Bookmark buttons
The record Palindromic Primes
[ Submitted March 5, 2014 ]
This palindromic prime is from David Broadhurst
Phi(3, 10^{160118}) +
(137 * 10^{160119} + 731 * 10^{159275}) * (10^{843} – 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, 10^{160048}) +
(137 * 10^{160049} + 731 * 10^{157453}) * (10^{2595} – 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)*(10^31)/999" 13707310000000
Pfgw64 f0 od q"(137*10^15+731*10^8)*(10^61)/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^A1)/999
(10^A1)/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 13, 2012 ]
This palindromic prime is from David Broadhurst
who devised the next beautiful formula
Phi(3, 10^{137747}) +
(137 * 10^{137748} + 731 * 10^{129293}) * (10^{8454} – 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 64bit 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 )

Former Palindromic Prime Records computed by Harvey Dubner & ass.
Harvey Dubner (19282019 †)
Picture borrowed from Ivars Peterson webpage Palindromic Primes

[ Submitted November 14, 1999 ]
Harvey Dubner announced via the NMBRTHRY bulletin board
a new palindromic prime record.
10^{35352} + 2049402 * 10^{17673} + 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 October 5, 2001 ]
David Broadhurst and Harvey Dubner announced via the
NMBRTHRY bulletin board a new palindromic prime
N = (1989191989)_{1560}1
(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 KonyaginPomerance proof of primality of
the 15601digit base10 palindrome N = (1989191989)_{1560}1
1) Larger palindromic primes have been proven by applying
BrillhartLehmerSelfridge (BLS) tests to numbers such as
(9)_{23034}8(9)_{23034} and
1(0)_{19509}4538354(0)_{19509}1,
but this method appears to be limited to base10
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 KonyaginPomerance (KP) proof was set by
(1579393975)_{1386}1, recently proven by collaboration
of members of the PrimeNumbers egroup.
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 10^156001 had been factorized into proven primes.
4) We noted that the 1914digit probable prime
prp1914 = Phi_{5200}(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 P1 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 10^156001, it is
straightforward to generate and prove further palindromic
primes between 10^15600 and 2*10^15600. For example
(1854050458)_{1560}1, (1844454448)_{1560}1,
(1413323314)_{1560}1, (1120373021)_{1560}1,
and 107870_{2600}1 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 egroups, 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 .
10^{11840} + 42924 * 10^{5918} + 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 recordholder 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^(n3)+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*b  b  k  digits  comments 
10080  6  110101  10081  palindrome, all 1's and 0's 
10080  6  159795  10081  palindrome, odd digits only 
10080  10  1165000561  10081  palindrome 
10080  10  1816606618  10081  palindrome 
10080  10  1818535818  10081  palindrome 
(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.
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
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
[ TOP OF PAGE]