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The record Palindromic Primes


flash [ Submitted October 18, 2021 ]

The newest palindromic prime record is from Propper and Batalov
101888529 – 10944264 – 1
(1888529 digits)


Written out, it's 944264 nines, an eight, and 944264 more nines
and is therefore a PWP or a Palindromic Wing Prime.


Source (The PrimePages, NMBRTHRY, ...)
http://primes.utm.edu/primes/page.php?id=132851
Top Twenty Record Palindromic Primes


[ Submitted September 15, 2021 ]

This palindromic prime is from Propper and Batalov
101234567 – 20342924302 * 10617278 – 1
(1234567 digits)


Source (The PrimePages, NMBRTHRY, ...)
https://primes.utm.edu/primes/page.php?id=132715


[ Submitted September 29, 2021 ]

This palindromic prime is from Propper and Batalov
101234567 – 3626840486263 * 10617277 – 1
(1234567 digits)


Source (The PrimePages, NMBRTHRY, ...)
https://primes.utm.edu/primes/page.php?id=132766


[ Submitted September 29, 2021 ]

This palindromic prime is from Propper and Batalov
101234567 – 4708229228074 * 10617277 – 1
(1234567 digits)


Source (The PrimePages, NMBRTHRY, ...)
https://primes.utm.edu/primes/page.php?id=132767


[ Submitted August 6, 2021 ]

This palindromic prime is from Makoto Morimoto
10490000 + 3 * (107383 – 1)/9 * 10241309 + 1
(490001 digits)


Source (The PrimePages, NMBRTHRY, ...)
http://primes.utm.edu/primes/page.php?id=132591
Nieuw (voorlopig) grootste palindroom-priemgetal ontdekt
https://www.mersenneforum.org/showthread.php?p=579376


[ Submitted November 16, 2014 ]

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


Source (The PrimePages, NMBRTHRY, ...)
http://primes.utm.edu/primes/page.php?id=118775
http://groups.yahoo.com/group/primeform/message/11538


[ Submitted July 27, 2021 ]

This palindromic prime is from Makoto Morimoto
10400000 + 4 * (10102381 – 1)/9 * 10148810 + 1
(400001 digits)


Source (The PrimePages, NMBRTHRY, ...)
https://primes.utm.edu/primes/page.php?id=132557


[ Submitted November 16, 2014 ]

This palindromic prime is from Serge Batalov
10390636 + 999 * 10195317 + 1
(390637 digits)


Source (The PrimePages, NMBRTHRY, ...)
http://primes.utm.edu/primes/page.php?id=118773
http://groups.yahoo.com/group/primeform/message/11536


[ Submitted November 15, 2014 ]

This palindromic prime is from Serge Batalov
10362600 + 666 * 10181299 + 1
(362601 digits)


Source (The PrimePages, NMBRTHRY, ...)
http://primes.utm.edu/primes/page.php?id=118770
http://groups.yahoo.com/group/primeform/message/11535


[ Submitted March 5, 2014 ]

This palindromic prime 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 (The PrimePages, NMBRTHRY, ...)
http://primes.utm.edu/primes/page.php?id=117373
Message 11475 Big Palindrome + Reply Message 11476

The function Phi used here in the first part stands for Pfgw's 'Cyclotomic Number'
It provides us the left, the middle and the right digit 1 in the decimal expansion.
Pfgw64 -f0 -od -q"Phi(3,10^2)" = 10101
Pfgw64 -f0 -od -q"Phi(3,10^3)" = 1001001
Pfgw64 -f0 -od -q"Phi(3,10^4)" = 100010001
Pfgw64 -f0 -od -q"Phi(3,10^160118)" = 100.. ..00100.. ..001
Change the first parameter into e.g. 5 and you get
Pfgw64 -f0 -od -q"Phi(5,10^4)" = 10001000100010001

This cyclotomic number seems to be related to Euler's phi (totient) function ?
Pfgw64 Phi(10^4): 4000
Pari/gp (16:59) gp > eulerphi(10^4) %1 = 4000

What is the role of the first parameter in pfgw64's Phi ?


[ Submitted March 8, 2014 ]

This palindromic prime is from David Broadhurst
Phi(3, 10160048) +
(137 * 10160049 + 731 * 10157453) * (102595 – 1)/999

(320097 digits)
Written out it looks like this :
1_(0)157452_(137)865_1_(731)865_(0)157452_1


Source (The PrimePages, NMBRTHRY, ...)
http://primes.utm.edu/primes/page.php?id=117386

Analyse of the second part of the equation

It provides us the left and the right string of digits around the central digit 1 in the expansion.
Pfgw64 -f0 -od -q"(137*10^11+731*10^7)*(10^3-1)/999" 13707310000000
Pfgw64 -f0 -od -q"(137*10^15+731*10^8)*(10^6-1)/999" 137137073173100000000
Notice the zero between the 137 and 731 string where the middle digit '1' of the Phi function will be added to.

General formula is (137*10^K+731*10^L)*(10^A-1)/999
(10^A-1)/999 is the repeater; with A multiple of 3 (137 and 731 are each three digits long).
L is the number of trailing zeros you want implemented (The last zero will change to a '1' when the right digit '1' of the Phi function is added).
K = L + A + 1


[ Submitted January 7, 2013 ]

This palindromic prime is 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 (The PrimePages, NMBRTHRY, ...)
https://primes.utm.edu/primes/page.php?id=110658
http://zerosink.blogspot.com/2013/01/
http://groups.yahoo.com/group/primeform/message/11343


[ Submitted June 17, 2021 ]

This palindromic prime is from Makoto Morimoto
10300000 + 5 * (1048153 – 1)/9 * 10125924 + 1
(300001 digits)


Source (The PrimePages, NMBRTHRY, ...)
http://primes.utm.edu/primes/page.php?id=132404


[ Submitted April 11, 2012 ]

This palindromic prime 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 (The PrimePages, NMBRTHRY, ...)
http://primes.utm.edu/primes/page.php?id=106219
http://zerosink.blogspot.com/2012/04/
http://groups.yahoo.com/group/primeform/message/11254 http://groups.yahoo.com/group/primeform/message/11280


[ Submitted May 19, 2020 ]

This palindromic prime is from Mohamed Reda Kebbaj
10283355 – 737 * 10141676 – 1
(283355 digits)


Source (The PrimePages, NMBRTHRY, ...)
https://primes.utm.edu/primes/page.php?id=130908


[ Submitted January 13, 2012 ]

This palindromic prime 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 (The PrimePages, NMBRTHRY, ...)
http://primes.utm.edu/primes/page.php?id=103926
http://tech.groups.yahoo.com/group/primeform/message/11203
(Reply Message 11204 )


[ Submitted February 29, 2012 ]

This palindromic prime is from Darren Bedwell
10269479 – 7 * 10134739 – 1
(269479 digits)


Written out, it's 134739 nines, a two, and 134739 more nines
and is therefore a PWP or a Palindromic Wing Prime.


Source (The PrimePages, NMBRTHRY, ...)
http://primes.utm.edu/primes/page.php?id=105258
http://zerosink.blogspot.com/2012/04/


[ Submitted June 3, 2021 ]

This palindromic prime is from Makoto Morimoto
10262144 + 7 * (105193 – 1)/9 * 10128476 + 1
(262144 digits)


Source (The PrimePages, NMBRTHRY, ...)
http://primes.utm.edu/primes/page.php?id=132365


[ Submitted January 7, 2016 ]

This palindromic prime is from Serge Batalov
10223663 – 454 * 10111830 – 1
(223663 digits)


Source (The PrimePages, NMBRTHRY, ...)
http://primes.utm.edu/primes/page.php?id=120863


[ Submitted January 5, 2016 ]

This palindromic prime is from Serge Batalov
10220285 – 949 * 10110141 – 1
(220285 digits)


Source (The PrimePages, NMBRTHRY, ...)
http://primes.utm.edu/primes/page.php?id=120848


[ Submitted September 17, 2010 ]

This palindromic prime is from Bernardo Boncompagni
10200000 + 47960506974 * 1099995 + 1
(200001 digits)


Source (The PrimePages, NMBRTHRY, ...)
http://primes.utm.edu/primes/page.php?id=94993
http://tech.groups.yahoo.com/group/primeform/message/10535


[ Submitted May 23, 2010 ]

This palindromic prime is from Harvey Dubner
10190004 + 214757412 * 1094998 + 1
(190005 digits)


Source (The PrimePages, NMBRTHRY, ...)
http://primes.utm.edu/primes/page.php?id=92828


[ Submitted May 2, 2010 ]

This palindromic prime is from Harvey Dubner
10185008 + 130525031 * 1092500 + 1
(185009 digits)


Source (The PrimePages, NMBRTHRY, ...)
http://primes.utm.edu/primes/page.php?id=92569


[ Submitted September 8, 2007 ]

Darren Bedwell re-establishes a new
world palindromic prime record
10180054 + 8*R(58567) * 1060744 + 1
(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 (The PrimePages, NMBRTHRY, ...)
http://primes.utm.edu/primes/page.php?id=89907
http://zerosink.blogspot.com/2009/09/


[ Submitted August 8, 2007 ]

Harvey Dubner re-establishes
a palindromic prime record
10180004 + 248797842 * 1089998 + 1
(180005 digits)


Source (The PrimePages, NMBRTHRY, ...)
http://primes.utm.edu/primes/page.php?id=81904
http://tech.groups.yahoo.com/group/primeform/message/9261


[ Submitted June 9, 2007 ]

Harvey Dubner re-establishes
a palindromic prime record
10175108 + 230767032 * 1087550 + 1
(175109 digits)


Source (The PrimePages, NMBRTHRY, ...)
http://primes.utm.edu/primes/page.php?id=80980
http://tech.groups.yahoo.com/group/primeform/message/8673


[ Submitted October 21, 2006 ]

Harvey Dubner re-establishes
a palindromic prime record
10170006 + 3880883 * 1085000 + 1
(170007 digits)


Source (The PrimePages, NMBRTHRY, ...)
http://primes.utm.edu/primes/page.php?id=78702


[ Submitted May 19, 2006 ]

Harvey Dubner re-establishes
a palindromic prime record
10160016 + 8231328 * 1080005 + 1
(160017 digits)


Source (The PrimePages, NMBRTHRY, ...)
http://primes.utm.edu/primes/page.php?id=77815
Number Theory List - Message of 1 Jun 2006


[ Submitted February 1, 2006 ]

Harvey Dubner retakes
the palindromic prime record
10150008 + 4798974 * 1075001 + 1
(150009 digits)


Source (The PrimePages, NMBRTHRY, ...)
http://primes.utm.edu/primes/page.php?id=76932
http://groups.yahoo.com/group/primeform/message/6888


[ Submitted December 26, 2005 ]

Paul Jobling tops the list now
with his palindromic prime record
10150006 + 7426247 * 1075000 + 1
(150007 digits)


Source (The PrimePages, NMBRTHRY, ...)
http://primes.utm.edu/primes/page.php?id=76550
http://groups.yahoo.com/group/primeform/message/6757
(Reply Messages 6764, 6765, 6766, 6767, 6768, 6769)


[ Submitted November 25, 2010 ]

Darren Bedwell announced this
NearRepDigit palindromic prime record
10134809 – 1067404 – 1
(134809 digits)


Source (The PrimePages, NMBRTHRY, ...)
http://groups.yahoo.com/group/primeform/message/10769


[ Submitted October 31, 2010 ]

Darren Bedwell sent in this former
NearRepDigit palindromic prime record
10125877 – 7 * 1062938 – 1
(125877 digits)


Source (The PrimePages, NMBRTHRY, ...)
https://primes.utm.edu/primes/page.php?id=95802
http://groups.yahoo.com/group/primeform/message/10743


[ Submitted September 4, 2010 ]

Darren Bedwell sent in (rather late)
this former palindromic prime record
10111725 – 4 * 1055862 – 1
(111725 digits)


Source (The PrimePages, NMBRTHRY, ...)
https://primes.utm.edu/primes/page.php?id=94508
http://groups.yahoo.com/group/primeform/message/10535


Former Palindromic Prime Records computed by Harvey Dubner & ass.


Harvey Dubner (1928-2019 )

Harvey Dubner


Picture borrowed from Ivars Peterson webpage Palindromic Primes


[ Submitted December 6, 2005 ]

Harvey Dubner re-establishes
a personal palindromic prime record
10140008 + 4546454 * 1070001 + 1
(140009 digits)


Source (The PrimePages, NMBRTHRY, ...)
https://primes.utm.edu/primes/page.php?id=76391
http://groups.yahoo.com/group/primeform/message/6719


[ Submitted November 8, 2004 ]

Harvey Dubner establishes again
a new palindromic prime record
10130022 + 3761673 * 1065008 + 1
(130023 digits)


Source (The PrimePages, NMBRTHRY, ...)
https://primes.utm.edu/primes/page.php?id=72332
Number Theory List - Message from 19 Nov 2004
http://groups.yahoo.com/group/primeform/message/4999


[ Submitted April 5, 2004 ]

Let us acclaim Harvey Dubner's
latest palindromic prime record
10120016 + 1726271 * 1060005 + 1
(120017 digits)


Source (The PrimePages, NMBRTHRY, ...)
https://primes.utm.edu/primes/page.php?id=69756
Number Theory List - Message from 12 Apr 2004
http://groups.yahoo.com/group/primeform/message/4273


[ Submitted April 2, 2004 ]

Let us acclaim Harvey Dubner's
latest palindromic prime record
10120002 + 1617161 * 1059998 + 1
(120003 digits)


Source (The PrimePages, NMBRTHRY, ...)
https://primes.utm.edu/primes/page.php?id=69729
Number Theory List- Message from 12 Apr 2004
http://groups.yahoo.com/group/primeform/message/4269


[ Submitted May 3, 2004 ]

The largest palindromic prime (from Andersen and Dubner)
with a palindromic prime length is
1098689 – 429151924 * 1049340 – 1
(98689 digits)


A doubly palindromic prime !


Source (The PrimePages, NMBRTHRY, ...)
https://primes.utm.edu/primes/page.php?id=70157
Number Theory List - Message from 18 May 2004
http://groups.yahoo.com/group/primeform/message/4320


[ Submitted February 15, 2004 ]

Harvey Dubner announced via the NMBRTHRY bulletin board
a new  palindromic prime  record.
1091018 + 126696621 * 1045505 + 1
(91019 digits)


A doubly palindromic prime !


Source (The PrimePages, NMBRTHRY, ...)
Number Theory List - Message from 12 Apr 2004
https://primes.utm.edu/primes/page.php?id=68843


[ Submitted April 8, 2004 ]

Harvey Dubner announced via the NMBRTHRY bulletin board
a new  palindromic prime  record.
1051000 + R(4133) * 1023434 + 1
(51001 digits)


R(4133) is a repunit consisting of 4133 1's.
This is a tetradic prime or a 4-way prime that
reads the same left, right, up & down and rotated.
(previous record had 30803 digits).


Source (The PrimePages, NMBRTHRY, ...)
Number Theory List - Message from 12 Apr 2004
https://primes.utm.edu/primes/page.php?id=69773


[ Submitted July 17, 2001 ]

Harvey Dubner announced via the NMBRTHRY bulletin board
a new  palindromic prime  record.
1039026 + 4538354 * 1019510 + 1
(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 !


Source (The PrimePages, NMBRTHRY, ...)
https://primes.utm.edu/primes/page.php?id=2507
Number Theory List - Message from 17 Jul 2001


[ Submitted November 14, 1999 ]

Harvey Dubner announced via the NMBRTHRY bulletin board
a new  palindromic prime  record.
1035352 + 2049402 * 1017673 + 1
(35353 digits)


The number of digits, 35353, is also
a palindromic prime (no accident).


Source (The PrimePages, NMBRTHRY, ...)
https://primes.utm.edu/primes/page.php?id=2783
Number Theory List - Message from 14 Nov 1999

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.


[ Submitted April 28, 1999 ]

Harvey Dubner announced via the NMBRTHRY bulletin board
a new  palindromic prime  record.
1030802 + 1110111 * 1015398 + 1
(30803 digits)


The number of digits, 30803, is also
a palindromic prime (no accident).
It consists of only 1's and 0's (definitely an accident).


Source (The PrimePages, NMBRTHRY, ...)
Number Theory List - Message from 28 Apr 1999

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 accident!).
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.


[ Submitted February 23, 1999 ]

Harvey Dubner announced via the NMBRTHRY bulletin board
a new  palindromic prime  record.
1019390 + 4300034 * 109692 + 1
(19391 digits)


The number of digits, 19391, is also
a palindromic prime (no accident).


Source (The PrimePages, NMBRTHRY, ...)
Number Theory List - Message from 23 Feb 1999


[ Submitted February 2002 ]

David Broadhurst and Harvey Dubner announced via the
NMBRTHRY bulletin board a new  palindromic prime 
N = (1808010808)15601
(15601 digits)


This is a tetratic (4-way) prime: it is a palindrome
as well as the same upside down and mirror reflected.


Source (The PrimePages, NMBRTHRY, ...)
Number Theory List - Message from 16 Feb 2002


[ Submitted October 5, 2001 ]

David Broadhurst and Harvey Dubner announced via the
NMBRTHRY bulletin board a new  palindromic prime 
N = (1989191989)15601
(15601 digits)


This is not a new record however this palindrome is of
a more varied form and hence considerably
more demanding to prove !


Source (The PrimePages, NMBRTHRY, ...)
https://primes.utm.edu/primes/page.php?id=11776
Wonplate 126
Number Theory List - Message from 4 Oct 2001

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 


[ Submitted January 29, 2000 ]

Warut Roonguthai found using PrimeForm
a new  palindromic prime .
1011840 + 42924 * 105918 + 1
(11841 digits)


Source (The PrimePages, NMBRTHRY, ...)

[ January 29, 2000 ]
Not afraid to search for larger palprimes himself Warut Roonguthai
used PrimeForm 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].


[ Submitted February 13, 1997 ]

Harvey Dubner announced via the NMBRTHRY bulletin board
some new  palindromic prime  records.

a*bbkdigitscomments
10080611010110081palindrome, all 1's and 0's
10080615979510081palindrome, odd digits only
1008010116500056110081palindrome
1008010181660661810081palindrome
1008010181853581810081palindrome
(10081 digits)


All are of the form k * [ R(a*b) / R(b) ] * 10 + 1


Source (The PrimePages, NMBRTHRY, ...)
Number Theory List - Message from 13 Feb 1997


Sources

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

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 Palprime Records









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