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The record Palindromic Primes

flash [ November 16, 2014 ]

The newest palindromic prime record is from Serge Batalov
10474500 + 999 * 10237249 + 1
(474501 digits)

Also found prior to above record were the following palprimes
10390636 + 999 * 10195317 + 1 (390637 digits)
10362600 + 666 * 10181299 + 1 (362601 digits)



Source (Primeform, NMBRTHRY, ...)
From The Largest Known Primes!
Top Twenty Record Palindromic Primes


[ June 4, 2014 ]

The newest palindromic prime record is from David Broadhurst
Phi(3, 10160118) +
(137 * 10160119 + 731 * 10159275) * (10843 - 1)/999

(320237 digits)
Written out it looks like this :
1_(0)159274_(137)281_1_(731)281_(0)159274_1



Source (Primeform, NMBRTHRY, ...)
From The Largest Known Primes!
Top Twenty Record Palindromic Primes
Message 11475 Big Palindrome


[ January 8, 2013 ]

The newest palindromic prime record is again from Darren Bedwell
10314727 - 8 * 10157363 - 1
(314727 digits)
Written out, it's 157363 nines, a one, and 157363 more nines
and is therefore a PWP or a Palindromic Wing Prime.



Source (Primeform, NMBRTHRY, ...)
Top Twenty Record Palindromic Primes
FROM The Largest Known Primes!
http://zerosink.blogspot.com


[ April 11, 2012 ]

The newest palindromic prime record is from Darren Bedwell
10290253 - 2 * 10145126 - 1
(290253 digits)
Written out, it's 145126 nines, a seven, and 145126 more nines
and is therefore a PWP or a Palindromic Wing Prime.



Source (Primeform, NMBRTHRY, ...)
Top Twenty Record Palindromic Primes
FROM The Largest Known Primes!
http://zerosink.blogspot.com


[ January 2012 ]

The newest palindromic prime record is from David Broadhurst
who devised the next beautiful formula
Phi(3, 10137747) +
(137 * 10137748 + 731 * 10129293) * (108454 - 1)/999

(275495 digits)
Written in concatenated form we get
1_(0)129292_(137)2818_1_(731)2818_(0)129292_1
One can verify this with the following 64-bit PFGW commandline
C:\Pfgw64>pfgw64 -f0 -od -q"formula" >> palprim.txt



Source (Primeform, NMBRTHRY, ...)
Top Twenty Record Palindromic Primes
FROM The Largest Known Primes!
http://tech.groups.yahoo.com/group/primeform/message/11203


[ May & September, 2010 ]

After the following two records from Harvey Dubner made in May
10185008 + 130525031 * 1092500 + 1 (185009 digits)
10190004 + 214757412 * 1094998 + 1 (190005 digits)

came Bernardo Boncompagni in September with his latest world palindromic prime record
10200000 + 47960506974 * 1099995 + 1
Congratulations with this milestone of 200001 digits !



Source (Primeform, NMBRTHRY, ...)
Top Twenty Record Palindromic Primes
FROM The Largest Known Primes!


[ September, 2007 ]

Darren Bedwell re-establishes a new
world palindromic prime record
10180054 + 8*R(58567) * 1060744 + 1
Congratulations with this number of 180055 digits !

It is also tetradic, dihedral and strobogrammatic. Darren
discovered this particular number using PFGW with
the -f flag and a simple ABC2 file, only varying
the length of the center string of 8's.

Source (Primeform, NMBRTHRY, ...)
Top Twenty Record Palindromic Primes
FROM The Largest Known Primes!


[ August, 2007 ]

Harvey Dubner re-establishes
a palindromic prime record
10180004 + 248797842 * 1089998 + 1
Congratulations with this number of 180005 digits !

Source (Primeform, NMBRTHRY, ...)
Top Twenty Record Palindromic Primes
http://tech.groups.yahoo.com/group/primeform/message/9261
THE LARGEST KNOWN PRIMES


[ June, 2007 ]

Harvey Dubner re-establishes
a palindromic prime record
10175108 + 230767032 * 1087550 + 1
Congratulations with this number of 175109 digits !

Source (Primeform, NMBRTHRY, ...)
Top Twenty Record Palindromic Primes
http://tech.groups.yahoo.com/group/primeform/message/8673
THE LARGEST KNOWN PRIMES


[ October, 2006 ]

Harvey Dubner re-establishes
a palindromic prime record
10170006 + 3880883 * 1085000 + 1
Congratulations with this number of 170007 digits !

Source (Primeform, NMBRTHRY, ...)
Top Twenty Record Palindromic Primes
THE LARGEST KNOWN PRIMES


[ June 2, 2006 ]

Harvey Dubner re-establishes
a palindromic prime record
10160016 + 8231328 * 1080005 + 1
Congratulations with this number of 160017 digits !

Source (Primeform, NMBRTHRY, ...)
Top Twenty Record Palindromic Primes
Number Theory List
THE LARGEST KNOWN PRIMES


[ February 2, 2006 ]

Harvey Dubner retakes
the palindromic prime record
10150008 + 4798974 * 1075001 + 1
Congratulations with this number of 150009 digits !

Source (Primeform, NMBRTHRY, ...)
Top Twenty Record Palindromic Primes
http://groups.yahoo.com/group/primeform/message/6888 (last )
THE LARGEST KNOWN PRIMES


[ December 26, 2005 ]

Paul Jobling tops the list now
with his palindromic prime record
10150006 + 7426247 * 1075000 + 1
Congratulations with this number of 150007 digits !

Source (Primeform, NMBRTHRY, ...)
Top Twenty Record Palindromic Primes
http://primes.utm.edu/primes/page.php?id=76550
http://groups.yahoo.com/group/primeform/message/6757
http://groups.yahoo.com/group/primeform/message/6764
http://groups.yahoo.com/group/primeform/message/6765
http://groups.yahoo.com/group/primeform/message/6766
http://groups.yahoo.com/group/primeform/message/6767
http://groups.yahoo.com/group/primeform/message/6768
http://groups.yahoo.com/group/primeform/message/6769
THE LARGEST KNOWN PRIMES


Former Palindromic Prime Records computed by Harvey Dubner & ass.



Harvey Dubner( picture borrowed from Ivars Peterson webpage Palindromic Primes )


[ December 7, 2005 ]

Harvey Dubner re-establishes
a personal palindromic prime record
10140008 + 4546454 * 1070001 + 1
Congratulations with this number of 140009 digits !

Source (Primeform, NMBRTHRY, ...)
Top Twenty Record Palindromic Primes
http://groups.yahoo.com/group/primeform/message/6719
THE LARGEST KNOWN PRIMES


[ November 7, 2004 ]

Harvey Dubner establishes again
a new palindromic prime record
10130022 + 3761673 * 1065008 + 1
Congratulations with this number of 130023 digits !

Source (Primeform, NMBRTHRY, ...)
Number Theory List
http://groups.yahoo.com/group/primeform/message/4999
THE LARGEST KNOWN PRIMES


[ May 4, 2004 ]

The largest palindromic prime with
a palindromic prime length is
1098689 – 429151924 * 1049340 – 1
a palprime with a length of 98689 digits!
A doubly palindromic prime.

Sources
Number Theory List
http://groups.yahoo.com/group/primeform/message/4320
Congrats to Jens (Kruse Andersen) and Harvey (Dubner) for

10^98689-429151924*10^49340-1 98689 p138 04 Palindrome 

It will be a long time before anyone finds a larger
base-10 palindromic prime with a number of digits
that is also a palindromic prime. This is the largest
such prime less than 10^(10^6).

David Broadhurst


[ April 5, 2004 ]

Let us acclaim Harvey Dubner's
latest palindromic prime record
10120016 + 1726271 * 1060005 + 1
Congratulations with this number of 120017 digits !

Source (Primeform, NMBRTHRY, ...)
Number Theory List
http://groups.yahoo.com/group/primeform/message/4273
THE LARGEST KNOWN PRIMES


[ April 3, 2004 ]

Let us acclaim Harvey Dubner's
latest palindromic prime record
10120002 + 1617161 * 1059998 + 1
Congratulations with this number of 120003 digits !

Source from Primeform List Record palindrome


[ October 5, 2001 ]
David Broadhurst and Harvey Dubner announced via the
NMBRTHRY bulletin board a new  palindromic prime 
N = (1989191989)15601 containing 15601 digits.
This is not a new record however this palindrome is of
a more varied form and hence considerably
more demanding to prove !
We have found a Konyagin-Pomerance proof of primality of
the 15601-digit base-10 palindrome N = (1989191989)15601

1) Larger palindromic primes have been proven by applying
Brillhart-Lehmer-Selfridge (BLS) tests to numbers such as
(9)230348(9)23034 and
1(0)195094538354(0)195091,
but this method appears to be limited to base-10
strings consisting almost entirely of 9's or 0's.

2) It is considerably more demanding to prove gigantic
palindromic primes of a more varied form. The previous
record for a Konyagin-Pomerance (KP) proof was set by
(1579393975)13861, recently proven by collaboration
of members of the PrimeNumbers e-group.

3) For several years one of us (HD) has kept a database
of prime factors of primitive parts of 10^n +/- 1. Until
very recently this recorded that only 16.36% of the digits
of 10^15600-1 had been factorized into proven primes.

4) We noted that the 1914-digit probable prime
prp1914 = Phi5200(10)/5990401
was now easy to prove, thanks to Marcel Martin's Primo.
In the event, it took less than a day on a 1GHz machine.
That left us 214 digits short of the 30% threshold, required
by KP. This gap was made up by renewed P-1 and ECM efforts.

5) While not sufficient, the BLS tests are necessary.
There were done, with great efficiency, by OpenPfgw.

6) Pari-GP was used for the final cubic test.
All elements of the proof were checked by Greg Childers.

7) Now that one has 30% of the digits of 10^15600-1, it is
straightforward to generate and prove further palindromic
primes between 10^15600 and 2*10^15600. For example
(1854050458)15601, (1844454448)15601,
(1413323314)15601, (1120373021)15601,
and 10787026001 were proven by the same method. However,
to progress to significantly larger palindromes of such
a varied form, more factorization effort would be required.

We thank our colleagues in the PrimeNumbers, PrimeForm
and OpenPfgw e-groups, and in particular Greg Childers,
Jim Fougeron, Marcel Martin and Chris Nash.

David Broadhurst and Harvey Dubner 


[ July 18, 2001 ]
Harvey Dubner announced via the NMBRTHRY bulletin board
a new  palindromic prime  record.
1039026 + 4538354 * 1019510 + 1 containing 39027 digits.
Despite having a length that is not a prime nor a palindrome
this time, it still is quite an achievement. Well done, Harvey !


The site "The largest known primes" [http://www.utm.edu/research/primes/ftp/short.txt]
kept monthly updates also about palindromic primes.
The third largest one appearing in the list is 10^19390+4300034*10^9692+1 and is 19391 digits long !
Note that the length of this giant is palindromic too (no coincidence!).
The second largest one appearing in the list is 10^30802 + 1110111*10^15398 + 1 and is 30803 digits long !
It consists of only 1's and 0's (definitely an accident).
The estimated search time was 233 days of Pentium/200 equivalent.
It was found after only about 12 computer-days or 5% of the estimate.
"With this kind of luck maybe I should start playing the lottery" the author says
who is none other than Harvey Dubner and who already published various articles
about palindromic primes in the Journal of Recreational Mathematics.

The largest one in the list dating from [ November 14, 1999 ] is 10^35352+2049402*10^17673+1 and is 35353 digits long !
The number of digits, 35353, is a nice undulating palindromic prime (no accident).
I believe it is the largest known prime that is NOT of the form, a*b^n +/- 1.
The estimated search time was 121 days for a Pentium/400 equivalent.
Actual time was about 1/3 of this. It was found by my wife's P/166 computer
(running in the background), the slowest computer that was being used.

Most of Harvey's record palindromic primes are on display.
indicator The Top Ten by Rudolph Ondrejka
Be warned that your mind might boggle when trying to grasp the magnitude of the numbers involved.

Warut Roonguthai provided me the following link that displays the largest palindromic primes.
The current list of the largest known palindromic primes
indicator http://primes.utm.edu/top20/page.php?id=53


[ January 29, 2000 ] Not afraid to search for larger palprimes himself Warut Roonguthai
used PrimeForm (broken link) as a tool and this soon turned out to be successful, though he didn't become
record-holder yet...

I've just found the palindromic prime 10^11840+42924*10^5918+1 with
PrimeForm. To search, I used the expression:

(k*1000+100*(k/10)-990*(k/100)-99*(k/1000))*10^(n-3)+1+100^n

It was arranged in such a way that the values of k and n could be seen in
the log file. The formula
k*1000+100*(k/10)-990*(k/100)-99*(k/1000)
was used to generate the palindromic middle term from k; k could be any
integer from 1 to 9999. If the decimal expansion of k is abcd, then the
formula will produce the number abcdcba. Note that with PrimeForm, the
value of (x/y) is the greatest integer that is equal to or less than the
fraction x/y, i.e. (x/y) = [x/y].


Palprimes in 'Arithmetic Progression'

Albert H. Beiler wrote in his book "Recreations in the Theory of Numbers" Second Edition 1966 page 222 :
Primes can even be palindromic (reading the same backward as forward) and be in arithmetic progression.
Four such sets of four primes are:
(13931, 14741, 15551, 16361), (10301, 13331, 16361, 19391),
(70607, 73637, 76667, 79697), (94049, 94349, 94649, 94949).
The respective common differences are 810; 3030; 3030 and 300.

Warut Roonguthai
calculated the 5 smallest arithmetic progressions of 3, 4, 5, 6 and 7 palindromic primes
and even found one that was not mentioned by Beiler (case of 4 palprimes in AP).


Harvey Dubner sent me solutions for 5, 6, 8, 9 and 10 palprimes in arithmetic progression.
With this great contribution Harvey consolidates his position as the number one master in palindromic primes.

5
11492200000229411
11492210101229411
11492220202229411
11492230303229411
11492240404229411
The common difference is 10101000000.
Harvey can now find such fivefold palprimes at the rate of about one set every minute.
It is a good challenge problem to find a smaller set!
On [ June 21, 1999 ] Warut Roonguthai from Bangkok sent in the 5 smallest ones !!

6
1931602100012061391
1931602111112061391
1931602122212061391
1931602133312061391
1931602144412061391
1931602155512061391
The common difference is 11100000000.
Harvey thinks this is the only known set of 6 palindromic primes in AP.

8
14251705901010950715241
14251705911111950715241
14251705921212950715241
14251705931313950715241
14251705941414950715241
14251705951515950715241
14251705961616950715241
14251705971717950715241
The common difference is 10101000000000.

9
159056556140202041655650951
159056556141212141655650951
159056556142222241655650951
159056556143232341655650951
159056556144242441655650951
159056556145252541655650951
159056556146262641655650951
159056556147272741655650951
159056556148282841655650951
Date of submission : [ April 3, 1999 ].
The common difference is 1010100000000000.

10
742950290870000078092059247
742950290871010178092059247
742950290872020278092059247
742950290873030378092059247
742950290874040478092059247
742950290875050578092059247
742950290876060678092059247
742950290877070778092059247
742950290878080878092059247
742950290879090978092059247
Found [ April 23, 1999 ].
Common difference = 1010100000000000.
Primes were verified using the APRT-CLE program in UBASIC.
Ten primes in AP was the last series to look for as there doesn't appear to be any way of getting 11
or more, the expected search time was about two computer-years (Pentium/200 equivalent).
We were lucky and found the 10 palprimes in about 1/5 of the expected time.
'We' are the following persons that formed a search team in order to achieve the above result :
Harvey Dubner, Manfred Toplic [HomePage], Tony Forbes, Jonathan Johnson, Brian Schroeder
and Carlos Rivera.
About 20 PC's were used. Congratulations to all!


Palindromic 'Sophie Germain' Primes

More extensive info was available in Warut Roonguthai's (discontinued) site about
Palindromic Sophie Germain Primes
and Yves Gallot's Proth.exe and Cunningham Chains

On [ April 21, 1999 ] I received the following email from Harvey Dubner.

"I cannot resist looking for palindromic primes that have other interesting properties.
Naturally, I have to share the results with somebody, so here they are.

P, Q, R are palindromic primes. Q = 2P+1, R = 2Q+1.
Thus P, Q are Sophie Germain primes.

P, Q, R is also a Cunningham chain of Palindromic primes.
There cannot be a fourth palindromic member of the chain.

Smallest possible set - from J. Recreational Math., Vol.26(1), 1994,
"Palindromic Sophie Germain primes" by Harvey Dubner

23 digits

P = 19091918181818181919091
Q = 38183836363636363838183
R = 76367672727272727676367

Largest known set, recently found March 22, 1999
Also, number of digits is a palindromic prime.
Also, P is a tetradic prime (4-way prime)

P, Q, R all have 727 digits, the number of digits is a palindromic prime.

P =
1818181808080818080808180818081818081808181818081808080818180808080808_
0808080808080808080808080808080808080808080808080808080808080808080808_
0808080808080808080808080808080808080808080808080808080808080808080808_
0808080808080808080808080808080808080808080808080808080808080808080808_
0808080808080808080808080808080808080808080808080808080808080808080808_
0808080808080808080808080808080808080808080808080808080808080808080808_
0808080808080808080808080808080808080808080808080808080808080808080808_
0808080808080808080808080808080808080808080808080808080808080808080808_
0808080808080808080808080808080808080808080808080808080808080808080808_
0808080808080808080808080808080808080818180808081808181818081808181808_
180818080808180808081818181 

Q =
3636363616161636161616361636163636163616363636163616161636361616161616_
1616161616161616161616161616161616161616161616161616161616161616161616_
1616161616161616161616161616161616161616161616161616161616161616161616_
1616161616161616161616161616161616161616161616161616161616161616161616_
1616161616161616161616161616161616161616161616161616161616161616161616_
1616161616161616161616161616161616161616161616161616161616161616161616_
1616161616161616161616161616161616161616161616161616161616161616161616_
1616161616161616161616161616161616161616161616161616161616161616161616_
1616161616161616161616161616161616161616161616161616161616161616161616_
1616161616161616161616161616161616161636361616163616363636163616363616_
361636161616361616163636363 

R =
7272727232323272323232723272327272327232727272327232323272723232323232_
3232323232323232323232323232323232323232323232323232323232323232323232_
3232323232323232323232323232323232323232323232323232323232323232323232_
3232323232323232323232323232323232323232323232323232323232323232323232_
3232323232323232323232323232323232323232323232323232323232323232323232_
3232323232323232323232323232323232323232323232323232323232323232323232_
3232323232323232323232323232323232323232323232323232323232323232323232_
3232323232323232323232323232323232323232323232323232323232323232323232_
3232323232323232323232323232323232323232323232323232323232323232323232_
3232323232323232323232323232323232323272723232327232727272327232727232_
723272323232723232327272727 
Sorry, there is no shorthand way of expressing these palprimes.

Harvey Dubner"


Smallest Palprimes in 'Arithmetic Progression'
by Warut Roonguthai [ June 21-24, 1999 ]

Here are the 5 smallest arithmetic progressions of 3, 4, 5, 6 and 7
palindromic primes (ranked by the size of the last term):
Between brackets are the 'common difference' values.

3
3, 5, 7 (2)
3, 7, 11 (4)
11, 101, 191 (90)
727, 757, 787 (30)
10501, 12421, 14341 (1920)

4
13931, 14741, 15551, 16361 (810)
10301, 13331, 16361, 19391 (3030)
73637, 75557, 77477, 79397 (1920) - not mentioned by Beiler
70607, 73637, 76667, 79697 (3030)
94049, 94349, 94649, 94949 (300)

5
1150511, 1262621, 1374731, 1486841, 1598951 (112110)
1114111, 1335331, 1556551, 1777771, 1998991 (221220)
7190917, 7291927, 7392937, 7493947, 7594957 (101010)
9185819, 9384839, 9583859, 9782879, 9981899 (199020)
9585859, 9686869, 9787879, 9888889, 9989899 (101010)

6
10696969601, 11686968611, 12676967621, 13666966631, 14656965641, 15646964651 (989999010)
12374047321, 13364046331, 14354045341, 15344044351, 16334043361, 17324042371 (989999010)
13308880331, 14417771441, 15526662551, 16635553661, 17744444771, 18853335881 (1108891110)
19125452191, 19135553191, 19145654191, 19155755191, 19165856191, 19175957191 (10101000)
14282128241, 15272127251, 16262126261, 17252125271, 18242124281, 19232123291 (989999010)

7 - Three example of 7 palindromic primes in AP.
Warut said it's very likely that the first two are the smallest ones.

1251700071521
1352710172531
1453720273541
1554730374551
1655740475561
1756750576571
1857760677581
The common difference is 101010101010

7056994996507
7156984896517
7256974796527
7356964696537
7456954596547
7556944496557
7656934396567
The common difference is 99989900010

349263515362943
349264525462943
349265535562943
349266545662943
349267555762943
349268565862943
349269575962943
The common difference is 1010100000

Sent in on [ June 22-24, 1999 ]. Thanks Warut! Very nice work.


Some Straightforward 'Plateau' Palprimes
by Nicholas Angelou [ July 8, 2000 ]

Below are some prime numbers that are the form of 1kkk...k1
starting with 1 and ending with 1 and ALL the in_between digits
are the SAME digit 2 to 9 in repetition)

18...81  9 digits
19...91  9 digits
16...61 13 digits
18...81 15 digits
16...61 17 digits
16...61 19 digits
15...51 21 digits *
15...51 33 digits *
16...61 37 digits *
17...71 49 digits *
16...61 73 digits *
* N-1 ,Brillhart-Lehmer-Selfridge proving algorithm

For the full listing of these kind of primes (PDP's) see Plateau and Depression Primes






Sources

Read also this interesting article about Palindromic Primes by Ivars Peterson (email).






Contributions

Harvey Dubner (email) for his indefatigable computing work on palindromic primes.
go to topic on AP's - go to topic on Palprime Records

Warut Roonguthai (email) for his work on the smallest AP of palindromic primes.
go to topic

Nicholas Angelou (email) from Australia for his batch of straightforward plateau palprimes.
go to topic









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E-mail address : pdg@worldofnumbers.com