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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
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 :

(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 )

[ 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).

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

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

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.

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