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Plateau and Depression Primes
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101 131141151
161171181191313
323343353373383
717727747757767
787797919929989

Plateau and Depression Primes
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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://homepage2.nifty.com/m_kamada/math/factorizations.htm

Contributions of factorizations are welcome.

Cheers,
m_kamada@nifty.com
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.
The zip file contains a readme.txt detailing the method of proof and
the certificates.

Thanks,
Greg "

PDP Factorization Projects
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Last update : 30 December 2004
( n = w + 1 )

1(0)w1 = 10n+1 The Cunningham Project (search for '10^n+1') Complete up to n = 225
1(3)w1 = (4.10n-7)/3 Factorizations of 133...331 (M. Kamada) Complete up to n = 150
1(4)w1 = (13.10n-31)/9 Factorizations of 144...441 (M. Kamada) Complete up to n = 151
1(5)w1 = (14.10n-41)/9 Factorizations of 155...551 (M. Kamada) Complete up to n = 151
1(6)w1 = (5.10n-17)/3 Factorizations of 166...661 (M. Kamada) Complete up to n = 154
1(7)w1 = (16.10n-61)/9 Factorizations of 177...771 (M. Kamada) Complete up to n = 154
1(8)w1 = (17.10n-71)/9 Factorizations of 188...881 (M. Kamada) Complete up to n = 151
1(9)w1 = 2.10n-9 Factorizations of 199...991 (M. Kamada) Complete up to n = 150
3(1)w3 = (28.10n+17)/9 Factorizations of 311...113 (M. Kamada) Complete up to n = 150
3(2)w3 = (29.10n+7)/9 Factorizations of 322...223 (M. Kamada) Complete up to n = 150
3(4)w3 = (31.10n-13)/9 Factorizations of 344...443 (M. Kamada) Complete up to n = 150
3(5)w3 = (32.10n-23)/9 Factorizations of 355...553 (M. Kamada) Complete up to n = 150
3(7)w3 = (34.10n-43)/9 Factorizations of 377...773 (M. Kamada) Complete up to n = 152
3(8)w3 = (35.10n-53)/9 Factorizations of 388...883 (M. Kamada) Complete up to n = 152
7(1)w7 = (64.10n+53)/9 Factorizations of 711...117 (M. Kamada) Complete up to n = 150
7(2)w7 = (65.10n+43)/9 Factorizations of 722...227 (M. Kamada) Complete up to n = 151
7(3)w7 = 11·(2.10n+1)/3 Factorizations of 733...337 / 11 (M. Kamada) Range from 1 up to 152 complete
7(4)w7 = (67.10n+23)/9 Factorizations of 744...447 (M. Kamada) Complete up to n = 152
7(5)w7 = (68.10n+13)/9 Factorizations of 755...557 (M. Kamada) Complete up to n = 150
7(6)w7 = (23.10n+1)/3 Factorizations of 766...667 (M. Kamada) Complete up to n = 152
7(8)w7 = (71.10n-17)/9 Factorizations of 788...887 (M. Kamada) Complete up to n = 150
7(9)w7 = 8.10n-3 Factorizations of 799...997 (M. Kamada) Complete up to n = 150
9(1)w9 = (82.10n+71)/9 Factorizations of 911...119 (M. Kamada) Complete up to n = 150
9(2)w9 = (83.10n+61)/9 Factorizations of 922...229 (M. Kamada) Complete up to n = 150
9(4)w9 = (85.10n+41)/9 Factorizations of 944...449 (M. Kamada) Complete up to n = 150
9(5)w9 = (86.10n+31)/9 Factorizations of 955...559 (M. Kamada) Complete up to n = 150
9(7)w9 = 11·(8.10n+1)/9 Factorizations of 977...779 / 11 (M. Kamada) Range from 1 up to 151 complete
9(8)w9 = (83.10n+61)/9 Factorizations of 988...889 (M. Kamada) Complete up to n = 150

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
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The reference table for
Plateau and Depression Primes
This collection is complete for
probable primes up to 50001
digits and for proven
primes up to  3609  digits.
PDG = Patrick De Geest
JCR = Jean Claude Rosa
JKA = Jens Kruse Andersen
DB = David Broadhurst
GC = Greg Childers
PDPFormula
blue exp = # of digits		WhoWhenStatusOutput
Logs
 ¬   10n+1   [ n = (# of digits) - 1]   [ n > 2^17 or 131072 ]
1(0)11 0*(103-1)/9 + (102+1)
IMPORTANT NOTE		 JCR Oct 14 2002 PRIME View
A082697 ¬   (12*10n-21)/9 or 4*(10n-1)/3-1    [ n > 50000 ]
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		 In prep.
1(3)179551 (1017957-1)/3 - 2*(1017956+1) PDG Jan 14 2003 PROBABLE
PRIME		 In prep.
1(3)422611 (1042263-1)/3 - 2*(1042262+1) PDG Oct 03 2004 PROBABLE
PRIME		 In prep.
A082698 ¬   (13*10n-31)/9    [ n > 50000 ]
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		 In prep.
A082699 ¬   (14*10n-41)/9    [ n > 50000 ]
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		 In prep.
1(5)376831 5*(1037685-1)/9 - 4*(1037684+1) PDG Oct 11 2004 PROBABLE
PRIME		 In prep.
A082700 ¬   (15*10n-51)/9    [ n > 50000 ]
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		 In prep.
1(6)240271 2*(1024029-1)/3 - 5*(1024028+1) PDG Oct 15 2004 PROBABLE
PRIME		 In prep.
1(6)402211 2*(1040223-1)/3 - 5*(1040222+1) PDG Oct 17 2004 PROBABLE
PRIME		 In prep.
A082701 ¬   (16*10n-61)/9   [ n > 50000 ]
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		 In prep.
1(7)374451 7*(1037447-1)/9 - 6*(1037446+1) PDG Oct 21 2004 PROBABLE
PRIME		 In prep.
A082702 ¬   (17*10n-71)/9   [ n > 50000 ]
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		 In prep.
1(8)147671 8*(1014769-1)/9 - 7*(1014768+1) PDG Feb 06 2003 PROBABLE
PRIME		 In prep.
1(8)192571 8*(1019259-1)/9 - 7*(1019258+1) PDG Feb 07 2003 PROBABLE
PRIME		 In prep.
1(8)312331 8*(1031235-1)/9 - 7*(1031234+1) PDG Nov 17 2004 PROBABLE
PRIME		 In prep.
A082703 ¬   (18*10n-81)/9 or 2*10n-9   [ n > 50000 ]
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		 In prep.
1(9)286291 (1028631-1) - 8*(1028630+1) PDG Nov 26 2004 PROBABLE
PRIME		 In prep.
A082704 ¬   (28*10n+17)/9   [ n > 50000 ]
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		 In prep.
A082705 ¬   (29*10n+7)/9   [ n > 50000 ]
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		 In prep.
A082706 ¬   (31*10n-13)/9   [ n > 51000 ]
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		 In prep.
3(4)247553 4*(1024757-1)/9 - (1024756+1) PDG Apr 17 2005 PROBABLE
PRIME		 In prep.
A082707 ¬   (32*10n-23)/9 or 32*(10n-1)/9+1   [ n > 50000 ]
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		 In prep.
3(5)458593 5*(1045861-1)/9 - 2*(1045860+1) PDG May 16 2005 PROBABLE
PRIME		 In prep.
3(5)473033 5*(1047305-1)/9 - 2*(1047304+1) PDG May 17 2005 RECORD
PROBABLE
PRIME		 In prep.
A082708 ¬   (34*10n-43)/9 or 34*(10n-1)/9-1   [ n > 50000 ]
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 PROBABLE
PRIME		 In prep.
3(7)52073 7*(105209-1)/9 - 4*(105208+1) PDG Nov 28 2002 PROBABLE
PRIME		 In prep.
3(7)107453 7*(1010747-1)/9 - 4*(1010746+1) PDG Dec 20 2002 PROBABLE
PRIME		 In prep.
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		 In prep.
3(7)409573 7*(1040959-1)/9 - 4*(1040958+1) PDG May 20 2005 PROBABLE
PRIME		 In prep.
3(7)458033 7*(1045805-1)/9 - 4*(1045804+1) PDG May 22 2005 PROBABLE
PRIME		 In prep.
3(7)465653 7*(1046567-1)/9 - 4*(1046566+1) PDG May 22 2005 PROBABLE
PRIME		 In prep.
A082709 ¬   (35*10n-53)/9   [ n > 50000 ]
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		 In prep.
3(8)73613 8*(107363-1)/9 - 5*(107362+1) PDG Nov 29 2002 PROBABLE
PRIME		 In prep.
A082710 ¬   (64*10n+53)/9   [ n > 50000 ]
7(1)109057 (1010907-1)/9 + 6*(1010906+1) JKA Oct 17 2002 PROBABLE
PRIME		 In prep.
A082711 ¬   (65*10n+43)/9   [ n > 50000 ]
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		 In prep.
7(2)194617 2*(1019463-1)/9 + 5*(1019462+1) PDG Mar 16 2003 PROBABLE
PRIME		 In prep.
7(2)242137 2*(1024215-1)/9 + 5*(1024214+1) PDG Apr 21 2005 PROBABLE
PRIME		 In prep.
A082712 ¬   (67*10n+23)/9   [ n > 50000 ]
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		 In prep.
7(4)138097 4*(1013811-1)/9 + 3*(1013810+1) PDG Mar 17 2003 PROBABLE
PRIME		 In prep.
7(4)150937 4*(1015095-1)/9 + 3*(1015094+1) PDG Mar 18 2003 PROBABLE
PRIME		 In prep.
A082713 ¬   (68*10n+13)/9   [ n > 50000 ]
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 PROBABLE
PRIME		 In prep.
7(5)39817 5*(103983-1)/9 + 2*(103982+1) PDG Nov 30 2002 PROBABLE
PRIME		 In prep.
7(5)209237 5*(1020925-1)/9 + 2*(1020924+1) PDG Apr 23 2005 PROBABLE
PRIME		 In prep.
7(5)237857 5*(1023787-1)/9 + 2*(1023786+1) PDG Apr 23 2005 PROBABLE
PRIME		 In prep.
7(5)388517 5*(1038853-1)/9 + 2*(1038852+1) PDG May 04 2005 PROBABLE
PRIME		 In prep.
A082714 ¬   (69*10n+3)/9   [ n > 50000 ]
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		 In prep.
7(6)126237 2*(1012625-1)/3 + (1012624+1) PDG Mar 22 2003 PROBABLE
PRIME		 In prep.
7(6)194457 2*(1019447-1)/3 + (1019446+1) PDG Mar 25 2003 PROBABLE
PRIME		 In prep.
7(6)354597 2*(1035461-1)/3 + (1035460+1) PDG Jun 08 2005 PROBABLE
PRIME		 In prep.
A082715 ¬   (71*10n-17)/9   [ n > 50000 ]
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		 In prep.
A082716 ¬   (72*10n-27)/9 or 8*10n-3   [ n > 50000 ]
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