Plateau and Depression Primes

World!Of Numbers		HOME plate WON \|
Plateau and Depression Primes (PDP's)
1 2 3 4 5 6 Undulating Primes Palindromic Wing Primes Palindromic Merlon Primes Home Primes Circular Primes PDP-sorted Factorization of PDP 1(3)1 Factorization of PDP 3(1)3 Factorization of PDP 5(3)5

101		131	141	151
161	171	181	191	313
323	343	353	373	383
717	727	747	757	767
787	797	919	929	989

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 10^x+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 50 remaining Since half december 2023 also Factorization of 533...335 (M. Kamada)
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.
Note : since ca. half december 2023 the file also appears in Kamada's pages ! Factorization of 533...335 (M. Kamada)

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
PDP	Formula Blue exp = # of digits Accolades = prime exp	Who	When	Status	Output 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	JCR	Oct 14 2002	PRIME	View
A082697 ¬ A056244 ¬	(1210ⁿ–21)/9 or (410^n–7)/3 or 4*(10ⁿ–1)/3–1 [ n > 249,551 (by RC)]
1(3)₁1	(10^{3}–1)/3 – 2*(10²+1)	JCR	Oct 14 2002	PRIME	View
1(3)₃1	(10^{5}–1)/3 – 2*(10⁴+1)	JCR	Oct 14 2002	PRIME	View
1(3)₅1	(10^{7}–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	(10^{1471}–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	(10^{17957}–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
1(3)₂₄₉₅₄₉1	(10²⁴⁹⁵⁵¹–1)/3 – 2*(10²⁴⁹⁵⁵⁰+1)	SB	Jan 15 2023	PROBABLE PRIME	View
A082698 ¬ A056245 ¬	(13*10ⁿ–31)/9
1(4)₅1	4(10^{7}–1)/9 – 3(10⁶+1)	JCR	Oct 14 2002	PRIME	View
1(4)₆₅1	4(10^{67}–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(10^{3}–1)/9 – 4(10²+1)	JCR	Oct 14 2002	PRIME	View
1(5)₃1	5(10^{5}–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(10^{401}–1)/9 – 4(10⁴⁰⁰+1)	PDG	Nov 20 2002	PRIME	View
1(5)₅₆₁1	5(10^{563}–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	PRIME	View
1(5)₃₇₆₈₃1	5(10³⁷⁶⁸⁵–1)/9 – 4(10³⁷⁶⁸⁴+1)	PDG	Oct 11 2004	PROBABLE PRIME	View
1(5)₂₁₁₂₆₁1	5(10²¹¹²⁶³–1)/9 – 4(10²¹¹²⁶²+1)	SB	Jan 20 2023	PROBABLE PRIME	View
1(5)₂₂₂₇₁₇1	5(10²²²⁷¹⁹–1)/9 – 4(10²²²⁷¹⁸+1)	SB	Jan 21 2023	PROBABLE PRIME	View
1(5)₂₅₀₃₃₅1	5(10²⁵⁰³³⁷–1)/9 – 4(10²⁵⁰³³⁶+1)	SB	Jan 22 2023	PROBABLE PRIME	View
A082700 ¬ A056247 ¬	(1510ⁿ–51)/9 or (510ⁿ–17)/3 [ n > 200,000 (by RC)]
1(6)₃1	2(10^{5}–1)/3 – 5(10⁴+1)	JCR	Oct 14 2002	PRIME	View
1(6)₁₁1	2(10^{13}–1)/3 – 5(10¹²+1)	JCR	Oct 14 2002	PRIME	View
1(6)₁₅1	2(10^{17}–1)/3 – 5(10¹⁶+1)	JCR	Oct 14 2002	PRIME	View
1(6)₁₇1	2(10^{19}–1)/3 – 5(10¹⁸+1)	JCR	Oct 14 2002	PRIME	View
1(6)₃₅1	2(10^{37}–1)/3 – 5(10³⁶+1)	JCR	Oct 14 2002	PRIME	View
1(6)₅₁1	2(10^{53}–1)/3 – 5(10⁵²+1)	JCR	Oct 14 2002	PRIME	View
1(6)₇₁1	2(10^{73}–1)/3 – 5(10⁷²+1)	JCR	Oct 14 2002	PRIME	View
1(6)₉₉1	2(10^{101}–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	PRIME	View
1(6)₂₄₀₂₇1	2(10^{24029}–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(10^{7}–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(10^{103}–1)/9 – 6(10¹⁰²+1)	JCR	Oct 14 2002	PRIME	View
1(7)₁₉₁1	7(10^{193}–1)/9 – 6(10¹⁹²+1)	JCR	Oct 14 2002	PRIME	View
1(7)₃₆₅1	7(10^{367}–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(10^{37447}–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(10^{3}–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(10^{41}–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(10^{19259}–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 ¬	(1810ⁿ–81)/9 or 210ⁿ–9 [ n > 200,000 (by RC)]
1(9)₁1	(10^{3}–1) – 8*(10²+1)	JCR	Oct 14 2002	PRIME	View
1(9)₃1	(10^{5}–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	(10^{41}–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	(10^{28631}–1) – 8*(10²⁸⁶³⁰+1)	PDG	Nov 26 2004	PROBABLE PRIME	View
A082704 ¬ A056251 ¬	(28*10ⁿ+17)/9
3(1)₁3	(10^{3}–1)/9 + 2*(10²+1)	JCR	Oct 14 2002	PRIME	View
3(1)₁₁3	(10^{13}–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	(10^{31}–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	(10^{127}–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	(10^{601}–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*(10^{7}–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*(10^{3037}–1)/9 + (10³⁰³⁶+1)	PDG	Aug 02 2003	PRIME	View
3(2)₂₁₁₅₅3	2*(10^{21157}–1)/9 + (10²¹¹⁵⁶+1)	PDG	Apr 16 2005	PROBABLE PRIME	View
A082706 ¬ A056253 ¬	(31*10ⁿ–13)/9
3(4)₅3	4*(10^{7}–1)/9 – (10⁶+1)	JCR	Oct 14 2002	PRIME	View
3(4)₁₁3	4*(10^{13}–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*(10^{5569}–1)/9 – (10⁵⁵⁶⁸+1)	PDG	Nov 26 2002	PRIME	View
3(4)₂₄₇₅₅3	4*(10²⁴⁷⁵⁷–1)/9 – (10²⁴⁷⁵⁶+1)	PDG	Apr 17 2005	PROBABLE PRIME	View
A082707 ¬ A056254 ¬	(3210ⁿ–23)/9 or 32(10ⁿ–1)/9+1
3(5)₁3	5(10^{3}–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(10^{463}–1)/9 – 2(10⁴⁶²+1)	PDG	Nov 27 2002	PRIME	View
3(5)₇₂₅3	5(10^{727}–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	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
3(5)₂₈₃₈₂₅3	5(10²⁸³⁸²⁷–1)/9 – 2(10²⁸³⁸²⁶+1)	SB	Jan 17 2023	PROBABLE PRIME	View
A082708 ¬ A056255 ¬	(3410ⁿ–43)/9 or 34(10ⁿ–1)/9–1
3(7)₁3	7(10^{3}–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(10^{157}–1)/9 – 4(10¹⁵⁶+1)	JCR	Oct 14 2002	PRIME	View
3(7)₂₇₆₅3	7(10^{2767}–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(10^{3877}–1)/9 – 4(10³⁸⁷⁶+1)	PDG	Nov 28 2002	PRIME	View RC
3(7)₅₂₀₇3	7(10^{5209}–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(10^{46567}–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(10^{3}–1)/9 – 5(10²+1)	JCR	Oct 14 2002	PRIME	View
3(8)₁₁3	8(10^{13}–1)/9 – 5(10¹²+1)	JCR	Oct 14 2002	PRIME	View
3(8)₂₉3	8(10^{31}–1)/9 – 5(10³⁰+1)	JCR	Oct 14 2002	PRIME	View
3(8)₅₉3	8(10^{61}–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	PRIME	View
3(8)₇₃₆₁3	8(10⁷³⁶³–1)/9 – 5(10⁷³⁶²+1)	PDG	Nov 29 2002	PRIME	View
A082710 ¬ ¬	(64*10ⁿ+53)/9 [ n > 1,200,000 (by SB)]
7(1)₁₀₉₀₅7	(10¹⁰⁹⁰⁷–1)/9 + 6*(10¹⁰⁹⁰⁶+1)	JKA	Oct 17 2002	PRIME	View
7(1)₄₉₉₂₀₉7	(10⁴⁹⁹²¹¹–1)/9 + 6*(10⁴⁹⁹²¹⁰+1)	SB	Mar 01 2015	PROBABLE PRIME	View
A082711 ¬ A056257 ¬	(65*10ⁿ+43)/9
7(2)₁7	2(10^{3}–1)/9 + 5(10²+1)	JCR	Oct 14 2002	PRIME	View
7(2)₃7	2(10^{5}–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(10^{29}–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(10^{1787}–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	PRIME	View
7(2)₁₉₄₆₁7	2(10^{19463}–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
7(2)₁₃₁₃₉₁7	2(10¹³¹³⁹³–1)/9 + 5(10¹³¹³⁹²+1)	TB	Jan 11 2023	PROBABLE PRIME	View
A082712 ¬ A056258 ¬	(67*10ⁿ+23)/9
7(4)₉7	4(10^{11}–1)/9 + 3(10¹⁰+1)	JCR	Oct 14 2002	PRIME	View
7(4)₂₉7	4(10^{31}–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(10^{1487}–1)/9 + 3(10¹⁴⁸⁶+1)	PDG	Jul 05 2003	PRIME	View
7(4)₁₅₇₇7	4(10^{1579}–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
7(4)₂₀₇₅₅₅7	4(10^{207557}–1)/9 + 3(10²⁰⁷⁵⁵⁶+1)	SB	Jan 13 2023	PROBABLE PRIME	View
7(4)_11166757	4(10^{1116677}–1)/9 + 3(10^1116676+1)	RPSB	Jan 22 2023	RECORD PROBABLE PRIME	View
A082713 ¬ A056259 ¬	(68*10ⁿ+13)/9
7(5)₁7	5(10^{3}–1)/9 + 2(10²+1)	JCR	Oct 14 2002	PRIME	View
7(5)₃7	5(10^{5}–1)/9 + 2(10⁴+1)	JCR	Oct 14 2002	PRIME	View
7(5)₉7	5(10^{11}–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(10^{23}–1)/9 + 2(10²²+1)	JCR	Oct 14 2002	PRIME	View
7(5)₅₇7	5(10^{59}–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(10^{83}–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
7(5)₂₀₅₅₀₉7	5(10²⁰⁵⁵¹¹–1)/9 + 2(10²⁰⁵⁵¹⁰+1)	SB	Jan 13 2023	PROBABLE PRIME	View
A082714 ¬ A056260 ¬	(6910ⁿ+3)/9 or (2310ⁿ+1)/3 [ n > 700,000 (by RC)]
7(6)₃7	2*(10^{5}–1)/3 + (10⁴+1)	JCR	Oct 14 2002	PRIME	View
7(6)₅7	2*(10^{7}–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*(10^{97}–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*(10^{19447}–1)/3 + (10¹⁹⁴⁴⁶+1)	PDG	Mar 25 2003	PROBABLE PRIME	View
7(6)₃₅₄₅₉7	2*(10^{35461}–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*(10^{95327}–1)/3 + (10⁹⁵³²⁶+1)	SB	May 27 2009	PROBABLE PRIME	View
A082715 ¬ A056262 ¬	(71*10ⁿ–17)/9
7(8)₁7	8*(10^{3}–1)/9 – (10²+1)	JCR	Oct 14 2002	PRIME	View
7(8)₃7	8*(10^{5}–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*(10^{113}–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*(10^{115811}–1)/9 – (10¹¹⁵⁸¹⁰+1)	RC	Aug 05 2011	PROBABLE PRIME	View
7(8)₁₆₅₇₁₅7	8*(10¹⁶⁵⁷¹⁷–1)/9 – (10¹⁶⁵⁷¹⁶+1)	SB	Jan 19 2023	PROBABLE PRIME	View
A082716 ¬ A056263 ¬	(7210ⁿ–27)/9 or 810ⁿ–3 [ n > 219,740 (by RC)]
7(9)₁7	(10^{3}–1) – 2*(10²+1)	JCR	Oct 14 2002	PRIME	View
7(9)₃7	(10^{5}–1) – 2*(10⁴+1)	JCR	Oct 14 2002	PRIME	View
7(9)₂₇7	(10^{29}–1) – 2*(10²⁸+1)	JCR	Oct 14 2002	PRIME	View
7(9)₁₅₅7	(10^{157}–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	(10^{353}–1) – 2*(10³⁵²+1)	PDG	Dec 02 2002	PRIME	View
7(9)₁₂₁₁7	(10^{1213}–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	(10^{15193}–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	(10^{148861}–1) – 2*(10¹⁴⁸⁸⁶⁰+1)	RC	Apr 9 2011	PROBABLE PRIME	View
7(9)₂₁₉₇₃₉7	(10²¹⁹⁷⁴¹–1) – 2*(10²¹⁹⁷⁴⁰+1)	SB	Jan 14 2023	PROBABLE PRIME	View
A082717 ¬ A056264 ¬	(82*10ⁿ+71)/9
9(1)₁9	(10^{3}–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(10^{3}–1)/9 + 7(10²+1)	JCR	Oct 15 2002	PRIME	View
9(2)₅9	2(10^{7}–1)/9 + 7(10⁶+1)	JCR	Oct 15 2002	PRIME	View
9(2)₁₁9	2(10^{13}–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
9(2)₁₈₁₅₄₃9	2(10¹⁸¹⁵⁴⁵–1)/9 + 7(10¹⁸¹⁵⁴⁴+1)	SB	Jan 19 2023	PROBABLE PRIME	View
A082719 ¬ A056266 ¬	(89*10ⁿ+1)/9 [ n > 700,000 (by SB)]
9(8)₅9	8*(10^{7}–1)/9 + (10⁶+1)	JCR	Oct 15 2002	PRIME	View
9(8)₇₁9	8*(10^{73}–1)/9 + (10⁷²+1)	JCR	Oct 15 2002	PRIME	View
9(8)₉₅9	8*(10^{97}–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	PROBABLE 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.

Form	prime 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

10001 depression composite

13331

16661

19991

50005 depression composite

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

[ TOP OF PAGE]

Patrick De Geest - Belgium

- Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com

101		131	141	151
161	171	181	191	313
323	343	353	373	383
717	727	747	757	767
787	797	919	929	989

101		131	141	151
161	171	181	191	313
323	343	353	373	383
717	727	747	757	767
787	797	919	929	989

101		131	141	151
161	171	181	191	313
323	343	353	373	383
717	727	747	757	767
787	797	919	929	989