PDP's sorted by length

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

general formula (1)_k x 11 ; ( k >= 2 )

general formula (6)_k7 x 11 ; ( k >= 1 )

general formula (8)_k9 x 11 ; ( k >= 1 )

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

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

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

E.g.: w = 10905 m = 1817 (J. K. Andersen)

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

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...

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 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.

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://stdkmd.net/nrr/

Contributions of factorizations 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.

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