or 2147483648 (by X. Chen)]
1(0)11 0*(10{3}–1)/9 + (102+1)
IMPORTANT NOTE
JCROct 14 2002PRIME View
A082697 ¬
A056244 ¬
 
  (12*10n–21)/9 or (4*10^n–7)/3 or 4*(10n–1)/3–1
[ n > 249,551 (by RC)]
1(3)11 (10{3}–1)/3 – 2*(102+1) JCROct 14 2002PRIME View
1(3)31 (10{5}–1)/3 – 2*(104+1) JCROct 14 2002PRIME View
1(3)51 (10{7}–1)/3 – 2*(106+1) JCROct 14 2002PRIME View
1(3)931 (1095–1)/3 – 2*(1094+1) JCROct 14 2002PRIME View
1(3)1591 (10161–1)/3 – 2*(10160+1) JCROct 14 2002PRIME View
1(3)3591 (10361–1)/3 – 2*(10360+1) PDGNov 19 2002PRIME View
1(3)14691 (10{1471}–1)/3 – 2*(101470+1) DBJan 23 2003PRIME View
1(3)28971 (102899–1)/3 – 2*(102898+1) DBJan 23 2003PRIME View
1(3)30931 (103095–1)/3 – 2*(103094+1) PDGAug 20 2003PRIME View
1(3)31111 (103113–1)/3 – 2*(103112+1) PDGSep 01 2003PRIME View
1(3)156971 (1015699–1)/3 – 2*(1015698+1) PDGJan 13 2003PROBABLE
PRIME
View
1(3)179551 (10{17957}–1)/3 – 2*(1017956+1) PDGJan 14 2003PROBABLE
PRIME
View
1(3)422611 (1042263–1)/3 – 2*(1042262+1) PDGOct 03 2004PROBABLE
PRIME
View
1(3)1110311 (10111033–1)/3 – 2*(10111032+1) RCApr 14 2011PROBABLE
PRIME
View
1(3)2495491 (10249551–1)/3 – 2*(10249550+1) SBJan 15 2023PROBABLE
PRIME
View
A082698 ¬
A056245 ¬
 
  (13*10n–31)/9
1(4)51 4*(10{7}–1)/9 – 3*(106+1) JCROct 14 2002PRIME View
1(4)651 4*(10{67}–1)/9 – 3*(1066+1) JCROct 14 2002PRIME View
1(4)12531 4*(101255–1)/9 – 3*(101254+1) PDGJul 02 2003PRIME View
1(4)84051 4*(108407–1)/9 – 3*(108406+1) PDGNov 20 2002PROBABLE
PRIME
View
1(4)670371 4*(1067039–1)/9 – 3*(1067038+1) SBNov 2 2008PROBABLE
PRIME
View
A082699 ¬
A056246 ¬
 
  (14*10n–41)/9
1(5)11 5*(10{3}–1)/9 – 4*(102+1) JCROct 14 2002PRIME View
1(5)31 5*(10{5}–1)/9 – 4*(104+1) JCROct 14 2002PRIME View
1(5)191 5*(1021–1)/9 – 4*(1020+1) JCROct 14 2002PRIME View
1(5)311 5*(1033–1)/9 – 4*(1032+1) JCROct 14 2002PRIME View
1(5)3991 5*(10{401}–1)/9 – 4*(10400+1) PDGNov 20 2002PRIME View
1(5)5611 5*(10{563}–1)/9 – 4*(10562+1) PDGNov 20 2002PRIME View
1(5)70151 5*(107017–1)/9 – 4*(107016+1) PDGNov 21 2002PRIME View
1(5)376831 5*(1037685–1)/9 – 4*(1037684+1) PDGOct 11 2004PROBABLE
PRIME
View
1(5)2112611 5*(10211263–1)/9 – 4*(10211262+1) SBJan 20 2023PROBABLE
PRIME
View
1(5)2227171 5*(10222719–1)/9 – 4*(10222718+1) SBJan 21 2023PROBABLE
PRIME
View
1(5)2503351 5*(10250337–1)/9 – 4*(10250336+1) SBJan 22 2023PROBABLE
PRIME
View
A082700 ¬
A056247 ¬
 
  (15*10n–51)/9 or (5*10n–17)/3
[ n > 200,000 (by RC)]
1(6)31 2*(10{5}–1)/3 – 5*(104+1) JCROct 14 2002PRIME View
1(6)111 2*(10{13}–1)/3 – 5*(1012+1) JCROct 14 2002PRIME View
1(6)151 2*(10{17}–1)/3 – 5*(1016+1) JCROct 14 2002PRIME View
1(6)171 2*(10{19}–1)/3 – 5*(1018+1) JCROct 14 2002PRIME View
1(6)351 2*(10{37}–1)/3 – 5*(1036+1) JCROct 14 2002PRIME View
1(6)511 2*(10{53}–1)/3 – 5*(1052+1) JCROct 14 2002PRIME View
1(6)711 2*(10{73}–1)/3 – 5*(1072+1) JCROct 14 2002PRIME View
1(6)991 2*(10{101}–1)/3 – 5*(10100+1) JCROct 14 2002PRIME View
1(6)62311 2*(106233–1)/3 – 5*(106232+1) PDGNov 22 2002PRIME View
1(6)240271 2*(10{24029}–1)/3 – 5*(1024028+1) PDGOct 15 2004PROBABLE
PRIME
View
1(6)402211 2*(1040223–1)/3 – 5*(1040222+1) PDGOct 17 2004PROBABLE
PRIME
View
1(6)663931 2*(1066395–1)/3 – 5*(1066394+1) SBMay 16 2009PROBABLE
PRIME
View
A082701 ¬
A056248 ¬
 
  (16*10n–61)/9
1(7)51 7*(10{7}–1)/9 – 6*(106+1) JCROct 14 2002PRIME View
1(7)471 7*(1049–1)/9 – 6*(1048+1) JCROct 14 2002PRIME View
1(7)1011 7*(10{103}–1)/9 – 6*(10102+1) JCROct 14 2002PRIME View
1(7)1911 7*(10{193}–1)/9 – 6*(10192+1) JCROct 14 2002PRIME View
1(7)3651 7*(10{367}–1)/9 – 6*(10366+1) PDGNov 22 2002PRIME View
1(7)10011 7*(101003–1)/9 – 6*(101002+1) DBJan 23 2003PRIME View
1(7)203631 7*(1020365–1)/9 – 6*(1020364+1) PDGOct 20 2004PROBABLE
PRIME
View
1(7)374451 7*(10{37447}–1)/9 – 6*(1037446+1) PDGOct 21 2004PROBABLE
PRIME
View
1(7)560811 7*(1056083–1)/9 – 6*(1056082+1) SBMay 17 2009PROBABLE
PRIME
View
A082702 ¬
A056249 ¬
 
  (17*10n–71)/9
1(8)11 8*(10{3}–1)/9 – 7*(102+1) JCROct 14 2002PRIME View
1(8)71 8*(109–1)/9 – 7*(108+1) JCROct 14 2002PRIME View
1(8)131 8*(1015–1)/9 – 7*(1014+1) JCROct 14 2002PRIME View
1(8)391 8*(10{41}–1)/9 – 7*(1040+1) JCROct 14 2002PRIME View
1(8)911 8*(1093–1)/9 – 7*(1092+1) JCROct 14 2002PRIME View
1(8)1271 8*(10129–1)/9 – 7*(10128+1) JCROct 14 2002PRIME View
1(8)8831 8*(10885–1)/9 – 7*(10884+1) PDGNov 23 2002PRIME View
1(8)94231 8*(109425–1)/9 – 7*(109424+1) PDGDec 11 2002PROBABLE
PRIME
View
1(8)147671 8*(1014769–1)/9 – 7*(1014768+1) PDGFeb 06 2003PROBABLE
PRIME
View
1(8)192571 8*(10{19259}–1)/9 – 7*(1019258+1) PDGFeb 07 2003PROBABLE
PRIME
View
1(8)312331 8*(1031235–1)/9 – 7*(1031234+1) PDGNov 17 2004PROBABLE
PRIME
View
A082703 ¬
A056250 ¬
 
  (18*10n–81)/9 or 2*10n–9   [ n > 200,000 (by RC)]
1(9)11 (10{3}–1) – 8*(102+1) JCROct 14 2002PRIME View
1(9)31 (10{5}–1) – 8*(104+1) JCROct 14 2002PRIME View
1(9)71 (109–1) – 8*(108+1) JCROct 14 2002PRIME View
1(9)391 (10{41}–1) – 8*(1040+1) JCROct 14 2002PRIME View
1(9)851 (1087–1) – 8*(1086+1) JCROct 14 2002PRIME View
1(9)1991 (10201–1) – 8*(10200+1) JCROct 14 2002PRIME View
1(9)7291 (10731–1) – 8*(10730+1) PDGNov 24 2002PRIME View
1(9)14591 (101461–1) – 8*(101460+1) PDGJul 04 2003PRIME View
1(9)236711 (1023673–1) – 8*(1023672+1) PDGNov 25 2004PROBABLE
PRIME
View
1(9)286291 (10{28631}–1) – 8*(1028630+1) PDGNov 26 2004PROBABLE
PRIME
View
A082704 ¬
A056251 ¬
 
  (28*10n+17)/9
3(1)13 (10{3}–1)/9 + 2*(102+1) JCROct 14 2002PRIME View
3(1)113 (10{13}–1)/9 + 2*(1012+1) JCROct 14 2002PRIME View
3(1)133 (1015–1)/9 + 2*(1014+1) JCROct 14 2002PRIME View
3(1)293 (10{31}–1)/9 + 2*(1030+1) JCROct 14 2002PRIME View
3(1)1033 (10105–1)/9 + 2*(10104+1) JCROct 14 2002PRIME View
3(1)1253 (10{127}–1)/9 + 2*(10126+1) JCROct 14 2002PRIME View
3(1)3413 (10343–1)/9 + 2*(10342+1) PDGNov 25 2002PRIME View
3(1)5993 (10{601}–1)/9 + 2*(10600+1) PDGNov 25 2002PRIME View
3(1)98233 (109825–1)/9 + 2*(109824+1) PDGDec 14 2002PROBABLE
PRIME
View
A082705 ¬
A056252 ¬
 
  (29*10n+7)/9
3(2)53 2*(10{7}–1)/9 + (106+1) JCROct 14 2002PRIME View
3(2)73 2*(109–1)/9 + (108+1) JCROct 14 2002PRIME View
3(2)8933 2*(10895–1)/9 + (10894+1) PDGNov 25 2002PRIME View
3(2)15233 2*(101525–1)/9 + (101524+1) PDGJul 06 2003PRIME View
3(2)30353 2*(10{3037}–1)/9 + (103036+1) PDGAug 02 2003PRIME View
3(2)211553 2*(10{21157}–1)/9 + (1021156+1) PDGApr 16 2005PROBABLE
PRIME
View
A082706 ¬
A056253 ¬
 
  (31*10n–13)/9
3(4)53 4*(10{7}–1)/9 – (106+1) JCROct 14 2002PRIME View
3(4)113 4*(10{13}–1)/9 – (1012+1) JCROct 14 2002PRIME View
3(4)4913 4*(10493–1)/9 – (10492+1) PDGNov 26 2002PRIME View
3(4)55673 4*(10{5569}–1)/9 – (105568+1) PDGNov 26 2002PRIME View
3(4)247553 4*(1024757–1)/9 – (1024756+1) PDGApr 17 2005PROBABLE
PRIME
View
A082707 ¬
A056254 ¬
 
  (32*10n–23)/9 or 32*(10n–1)/9+1
3(5)13 5*(10{3}–1)/9 – 2*(102+1) JCROct 14 2002PRIME View
3(5)73 5*(109–1)/9 – 2*(108+1) JCROct 14 2002PRIME View
3(5)1393 5*(10141–1)/9 – 2*(10140+1) JCROct 14 2002PRIME View
3(5)2293 5*(10231–1)/9 – 2*(10230+1) JCROct 14 2002PRIME View
3(5)4253 5*(10427–1)/9 – 2*(10426+1) PDGNov 27 2002PRIME View
3(5)4613 5*(10{463}–1)/9 – 2*(10462+1) PDGNov 27 2002PRIME View
3(5)7253 5*(10{727}–1)/9 – 2*(10726+1) PDGNov 27 2002PRIME View
3(5)19733 5*(101975–1)/9 – 2*(101974+1) DBJan 23 2003PRIME View
3(5)72293 5*(107231–1)/9 – 2*(107230+1) PDGNov 28 2002PRIME View
3(5)458593 5*(1045861–1)/9 – 2*(1045860+1) PDGMay 16 2005PROBABLE
PRIME
View
3(5)473033 5*(1047305–1)/9 – 2*(1047304+1) PDGMay 17 2005PROBABLE
PRIME
View
3(5)2838253 5*(10283827–1)/9 – 2*(10283826+1) SBJan 17 2023PROBABLE
PRIME
View
A082708 ¬
A056255 ¬
 
  (34*10n–43)/9 or 34*(10n–1)/9–1
3(7)13 7*(10{3}–1)/9 – 4*(102+1) JCROct 14 2002PRIME View
3(7)133 7*(1015–1)/9 – 4*(1014+1) JCROct 14 2002PRIME View
3(7)533 7*(1055–1)/9 – 4*(1054+1) JCROct 14 2002PRIME View
3(7)673 7*(1069–1)/9 – 4*(1068+1) JCROct 14 2002PRIME View
3(7)833 7*(1085–1)/9 – 4*(1084+1) JCROct 14 2002PRIME View
3(7)853 7*(1087–1)/9 – 4*(1086+1) JCROct 14 2002PRIME View
3(7)1553 7*(10{157}–1)/9 – 4*(10156+1) JCROct 14 2002PRIME View
3(7)27653 7*(10{2767}–1)/9 – 4*(102766+1) PDGJul 21 2003PRIME View
3(7)33793 7*(103381–1)/9 – 4*(103380+1) PDGOct 09 2003PRIME View
3(7)38753 7*(10{3877}–1)/9 – 4*(103876+1) PDGNov 28 2002PRIME View RC
3(7)52073 7*(10{5209}–1)/9 – 4*(105208+1) PDGNov 28 2002PRIME View RC
3(7)107453 7*(1010747–1)/9 – 4*(1010746+1) PDGDec 20 2002PROBABLE
PRIME
View
3(7)157673 7*(1015769–1)/9 – 4*(1015768+1) GCFeb 28 2006RECORD
PROVEN
PRIME
View
3(7)313153 7*(1031317–1)/9 – 4*(1031316+1) PDGMay 18 2005PROBABLE
PRIME
View
3(7)409573 7*(1040959–1)/9 – 4*(1040958+1) PDGMay 20 2005PROBABLE
PRIME
View
3(7)458033 7*(1045805–1)/9 – 4*(1045804+1) PDGMay 22 2005PROBABLE
PRIME
View
3(7)465653 7*(10{46567}–1)/9 – 4*(1046566+1) PDGMay 22 2005PROBABLE
PRIME
View
3(7)510073 7*(1051009–1)/9 – 4*(1051008+1) RCSep 20 2010PROBABLE
PRIME
View
3(7)801613 7*(1080163–1)/9 – 4*(1080162+1) RCDec 13 2010PROBABLE
PRIME
View
A082709 ¬
A056256 ¬
 
  (35*10n–53)/9
3(8)13 8*(10{3}–1)/9 – 5*(102+1) JCROct 14 2002PRIME View
3(8)113 8*(10{13}–1)/9 – 5*(1012+1) JCROct 14 2002PRIME View
3(8)293 8*(10{31}–1)/9 – 5*(1030+1) JCROct 14 2002PRIME View
3(8)593 8*(10{61}–1)/9 – 5*(1060+1) JCROct 14 2002PRIME View
3(8)1153 8*(10117–1)/9 – 5*(10116+1) JCROct 14 2002PRIME View
3(8)2893 8*(10291–1)/9 – 5*(10290+1) JCROct 14 2002PRIME View
3(8)6313 8*(10633–1)/9 – 5*(10632+1) PDGNov 29 2002PRIME View
3(8)10633 8*(101065–1)/9 – 5*(101064+1) PDGFeb 02 2003PRIME View
3(8)14933 8*(101495–1)/9 – 5*(101494+1) PDGJul 05 2003PRIME View
3(8)54313 8*(105433–1)/9 – 5*(105432+1) PDGNov 29 2002PRIME View
3(8)73613 8*(107363–1)/9 – 5*(107362+1) PDGNov 29 2002PRIME View
A082710 ¬
¬
 
  (64*10n+53)/9   [ n > 1,200,000 (by SB)]
7(1)109057 (1010907–1)/9 + 6*(1010906+1) JKAOct 17 2002PRIME View
7(1)4992097 (10499211–1)/9 + 6*(10499210+1) SBMar 01 2015PROBABLE
PRIME
View
A082711 ¬
A056257 ¬
 
  (65*10n+43)/9
7(2)17 2*(10{3}–1)/9 + 5*(102+1) JCROct 14 2002PRIME View
7(2)37 2*(10{5}–1)/9 + 5*(104+1) JCROct 14 2002PRIME View
7(2)77 2*(109–1)/9 + 5*(108+1) JCROct 14 2002PRIME View
7(2)277 2*(10{29}–1)/9 + 5*(1028+1) JCROct 14 2002PRIME View
7(2)637 2*(1065–1)/9 + 5*(1064+1) JCROct 14 2002PRIME View
7(2)7237 2*(10725–1)/9 + 5*(10724+1) PDGNov 29 2002PRIME View
7(2)17857 2*(10{1787}–1)/9 + 5*(101786+1) PDGJul 09 2003PRIME View
7(2)72757 2*(107277–1)/9 + 5*(107276+1) PDGNov 30 2002PRIME View
7(2)194617 2*(10{19463}–1)/9 + 5*(1019462+1) PDGMar 16 2003PROBABLE
PRIME
View
7(2)242137 2*(1024215–1)/9 + 5*(1024214+1) PDGApr 21 2005PROBABLE
PRIME
View
7(2)517777 2*(1051779–1)/9 + 5*(1051778+1) RCSep 21 2010PROBABLE
PRIME
View
7(2)1313917 2*(10131393–1)/9 + 5*(10131392+1) TBJan 11 2023PROBABLE
PRIME
View
A082712 ¬
A056258 ¬
 
  (67*10n+23)/9
7(4)97 4*(10{11}–1)/9 + 3*(1010+1) JCROct 14 2002PRIME View
7(4)297 4*(10{31}–1)/9 + 3*(1030+1) JCROct 14 2002PRIME View
7(4)1197 4*(10121–1)/9 + 3*(10120+1) JCROct 14 2002PRIME View
7(4)4837 4*(10485–1)/9 + 3*(10484+1) PDGNov 30 2002PRIME View
7(4)14857 4*(10{1487}–1)/9 + 3*(101486+1) PDGJul 05 2003PRIME View
7(4)15777 4*(10{1579}–1)/9 + 3*(101578+1) PDGJul 06 2003PRIME View
7(4)136717 4*(1013673–1)/9 + 3*(1013672+1) PDGMar 17 2003PROBABLE
PRIME
View
7(4)138097 4*(1013811–1)/9 + 3*(1013810+1) PDGMar 17 2003PROBABLE
PRIME
View
7(4)150937 4*(1015095–1)/9 + 3*(1015094+1) PDGMar 18 2003PROBABLE
PRIME
View
7(4)727717 4*(1072773–1)/9 + 3*(1072772+1) RCNov 12 2010PROBABLE
PRIME
View
7(4)942117 4*(1094213–1)/9 + 3*(1094212+1) RCFeb 22 2011PROBABLE
PRIME
View
7(4)2075557 4*(10{207557}–1)/9 + 3*(10207556+1) SBJan 13 2023PROBABLE
PRIME
View
7(4)11166757 4*(10{1116677}–1)/9 + 3*(101116676+1) RPSBJan 22 2023RECORD
PROBABLE
PRIME
View
A082713 ¬
A056259 ¬
 
  (68*10n+13)/9
7(5)17 5*(10{3}–1)/9 + 2*(102+1) JCROct 14 2002PRIME View
7(5)37 5*(10{5}–1)/9 + 2*(104+1) JCROct 14 2002PRIME View
7(5)97 5*(10{11}–1)/9 + 2*(1010+1) JCROct 14 2002PRIME View
7(5)197 5*(1021–1)/9 + 2*(1020+1) JCROct 14 2002PRIME View
7(5)217 5*(10{23}–1)/9 + 2*(1022+1) JCROct 14 2002PRIME View
7(5)577 5*(10{59}–1)/9 + 2*(1058+1) JCROct 14 2002PRIME View
7(5)737 5*(1075–1)/9 + 2*(1074+1) JCROct 14 2002PRIME View
7(5)817 5*(10{83}–1)/9 + 2*(1082+1) JCROct 14 2002PRIME View
7(5)2077 5*(10209–1)/9 + 2*(10208+1) JCROct 14 2002PRIME View
7(5)3497 5*(10351–1)/9 + 2*(10350+1) PDGNov 30 2002PRIME View
7(5)4217 5*(10423–1)/9 + 2*(10422+1) PDGNov 30 2002PRIME View
7(5)38117 5*(103813–1)/9 + 2*(103812+1) PDGNov 30 2002PRIME View RC
7(5)39817 5*(103983–1)/9 + 2*(103982+1) PDGNov 30 2002PRIME View RC
7(5)209237 5*(1020925–1)/9 + 2*(1020924+1) PDGApr 23 2005PROBABLE
PRIME
View
7(5)237857 5*(1023787–1)/9 + 2*(1023786+1) PDGApr 23 2005PROBABLE
PRIME
View
7(5)388517 5*(1038853–1)/9 + 2*(1038852+1) PDGMay 04 2005PROBABLE
PRIME
View
7(5)560417 5*(1056043–1)/9 + 2*(1056042+1) RCSep 29 2010PROBABLE
PRIME
View
7(5)685037 5*(1068505–1)/9 + 2*(1068504+1) RCOct 30 2010PROBABLE
PRIME
View
7(5)744337 5*(1074435–1)/9 + 2*(1074434+1) RCNov 18 2010PROBABLE
PRIME
View
7(5)2055097 5*(10205511–1)/9 + 2*(10205510+1) SBJan 13 2023PROBABLE
PRIME
View
A082714 ¬
A056260 ¬
 
  (69*10n+3)/9  or (23*10n+1)/3
[ n > 700,000 (by RC)]
7(6)37 2*(10{5}–1)/3 + (104+1) JCROct 14 2002PRIME View
7(6)57 2*(10{7}–1)/3 + (106+1) JCROct 14 2002PRIME View
7(6)537 2*(1055–1)/3 + (1054+1) JCROct 14 2002PRIME View
7(6)957 2*(10{97}–1)/3 + (1096+1) JCROct 14 2002PRIME View
7(6)4537 2*(10455–1)/3 + (10454+1) PDGDec 01 2002PRIME View
7(6)5737 2*(10575–1)/3 + (10574+1) PDGDec 01 2002PRIME View
7(6)33837 2*(103385–1)/3 + (103384+1) PDGOct 25 2003PRIME View
7(6)114397 2*(1011441–1)/3 + (1011440+1) PDGMar 22 2003PROBABLE
PRIME
View
7(6)126237 2*(1012625–1)/3 + (1012624+1) PDGMar 22 2003PROBABLE
PRIME
View
7(6)194457 2*(10{19447}–1)/3 + (1019446+1) PDGMar 25 2003PROBABLE
PRIME
View
7(6)354597 2*(10{35461}–1)/3 + (1035460+1) PDGJun 08 2005PROBABLE
PRIME
View
7(6)812137 2*(1081215–1)/3 + (1081214+1) SBJun 08 2009PROBABLE
PRIME
View
7(6)953257 2*(10{95327}–1)/3 + (1095326+1) SBMay 27 2009PROBABLE
PRIME
View
A082715 ¬
A056262 ¬
 
  (71*10n–17)/9
7(8)17 8*(10{3}–1)/9 – (102+1) JCROct 14 2002PRIME View
7(8)37 8*(10{5}–1)/9 – (104+1) JCROct 14 2002PRIME View
7(8)857 8*(1087–1)/9 – (1086+1) JCROct 14 2002PRIME View
7(8)1117 8*(10{113}–1)/9 – (10112+1) JCROct 14 2002PRIME View
7(8)1697 8*(10171–1)/9 – (10170+1) JCROct 14 2002PRIME View
7(8)5657 8*(10567–1)/9 – (10566+1) PDGDec 02 2002PRIME View
7(8)16877 8*(101689–1)/9 – (101688+1) PDGJul 07 2003PRIME View
7(8)89017 8*(108903–1)/9 – (108902+1) PDGJan 03 2003PROBABLE
PRIME
View
7(8)1158097 8*(10{115811}–1)/9 – (10115810+1) RCAug 05 2011PROBABLE
PRIME
View
7(8)1657157 8*(10165717–1)/9 – (10165716+1) SBJan 19 2023PROBABLE
PRIME
View
A082716 ¬
A056263 ¬
 
  (72*10n–27)/9 or 8*10n–3   [ n > 219,740 (by RC)]
7(9)17 (10{3}–1) – 2*(102+1) JCROct 14 2002PRIME View
7(9)37 (10{5}–1) – 2*(104+1) JCROct 14 2002PRIME View
7(9)277 (10{29}–1) – 2*(1028+1) JCROct 14 2002PRIME View
7(9)1557 (10{157}–1) – 2*(10156+1) JCROct 14 2002PRIME View
7(9)3217 (10323–1) – 2*(10322+1) PDGDec 02 2002PRIME View
7(9)3517 (10{353}–1) – 2*(10352+1) PDGDec 02 2002PRIME View
7(9)12117 (10{1213}–1) – 2*(101212+1) PDGJun 30 2003PRIME View
7(9)12837 (101285–1) – 2*(101284+1) PDGJul 03 2003PRIME View
7(9)79837 (107985–1) – 2*(107984+1) PDGJan 07 2003PROBABLE
PRIME
View
7(9)151917 (10{15193}–1) – 2*(1015192+1) PDGMar 28 2003PROBABLE
PRIME
View
7(9)847717 (1084773–1) – 2*(1084772+1) RCJan 3 2011PROBABLE
PRIME
View
7(9)1199297 (10119931–1) – 2*(10119930+1) RCApr 1 2011PROBABLE
PRIME
View
7(9)1488597 (10{148861}–1) – 2*(10148860+1) RCApr 9 2011PROBABLE
PRIME
View
7(9)2197397 (10219741–1) – 2*(10219740+1) SBJan 14 2023PROBABLE
PRIME
View
A082717 ¬
A056264 ¬
 
  (82*10n+71)/9
9(1)19 (10{3}–1)/9 + 8*(102+1) JCROct 15 2002PRIME View
9(1)2459 (10247–1)/9 + 8*(10246+1) JCROct 15 2002PRIME View
9(1)11399 (101141–1)/9 + 8*(101140+1) DBJan 23 2003PRIME View
9(1)103939 (1010395–1)/9 + 8*(1010394+1) PDGJan 16 2003PROBABLE
PRIME
View
9(1)438799 (1043881–1)/9 + 8*(1043880+1) PDGJun 23 2005PROBABLE
PRIME
View
A082718 ¬
A056265 ¬
 
  (83*10n+61)/9
9(2)19 2*(10{3}–1)/9 + 7*(102+1) JCROct 15 2002PRIME View
9(2)59 2*(10{7}–1)/9 + 7*(106+1) JCROct 15 2002PRIME View
9(2)119 2*(10{13}–1)/9 + 7*(1012+1) JCROct 15 2002PRIME View
9(2)1099 2*(10111–1)/9 + 7*(10110+1) JCROct 15 2002PRIME View
9(2)36079 2*(103609–1)/9 + 7*(103608+1) PDGNov 14 2003PRIME View
9(2)377839 2*(1037785–1)/9 + 7*(1037784+1) PDGJun 26 2005PROBABLE
PRIME
View
9(2)1815439 2*(10181545–1)/9 + 7*(10181544+1) SBJan 19 2023PROBABLE
PRIME
View
A082719 ¬
A056266 ¬
 
  (89*10n+1)/9   [ n > 700,000 (by SB)]
9(8)59 8*(10{7}–1)/9 + (106+1) JCROct 15 2002PRIME View
9(8)719 8*(10{73}–1)/9 + (1072+1) JCROct 15 2002PRIME View
9(8)959 8*(10{97}–1)/9 + (1096+1) JCROct 15 2002PRIME View
9(8)1139 8*(10115–1)/9 + (10114+1) JCROct 15 2002PRIME View
9(8)2039 8*(10205–1)/9 + (10204+1) JCROct 15 2002PRIME View
9(8)9839 8*(10985–1)/9 + (10984+1) PDGDec 04 2002PRIME View
9(8)12259 8*(101227–1)/9 + (101226+1) PDGJul 01 2003PRIME View
9(8)47939 8*(104795–1)/9 + (104794+1) PDGDec 04 2002PRIME View RC
9(8)207199 8*(1020721–1)/9 + (1020720+1) PDGApr 05 2003PROBABLE
PRIME
View
9(8)1335799 8*(10133581–1)/9 + (10133580+1) SBMay 15 2010PROBABLE
PRIME
View
9(8)4115899 8*(10411591–1)/9 + (10411590+1) SBSep 21 2014PROBABLE
PRIME
View


Data table for PDP's becoming prime when removing all prime factors 2 and 5

[ May 13, 2023 ]
Data table for the PDP's ending with digits 2, 4, 5, 6 and 8 becoming prime when removing
all the prime factors 2 and 5 (i.e. A132740).
By Xinyao Chen.

Formprime at n
(2(1^^n)2)/(2^3)25, 133, 193, 289, 511, 1075, ...
(2(3^^n)2)/(2^2)7, 11, 37, 1743, 2023, 10123, ...
(2(5^^n)2)/(2^5)5, 9, 21, 111, 153, 303, 339, 531, 965, ...
(2(7^^n)2)/(2^2)7, 147, 301, 309, 1203, ...
(2(9^^n)2)/(2^3)59, 107, 139, 251, 463, 1051, ...
(4(1^^n)4)/(2^1) ⩾ 300833 (none found, but a covering set does not appear)
(4(3^^n)4)/(2^1)11, 39, 63, 113, 129, 323, 393, 905, ...
(4(5^^n)4)/(2^1)1, 3, 19, 12475, ...
(4(7^^n)4)/(2^1)3, 5, 23, 195, ...
(4(9^^n)4)/(2^1)5, 17, 41, ...
(5(1^^n)5)/(5^1)0, 1, 3, 9, 13, 31, 139, 211, 1203, ...
(5(2^^n)5)/(5^2)3, 5, 527, ...
(5(3^^n)5)/(5^1)0, 1, 3, 133, 139, ...
(5(4^^n)5)/(5^1)0, 1, 3, 37, 63, 153, 283, 1179, ...
(5(6^^n)5)/(5^1)0, 1, 5, 7, 25, 1157, 2609, ...
(5(7^^n)5)/(5^2)1, 3, 19, 25, 85, 87, 103, 121, 4303, 23269, ...
(5(8^^n)5)/(5^1)0, 3, 9, 23, 47, 59, 489, 4695, ...
(5(9^^n)5)/(5^1)0, 5, 3191, 3785, 5513, 14717, ...
(6(1^^n)6)/(2^2)none exists (always divisible by 11)
(6(5^^n)6)/(2^2)1129, ...
(6(7^^n)6)/(2^2)11, 77, 911, ...
(8(1^^n)8)/(2^1)1, 3, 69, 85, 399, ...
(8(3^^n)8)/(2^1)1, 3, 63, 73, 183, 237, 835, 907, ...
(8(5^^n)8)/(2^1)none exists (always divisible by 11)
(8(7^^n)8)/(2^1)1, 3, 7, 67, 133, 583, 703, 861, ...
(8(9^^n)8)/(2^1)1, 11959, ...


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 Plateau and depression numbers. under A0?????
%N Plateau and depression primes. under A056728
%N Plateau and depression primes exist for digitlengths a(n). under A082720
%N Primes which are a sandwich of numbers using at most one digit between two 1's. under A068685
%N Primes which are a sandwich of numbers made of only one digit between two 3's. under A068687
%N Primes which are a sandwich of numbers made of only one digit between two 7's. under A068688
%N Primes which are a sandwich of numbers made of only one digit between two 9's. under A068689
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.


C. Rivera, J.C. Rosa and J.K. Andersen, Puzzle 197. Always composite numbers?

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
10001 depression composite
13331
16661
19991
50005 depression composite
76667
1777771
188888881
722222227
1666666666661
3111111111113
311111111111113
31111...11113 (31-digits)
15555...15555 (33-digits)
78888...88887 (87-digits)
18888...88881 (93-digits)
13333...33331 (95-digits)
98888...88889 (97-digits)
16666...66661 (101-digits)
31111...11113 (105-digits)
91111...11119 (247-digits)
18888...88881 (885-digits)
98888...88889 (985-digits)
17777...77771 (1003-digits)
91111...11119 (1141-digits)
32222...22223 (1525-digits)
13333...33331 (2899-digits)
32222...22223 (3037-digits)

I (PDG) also submitted all probable primes above 10000 digits
to the PRP TOP records table maintained by Henri & Renaud Lifchitz.
See : http://www.primenumbers.net/prptop/prptop.php










 

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Patrick De Geest - Belgium flag - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com