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101 131141151
161171181191313
323343353373383
717727747757767
787797919929989


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 (Polish PostScript file) by Andrzej Nowicki
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


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://perso.wanadoo.fr/yves.gallot/primes/math.html (theorem)
http://perso.wanadoo.fr/yves.gallot/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"


[ January 23, 2003 ]
David Broadhurst announced a new PDP record 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 factorizations 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.

Factorizations of near-repdigit numbers
http://mada.la.coocan.jp/nrr/

Contributions of factorizations are welcome.

Cheers,
(email)
http://homepage2.nifty.com/m_kamada/ "



[ 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/em150.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 (broken link).
The zip file contains a readme.txt detailing the method of proof and
the certificates.

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://mada.la.coocan.jp/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










Factorization Projects

( n = w + 1 )


1(0)w1 = 10n+1 The Cunningham Project (search for '10^n+1')
1(3)w1 = (4.10n-7)/3 Factorizations of 133...331 (M. Kamada)
1(4)w1 = (13.10n-31)/9 Factorizations of 144...441 (M. Kamada)
1(5)w1 = (14.10n-41)/9 Factorizations of 155...551 (M. Kamada)
1(6)w1 = (5.10n-17)/3 Factorizations of 166...661 (M. Kamada)
1(7)w1 = (16.10n-61)/9 Factorizations of 177...771 (M. Kamada)
1(8)w1 = (17.10n-71)/9 Factorizations of 188...881 (M. Kamada)
1(9)w1 = 2.10n-9 Factorizations of 199...991 (M. Kamada)
3(1)w3 = (28.10n+17)/9 Factorizations of 311...113 (M. Kamada)
3(2)w3 = (29.10n+7)/9 Factorizations of 322...223 (M. Kamada)
3(4)w3 = (31.10n-13)/9 Factorizations of 344...443 (M. Kamada)
3(5)w3 = (32.10n-23)/9 Factorizations of 355...553 (M. Kamada)
3(7)w3 = (34.10n-43)/9 Factorizations of 377...773 (M. Kamada)
3(8)w3 = (35.10n-53)/9 Factorizations of 388...883 (M. Kamada)
7(1)w7 = (64.10n+53)/9 Factorizations of 711...117 (M. Kamada)
7(2)w7 = (65.10n+43)/9 Factorizations of 722...227 (M. Kamada)
7(3)w7 = 11(2.10n+1)/3 Factorizations of 733...337 / 11 (M. Kamada)
7(4)w7 = (67.10n+23)/9 Factorizations of 744...447 (M. Kamada)
7(5)w7 = (68.10n+13)/9 Factorizations of 755...557 (M. Kamada)
7(6)w7 = (23.10n+1)/3 Factorizations of 766...667 (M. Kamada)
7(8)w7 = (71.10n-17)/9 Factorizations of 788...887 (M. Kamada)
7(9)w7 = 8.10n-3 Factorizations of 799...997 (M. Kamada)
9(1)w9 = (82.10n+71)/9 Factorizations of 911...119 (M. Kamada)
9(2)w9 = (83.10n+61)/9 Factorizations of 922...229 (M. Kamada)
9(4)w9 = (85.10n+41)/9 Factorizations of 944...449 (M. Kamada)
9(5)w9 = (86.10n+31)/9 Factorizations of 955...559 (M. Kamada)
9(7)w9 = 11(8.10n+1)/9 Factorizations of 977...779 / 11 (M. Kamada)
9(8)w9 = (83.10n+61)/9 Factorizations of 988...889 (M. Kamada)


1(3)w1 Factorizations of 133...331 (P. De Geest) For w <= 100
3(1)w3 Factorizations of 311...113 (J.C. Rosa) For w <= 100



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  3609  digits.
DB = David Broadhurst
GC = Greg Childers
JCR = Jean Claude Rosa
JKA = Jens Kruse Andersen
PDG = Patrick De Geest
RC = Ray Chandler
SB = Serge Batalov
PDPFormula
blue exp = # of digits
WhoWhenStatusOutput
Logs
 ¬   10n+1   [ n = (# of digits) - 1]
[ n > 2^22 or 4,194,304 (by RC)]
1(0)11 0*(103-1)/9 + (102+1)
IMPORTANT NOTE
JCR Oct 14 2002 PRIME View
A082697 ¬
A056244 ¬
  (12*10n-21)/9 or (4*10^n-7)/3 or 4*(10n-1)/3-1
[ n > 200,000 (by RC)]
1(3)11 (103-1)/3 - 2*(102+1) JCR Oct 14 2002 PRIME View
1(3)31 (105-1)/3 - 2*(104+1) JCR Oct 14 2002 PRIME View
1(3)51 (107-1)/3 - 2*(106+1) JCR Oct 14 2002 PRIME View
1(3)931 (1095-1)/3 - 2*(1094+1) JCR Oct 14 2002 PRIME View
1(3)1591 (10161-1)/3 - 2*(10160+1) JCR Oct 14 2002 PRIME View
1(3)3591 (10361-1)/3 - 2*(10360+1) PDG Nov 19 2002 PRIME View
1(3)14691 (101471-1)/3 - 2*(101470+1) DB Jan 23 2003 PRIME View
1(3)28971 (102899-1)/3 - 2*(102898+1) DB Jan 23 2003 PRIME View
1(3)30931 (103095-1)/3 - 2*(103094+1) PDG Aug 20 2003 PRIME View
1(3)31111 (103113-1)/3 - 2*(103112+1) PDG Sep 01 2003 PRIME View
1(3)156971 (1015699-1)/3 - 2*(1015698+1) PDG Jan 13 2003 PROBABLE
PRIME
View
1(3)179551 (1017957-1)/3 - 2*(1017956+1) PDG Jan 14 2003 PROBABLE
PRIME
View
1(3)422611 (1042263-1)/3 - 2*(1042262+1) PDG Oct 03 2004 PROBABLE
PRIME
View
1(3)1110311 (10111033-1)/3 - 2*(10111032+1) RC Apr 14 2011 PROBABLE
PRIME
View
A082698 ¬
A056245 ¬
  (13*10n-31)/9
1(4)51 4*(107-1)/9 - 3*(106+1) JCR Oct 14 2002 PRIME View
1(4)651 4*(1067-1)/9 - 3*(1066+1) JCR Oct 14 2002 PRIME View
1(4)12531 4*(101255-1)/9 - 3*(101254+1) PDG Jul 02 2003 PRIME View
1(4)84051 4*(108407-1)/9 - 3*(108406+1) PDG Nov 20 2002 PROBABLE
PRIME
View
1(4)670371 4*(1067039-1)/9 - 3*(1067038+1) SB Nov 2 2008 PROBABLE
PRIME
View
A082699 ¬
A056246 ¬
  (14*10n-41)/9
1(5)11 5*(103-1)/9 - 4*(102+1) JCR Oct 14 2002 PRIME View
1(5)31 5*(105-1)/9 - 4*(104+1) JCR Oct 14 2002 PRIME View
1(5)191 5*(1021-1)/9 - 4*(1020+1) JCR Oct 14 2002 PRIME View
1(5)311 5*(1033-1)/9 - 4*(1032+1) JCR Oct 14 2002 PRIME View
1(5)3991 5*(10401-1)/9 - 4*(10400+1) PDG Nov 20 2002 PRIME View
1(5)5611 5*(10563-1)/9 - 4*(10562+1) PDG Nov 20 2002 PRIME View
1(5)70151 5*(107017-1)/9 - 4*(107016+1) PDG Nov 21 2002 PROBABLE
PRIME
View
1(5)376831 5*(1037685-1)/9 - 4*(1037684+1) PDG Oct 11 2004 PROBABLE
PRIME
View
A082700 ¬
A056247 ¬
  (15*10n-51)/9 or (5*10n-17)/3
[ n > 200,000 (by RC)]
1(6)31 2*(105-1)/3 - 5*(104+1) JCR Oct 14 2002 PRIME View
1(6)111 2*(1013-1)/3 - 5*(1012+1) JCR Oct 14 2002 PRIME View
1(6)151 2*(1017-1)/3 - 5*(1016+1) JCR Oct 14 2002 PRIME View
1(6)171 2*(1019-1)/3 - 5*(1018+1) JCR Oct 14 2002 PRIME View
1(6)351 2*(1037-1)/3 - 5*(1036+1) JCR Oct 14 2002 PRIME View
1(6)511 2*(1053-1)/3 - 5*(1052+1) JCR Oct 14 2002 PRIME View
1(6)711 2*(1073-1)/3 - 5*(1072+1) JCR Oct 14 2002 PRIME View
1(6)991 2*(10101-1)/3 - 5*(10100+1) JCR Oct 14 2002 PRIME View
1(6)62311 2*(106233-1)/3 - 5*(106232+1) PDG Nov 22 2002 PROBABLE
PRIME
View
1(6)240271 2*(1024029-1)/3 - 5*(1024028+1) PDG Oct 15 2004 PROBABLE
PRIME
View
1(6)402211 2*(1040223-1)/3 - 5*(1040222+1) PDG Oct 17 2004 PROBABLE
PRIME
View
1(6)663931 2*(1066395-1)/3 - 5*(1066394+1) SB May 16 2009 PROBABLE
PRIME
View
A082701 ¬
A056248 ¬
  (16*10n-61)/9
1(7)51 7*(107-1)/9 - 6*(106+1) JCR Oct 14 2002 PRIME View
1(7)471 7*(1049-1)/9 - 6*(1048+1) JCR Oct 14 2002 PRIME View
1(7)1011 7*(10103-1)/9 - 6*(10102+1) JCR Oct 14 2002 PRIME View
1(7)1911 7*(10193-1)/9 - 6*(10192+1) JCR Oct 14 2002 PRIME View
1(7)3651 7*(10367-1)/9 - 6*(10366+1) PDG Nov 22 2002 PRIME View
1(7)10011 7*(101003-1)/9 - 6*(101002+1) DB Jan 23 2003 PRIME View
1(7)203631 7*(1020365-1)/9 - 6*(1020364+1) PDG Oct 20 2004 PROBABLE
PRIME
View
1(7)374451 7*(1037447-1)/9 - 6*(1037446+1) PDG Oct 21 2004 PROBABLE
PRIME
View
1(7)560811 7*(1056083-1)/9 - 6*(1056082+1) SB May 17 2009 PROBABLE
PRIME
View
A082702 ¬
A056249 ¬
  (17*10n-71)/9
1(8)11 8*(103-1)/9 - 7*(102+1) JCR Oct 14 2002 PRIME View
1(8)71 8*(109-1)/9 - 7*(108+1) JCR Oct 14 2002 PRIME View
1(8)131 8*(1015-1)/9 - 7*(1014+1) JCR Oct 14 2002 PRIME View
1(8)391 8*(1041-1)/9 - 7*(1040+1) JCR Oct 14 2002 PRIME View
1(8)911 8*(1093-1)/9 - 7*(1092+1) JCR Oct 14 2002 PRIME View
1(8)1271 8*(10129-1)/9 - 7*(10128+1) JCR Oct 14 2002 PRIME View
1(8)8831 8*(10885-1)/9 - 7*(10884+1) PDG Nov 23 2002 PRIME View
1(8)94231 8*(109425-1)/9 - 7*(109424+1) PDG Dec 11 2002 PROBABLE
PRIME
View
1(8)147671 8*(1014769-1)/9 - 7*(1014768+1) PDG Feb 06 2003 PROBABLE
PRIME
View
1(8)192571 8*(1019259-1)/9 - 7*(1019258+1) PDG Feb 07 2003 PROBABLE
PRIME
View
1(8)312331 8*(1031235-1)/9 - 7*(1031234+1) PDG Nov 17 2004 PROBABLE
PRIME
View
A082703 ¬
A056250 ¬
  (18*10n-81)/9 or 2*10n-9   [ n > 200,000 (by RC)]
1(9)11 (103-1) - 8*(102+1) JCR Oct 14 2002 PRIME View
1(9)31 (105-1) - 8*(104+1) JCR Oct 14 2002 PRIME View
1(9)71 (109-1) - 8*(108+1) JCR Oct 14 2002 PRIME View
1(9)391 (1041-1) - 8*(1040+1) JCR Oct 14 2002 PRIME View
1(9)851 (1087-1) - 8*(1086+1) JCR Oct 14 2002 PRIME View
1(9)1991 (10201-1) - 8*(10200+1) JCR Oct 14 2002 PRIME View
1(9)7291 (10731-1) - 8*(10730+1) PDG Nov 24 2002 PRIME View
1(9)14591 (101461-1) - 8*(101460+1) PDG Jul 04 2003 PRIME View
1(9)236711 (1023673-1) - 8*(1023672+1) PDG Nov 25 2004 PROBABLE
PRIME
View
1(9)286291 (1028631-1) - 8*(1028630+1) PDG Nov 26 2004 PROBABLE
PRIME
View
A082704 ¬
A056251 ¬
  (28*10n+17)/9
3(1)13 (103-1)/9 + 2*(102+1) JCR Oct 14 2002 PRIME View
3(1)113 (1013-1)/9 + 2*(1012+1) JCR Oct 14 2002 PRIME View
3(1)133 (1015-1)/9 + 2*(1014+1) JCR Oct 14 2002 PRIME View
3(1)293 (1031-1)/9 + 2*(1030+1) JCR Oct 14 2002 PRIME View
3(1)1033 (10105-1)/9 + 2*(10104+1) JCR Oct 14 2002 PRIME View
3(1)1253 (10127-1)/9 + 2*(10126+1) JCR Oct 14 2002 PRIME View
3(1)3413 (10343-1)/9 + 2*(10342+1) PDG Nov 25 2002 PRIME View
3(1)5993 (10601-1)/9 + 2*(10600+1) PDG Nov 25 2002 PRIME View
3(1)98233 (109825-1)/9 + 2*(109824+1) PDG Dec 14 2002 PROBABLE
PRIME
View
A082705 ¬
A056252 ¬
  (29*10n+7)/9
3(2)53 2*(107-1)/9 + (106+1) JCR Oct 14 2002 PRIME View
3(2)73 2*(109-1)/9 + (108+1) JCR Oct 14 2002 PRIME View
3(2)8933 2*(10895-1)/9 + (10894+1) PDG Nov 25 2002 PRIME View
3(2)15233 2*(101525-1)/9 + (101524+1) PDG Jul 06 2003 PRIME View
3(2)30353 2*(103037-1)/9 + (103036+1) PDG Aug 02 2003 PRIME View
3(2)211553 2*(1021157-1)/9 + (1021156+1) PDG Apr 16 2005 PROBABLE
PRIME
View
A082706 ¬
A056253 ¬
  (31*10n-13)/9
3(4)53 4*(107-1)/9 - (106+1) JCR Oct 14 2002 PRIME View
3(4)113 4*(1013-1)/9 - (1012+1) JCR Oct 14 2002 PRIME View
3(4)4913 4*(10493-1)/9 - (10492+1) PDG Nov 26 2002 PRIME View
3(4)55673 4*(105569-1)/9 - (105568+1) PDG Nov 26 2002 PROBABLE
PRIME
View
3(4)247553 4*(1024757-1)/9 - (1024756+1) PDG Apr 17 2005 PROBABLE
PRIME
View
A082707 ¬
A056254 ¬
  (32*10n-23)/9 or 32*(10n-1)/9+1
3(5)13 5*(103-1)/9 - 2*(102+1) JCR Oct 14 2002 PRIME View
3(5)73 5*(109-1)/9 - 2*(108+1) JCR Oct 14 2002 PRIME View
3(5)1393 5*(10141-1)/9 - 2*(10140+1) JCR Oct 14 2002 PRIME View
3(5)2293 5*(10231-1)/9 - 2*(10230+1) JCR Oct 14 2002 PRIME View
3(5)4253 5*(10427-1)/9 - 2*(10426+1) PDG Nov 27 2002 PRIME View
3(5)4613 5*(10463-1)/9 - 2*(10462+1) PDG Nov 27 2002 PRIME View
3(5)7253 5*(10727-1)/9 - 2*(10726+1) PDG Nov 27 2002 PRIME View
3(5)19733 5*(101975-1)/9 - 2*(101974+1) DB Jan 23 2003 PRIME View
3(5)72293 5*(107231-1)/9 - 2*(107230+1) PDG Nov 28 2002 PROBABLE
PRIME
View
3(5)458593 5*(1045861-1)/9 - 2*(1045860+1) PDG May 16 2005 PROBABLE
PRIME
View
3(5)473033 5*(1047305-1)/9 - 2*(1047304+1) PDG May 17 2005 PROBABLE
PRIME
View
A082708 ¬
A056255 ¬
  (34*10n-43)/9 or 34*(10n-1)/9-1
3(7)13 7*(103-1)/9 - 4*(102+1) JCR Oct 14 2002 PRIME View
3(7)133 7*(1015-1)/9 - 4*(1014+1) JCR Oct 14 2002 PRIME View
3(7)533 7*(1055-1)/9 - 4*(1054+1) JCR Oct 14 2002 PRIME View
3(7)673 7*(1069-1)/9 - 4*(1068+1) JCR Oct 14 2002 PRIME View
3(7)833 7*(1085-1)/9 - 4*(1084+1) JCR Oct 14 2002 PRIME View
3(7)853 7*(1087-1)/9 - 4*(1086+1) JCR Oct 14 2002 PRIME View
3(7)1553 7*(10157-1)/9 - 4*(10156+1) JCR Oct 14 2002 PRIME View
3(7)27653 7*(102767-1)/9 - 4*(102766+1) PDG Jul 21 2003 PRIME View
3(7)33793 7*(103381-1)/9 - 4*(103380+1) PDG Oct 09 2003 PRIME View
3(7)38753 7*(103877-1)/9 - 4*(103876+1) PDG Nov 28 2002 PRIME View RC
3(7)52073 7*(105209-1)/9 - 4*(105208+1) PDG Nov 28 2002 PRIME View RC
3(7)107453 7*(1010747-1)/9 - 4*(1010746+1) PDG Dec 20 2002 PROBABLE
PRIME
View
3(7)157673 7*(1015769-1)/9 - 4*(1015768+1) GC Feb 28 2006 RECORD
PROVEN
PRIME
View
3(7)313153 7*(1031317-1)/9 - 4*(1031316+1) PDG May 18 2005 PROBABLE
PRIME
View
3(7)409573 7*(1040959-1)/9 - 4*(1040958+1) PDG May 20 2005 PROBABLE
PRIME
View
3(7)458033 7*(1045805-1)/9 - 4*(1045804+1) PDG May 22 2005 PROBABLE
PRIME
View
3(7)465653 7*(1046567-1)/9 - 4*(1046566+1) PDG May 22 2005 PROBABLE
PRIME
View
3(7)510073 7*(1051009-1)/9 - 4*(1051008+1) RC Sep 20 2010 PROBABLE
PRIME
View
3(7)801613 7*(1080163-1)/9 - 4*(1080162+1) RC Dec 13 2010 PROBABLE
PRIME
View
A082709 ¬
A056256 ¬
  (35*10n-53)/9
3(8)13 8*(103-1)/9 - 5*(102+1) JCR Oct 14 2002 PRIME View
3(8)113 8*(1013-1)/9 - 5*(1012+1) JCR Oct 14 2002 PRIME View
3(8)293 8*(1031-1)/9 - 5*(1030+1) JCR Oct 14 2002 PRIME View
3(8)593 8*(1061-1)/9 - 5*(1060+1) JCR Oct 14 2002 PRIME View
3(8)1153 8*(10117-1)/9 - 5*(10116+1) JCR Oct 14 2002 PRIME View
3(8)2893 8*(10291-1)/9 - 5*(10290+1) JCR Oct 14 2002 PRIME View
3(8)6313 8*(10633-1)/9 - 5*(10632+1) PDG Nov 29 2002 PRIME View
3(8)10633 8*(101065-1)/9 - 5*(101064+1) PDG Feb 02 2003 PRIME View
3(8)14933 8*(101495-1)/9 - 5*(101494+1) PDG Jul 05 2003 PRIME View
3(8)54313 8*(105433-1)/9 - 5*(105432+1) PDG Nov 29 2002 PROBABLE
PRIME
View
3(8)73613 8*(107363-1)/9 - 5*(107362+1) PDG Nov 29 2002 PROBABLE
PRIME
View
A082710 ¬
¬
  (64*10n+53)/9   [ n > 400,000 (by SB)]
7(1)109057 (1010907-1)/9 + 6*(1010906+1) JKA Oct 17 2002 PROBABLE
PRIME
View
A082711 ¬
A056257 ¬
  (65*10n+43)/9
7(2)17 2*(103-1)/9 + 5*(102+1) JCR Oct 14 2002 PRIME View
7(2)37 2*(105-1)/9 + 5*(104+1) JCR Oct 14 2002 PRIME View
7(2)77 2*(109-1)/9 + 5*(108+1) JCR Oct 14 2002 PRIME View
7(2)277 2*(1029-1)/9 + 5*(1028+1) JCR Oct 14 2002 PRIME View
7(2)637 2*(1065-1)/9 + 5*(1064+1) JCR Oct 14 2002 PRIME View
7(2)7237 2*(10725-1)/9 + 5*(10724+1) PDG Nov 29 2002 PRIME View
7(2)17857 2*(101787-1)/9 + 5*(101786+1) PDG Jul 09 2003 PRIME View
7(2)72757 2*(107277-1)/9 + 5*(107276+1) PDG Nov 30 2002 PROBABLE
PRIME
View
7(2)194617 2*(1019463-1)/9 + 5*(1019462+1) PDG Mar 16 2003 PROBABLE
PRIME
View
7(2)242137 2*(1024215-1)/9 + 5*(1024214+1) PDG Apr 21 2005 PROBABLE
PRIME
View
7(2)517777 2*(1051779-1)/9 + 5*(1051778+1) RC Sep 21 2010 PROBABLE
PRIME
View
A082712 ¬
A056258 ¬
  (67*10n+23)/9
7(4)97 4*(1011-1)/9 + 3*(1010+1) JCR Oct 14 2002 PRIME View
7(4)297 4*(1031-1)/9 + 3*(1030+1) JCR Oct 14 2002 PRIME View
7(4)1197 4*(10121-1)/9 + 3*(10120+1) JCR Oct 14 2002 PRIME View
7(4)4837 4*(10485-1)/9 + 3*(10484+1) PDG Nov 30 2002 PRIME View
7(4)14857 4*(101487-1)/9 + 3*(101486+1) PDG Jul 05 2003 PRIME View
7(4)15777 4*(101579-1)/9 + 3*(101578+1) PDG Jul 06 2003 PRIME View
7(4)136717 4*(1013673-1)/9 + 3*(1013672+1) PDG Mar 17 2003 PROBABLE
PRIME
View
7(4)138097 4*(1013811-1)/9 + 3*(1013810+1) PDG Mar 17 2003 PROBABLE
PRIME
View
7(4)150937 4*(1015095-1)/9 + 3*(1015094+1) PDG Mar 18 2003 PROBABLE
PRIME
View
7(4)727717 4*(1072773-1)/9 + 3*(1072772+1) RC Nov 12 2010 PROBABLE
PRIME
View
7(4)942117 4*(1094213-1)/9 + 3*(1094212+1) RC Feb 22 2011 PROBABLE
PRIME
View
A082713 ¬
A056259 ¬
  (68*10n+13)/9
7(5)17 5*(103-1)/9 + 2*(102+1) JCR Oct 14 2002 PRIME View
7(5)37 5*(105-1)/9 + 2*(104+1) JCR Oct 14 2002 PRIME View
7(5)97 5*(1011-1)/9 + 2*(1010+1) JCR Oct 14 2002 PRIME View
7(5)197 5*(1021-1)/9 + 2*(1020+1) JCR Oct 14 2002 PRIME View
7(5)217 5*(1023-1)/9 + 2*(1022+1) JCR Oct 14 2002 PRIME View
7(5)577 5*(1059-1)/9 + 2*(1058+1) JCR Oct 14 2002 PRIME View
7(5)737 5*(1075-1)/9 + 2*(1074+1) JCR Oct 14 2002 PRIME View
7(5)817 5*(1083-1)/9 + 2*(1082+1) JCR Oct 14 2002 PRIME View
7(5)2077 5*(10209-1)/9 + 2*(10208+1) JCR Oct 14 2002 PRIME View
7(5)3497 5*(10351-1)/9 + 2*(10350+1) PDG Nov 30 2002 PRIME View
7(5)4217 5*(10423-1)/9 + 2*(10422+1) PDG Nov 30 2002 PRIME View
7(5)38117 5*(103813-1)/9 + 2*(103812+1) PDG Nov 30 2002 PRIME View RC
7(5)39817 5*(103983-1)/9 + 2*(103982+1) PDG Nov 30 2002 PRIME View RC
7(5)209237 5*(1020925-1)/9 + 2*(1020924+1) PDG Apr 23 2005 PROBABLE
PRIME
View
7(5)237857 5*(1023787-1)/9 + 2*(1023786+1) PDG Apr 23 2005 PROBABLE
PRIME
View
7(5)388517 5*(1038853-1)/9 + 2*(1038852+1) PDG May 04 2005 PROBABLE
PRIME
View
7(5)560417 5*(1056043-1)/9 + 2*(1056042+1) RC Sep 29 2010 PROBABLE
PRIME
View
7(5)685037 5*(1068505-1)/9 + 2*(1068504+1) RC Oct 30 2010 PROBABLE
PRIME
View
7(5)744337 5*(1074435-1)/9 + 2*(1074434+1) RC Nov 18 2010 PROBABLE
PRIME
View
A082714 ¬
A056260 ¬
  (69*10n+3)/9  or (23*10n+1)/3
[ n > 700,000 (by RC)]
7(6)37 2*(105-1)/3 + (104+1) JCR Oct 14 2002 PRIME View
7(6)57 2*(107-1)/3 + (106+1) JCR Oct 14 2002 PRIME View
7(6)537 2*(1055-1)/3 + (1054+1) JCR Oct 14 2002 PRIME View
7(6)957 2*(1097-1)/3 + (1096+1) JCR Oct 14 2002 PRIME View
7(6)4537 2*(10455-1)/3 + (10454+1) PDG Dec 01 2002 PRIME View
7(6)5737 2*(10575-1)/3 + (10574+1) PDG Dec 01 2002 PRIME View
7(6)33837 2*(103385-1)/3 + (103384+1) PDG Oct 25 2003 PRIME View
7(6)114397 2*(1011441-1)/3 + (1011440+1) PDG Mar 22 2003 PROBABLE
PRIME
View
7(6)126237 2*(1012625-1)/3 + (1012624+1) PDG Mar 22 2003 PROBABLE
PRIME
View
7(6)194457 2*(1019447-1)/3 + (1019446+1) PDG Mar 25 2003 PROBABLE
PRIME
View
7(6)354597 2*(1035461-1)/3 + (1035460+1) PDG Jun 08 2005 PROBABLE
PRIME
View
7(6)812137 2*(1081215-1)/3 + (1081214+1) SB Jun 08 2009 PROBABLE
PRIME
View
7(6)953257 2*(1095327-1)/3 + (1095326+1) SB May 27 2009 PROBABLE
PRIME
View
A082715 ¬
A056262 ¬
  (71*10n-17)/9
7(8)17 8*(103-1)/9 - (102+1) JCR Oct 14 2002 PRIME View
7(8)37 8*(105-1)/9 - (104+1) JCR Oct 14 2002 PRIME View
7(8)857 8*(1087-1)/9 - (1086+1) JCR Oct 14 2002 PRIME View
7(8)1117 8*(10113-1)/9 - (10112+1) JCR Oct 14 2002 PRIME View
7(8)1697 8*(10171-1)/9 - (10170+1) JCR Oct 14 2002 PRIME View
7(8)5657 8*(10567-1)/9 - (10566+1) PDG Dec 02 2002 PRIME View
7(8)16877 8*(101689-1)/9 - (101688+1) PDG Jul 07 2003 PRIME View
7(8)89017 8*(108903-1)/9 - (108902+1) PDG Jan 03 2003 PROBABLE
PRIME
View
7(8)1158097 8*(10115811-1)/9 - (10115810+1) RC Aug 05 2011 PROBABLE
PRIME
View
A082716 ¬
A056263 ¬
  (72*10n-27)/9 or 8*10n-3   [ n > 200,000 (by RC)]
7(9)17 (103-1) - 2*(102+1) JCR Oct 14 2002 PRIME View
7(9)37 (105-1) - 2*(104+1) JCR Oct 14 2002 PRIME View
7(9)277 (1029-1) - 2*(1028+1) JCR Oct 14 2002 PRIME View
7(9)1557 (10157-1) - 2*(10156+1) JCR Oct 14 2002 PRIME View
7(9)3217 (10323-1) - 2*(10322+1) PDG Dec 02 2002 PRIME View
7(9)3517 (10353-1) - 2*(10352+1) PDG Dec 02 2002 PRIME View
7(9)12117 (101213-1) - 2*(101212+1) PDG Jun 30 2003 PRIME View
7(9)12837 (101285-1) - 2*(101284+1) PDG Jul 03 2003 PRIME View
7(9)79837 (107985-1) - 2*(107984+1) PDG Jan 07 2003 PROBABLE
PRIME
View
7(9)151917 (1015193-1) - 2*(1015192+1) PDG Mar 28 2003 PROBABLE
PRIME
View
7(9)847717 (1084773-1) - 2*(1084772+1) RC Jan 3 2011 PROBABLE
PRIME
View
7(9)1199297 (10119931-1) - 2*(10119930+1) RC Apr 1 2011 PROBABLE
PRIME
View
7(9)1488597 (10148861-1) - 2*(10148860+1) RC Apr 9 2011 PROBABLE
PRIME
View
A082717 ¬
A056264 ¬
  (82*10n+71)/9
9(1)19 (103-1)/9 + 8*(102+1) JCR Oct 15 2002 PRIME View
9(1)2459 (10247-1)/9 + 8*(10246+1) JCR Oct 15 2002 PRIME View
9(1)11399 (101141-1)/9 + 8*(101140+1) DB Jan 23 2003 PRIME View
9(1)103939 (1010395-1)/9 + 8*(1010394+1) PDG Jan 16 2003 PROBABLE
PRIME
View
9(1)438799 (1043881-1)/9 + 8*(1043880+1) PDG Jun 23 2005 PROBABLE
PRIME
View
A082718 ¬
A056265 ¬
  (83*10n+61)/9
9(2)19 2*(103-1)/9 + 7*(102+1) JCR Oct 15 2002 PRIME View
9(2)59 2*(107-1)/9 + 7*(106+1) JCR Oct 15 2002 PRIME View
9(2)119 2*(1013-1)/9 + 7*(1012+1) JCR Oct 15 2002 PRIME View
9(2)1099 2*(10111-1)/9 + 7*(10110+1) JCR Oct 15 2002 PRIME View
9(2)36079 2*(103609-1)/9 + 7*(103608+1) PDG Nov 14 2003 PRIME View
9(2)377839 2*(1037785-1)/9 + 7*(1037784+1) PDG Jun 26 2005 PROBABLE
PRIME
View
A082719 ¬
A056266 ¬
  (89*10n+1)/9   [ n > 700,000 (by SB)]
9(8)59 8*(107-1)/9 + (106+1) JCR Oct 15 2002 PRIME View
9(8)719 8*(1073-1)/9 + (1072+1) JCR Oct 15 2002 PRIME View
9(8)959 8*(1097-1)/9 + (1096+1) JCR Oct 15 2002 PRIME View
9(8)1139 8*(10115-1)/9 + (10114+1) JCR Oct 15 2002 PRIME View
9(8)2039 8*(10205-1)/9 + (10204+1) JCR Oct 15 2002 PRIME View
9(8)9839 8*(10985-1)/9 + (10984+1) PDG Dec 04 2002 PRIME View
9(8)12259 8*(101227-1)/9 + (101226+1) PDG Jul 01 2003 PRIME View
9(8)47939 8*(104795-1)/9 + (104794+1) PDG Dec 04 2002 PRIME View RC
9(8)207199 8*(1020721-1)/9 + (1020720+1) PDG Apr 05 2003 PROBABLE
PRIME
View
9(8)1335799 8*(10133581-1)/9 + (10133580+1) SB May 15 2010 PROBABLE
PRIME
View
9(8)4115899 8*(10411591-1)/9 + (10411590+1) SB Sep 21 2014 RECORD
PROBABLE
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 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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(All rights reserved) - Last modified : November 25, 2014.
Patrick De Geest - Belgium flag - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com