Also found prior to above record were the following palprimes 10^{390636} + 999 * 10^{195317} + 1 (390637 digits) 10^{362600} + 666 * 10^{181299} + 1 (362601 digits)
Source (Primeform, NMBRTHRY, ...) From The Largest Known Primes! Top Twenty Record Palindromic Primes
Source (Primeform, NMBRTHRY, ...) From The Largest Known Primes! Top Twenty Record Palindromic Primes Message 11475 Big Palindrome
Source (Primeform, NMBRTHRY, ...) Top Twenty Record Palindromic Primes FROM The Largest Known Primes! http://zerosink.blogspot.com
Source (Primeform, NMBRTHRY, ...) Top Twenty Record Palindromic Primes FROM The Largest Known Primes! http://tech.groups.yahoo.com/group/primeform/message/11203
Source (Primeform, NMBRTHRY, ...) Top Twenty Record Palindromic Primes FROM The Largest Known Primes!
Former Palindromic Prime Records computed by Harvey Dubner & ass.
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
We have found a Konyagin-Pomerance proof of primality of the 15601-digit base-10 palindrome N = (1989191989)_{1560}1 1) Larger palindromic primes have been proven by applying Brillhart-Lehmer-Selfridge (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 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)_{1386}1, 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 = 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 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)_{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 e-groups, and in particular Greg Childers, Jim Fougeron, Marcel Martin and Chris Nash. David Broadhurst and Harvey Dubner
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. 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
[ 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 becomerecord-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].
(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'
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.
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 digitsP = 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"
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
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.
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 *
For the full listing of these kind of primes (PDP's) see Plateau and Depression Primes
Read also this interesting article about Palindromic Primes by Ivars Peterson (email).
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