## Palindrome |

The Prime Pages keeps a list of the 5000 largest known primes, plus a few each of certain selected archivable forms and classes. These forms are defined in this collection's home page. This page is about one of those forms. Comments and suggestions requested.

A **palindrome** (from the Greek *palindromos*
"running back again") is a word, verse, sentence, or
integer that reads the same forward or backward. For example,
"Able was I ere I saw Elba" or 333313333. Here is a little
longer one by Peter Hilton (a code-breaker on the British
team that cracked the German Enigma):

Doc, note. I dissent. A fast never prevents a fatness. I diet on cod.

Sotades the obscene of Maronea (3rd century BC) is credited with inventing the palindrome. Though today only eleven lines of his works still remain, he is thought to have recast the entire Illiad as palindromic verse. Sotades also wrote lines which when read backwards had the opposite meaning, now sometimes called Sotadic verses. Sotades attacked many with his unrestrained toungue, and eventually was jailed by Ptolemy II. Sotades eventually escaped, but Ptolemy's admiral Patroclus caught him, sealed him in a leaden chest and tossed him into the sea.

Though palindromic numbers have no significant role in modern mathematics, the survival of the old mysticism so often attached to numbers (perfect numbers, amicable numbers, abundant numbers...) insures the palindromes a secure place in the heart of the amateur numerologists.

rank prime digits who when comment 1 10^{1234567}- 20342924302 · 10^{617278}- 11234567 p423 Sep 2021 Palindrome 2 10^{490000}+ 3 · (10^{7383}- 1)/9 · 10^{241309}+ 1490001 p413 Aug 2021 Palindrome 3 10^{474500}+ 999 · 10^{237249}+ 1474501 p363 Nov 2014 Palindrome 4 10^{400000}+ 4 · (10^{102381}- 1)/9 · 10^{148810}+ 1400001 p413 Jul 2021 Palindrome 5 10^{390636}+ 999 · 10^{195317}+ 1390637 p363 Nov 2014 Palindrome 6 10^{362600}+ 666 · 10^{181299}+ 1362601 p363 Nov 2014 Palindrome 7 Phi(3, 10^{160118}) + (137 · 10^{160119}+ 731 · 10^{159275}) · (10^{843}- 1)/999320237 p44 Mar 2014 Palindrome 8 Phi(3, 10^{160048}) + (137 · 10^{160049}+ 731 · 10^{157453}) · (10^{2595}- 1)/999320097 p44 Mar 2014 Palindrome 9 10^{314727}- 8 · 10^{157363}- 1314727 p235 Jan 2013 Near - repdigit, palindrome 10 10^{300000}+ 5 · (10^{48153}- 1)/9 · 10^{125924}+ 1300001 p413 Jun 2021 Palindrome 11 10^{290253}- 2 · 10^{145126}- 1290253 p235 Apr 2012 Near - repdigit, Palindrome 12 10^{283355}- 737 · 10^{141676}- 1283355 p399 May 2020 Palindrome 13 Phi(3, 10^{137747}) + (137 · 10^{137748}+ 731 · 10^{129293}) · (10^{8454}- 1)/999275495 p44 Jan 2012 Palindrome 14 10^{269479}- 7 · 10^{134739}- 1269479 p235 Feb 2012 Near - repdigit, Palindrome 15 10^{262144}+ 7 · (10^{5193}- 1)/9 · 10^{128476}+ 1262145 p413 Jun 2021 Palindrome 16 10^{223663}- 454 · 10^{111830}- 1223663 p363 Jan 2016 Palindrome 17 10^{220285}- 949 · 10^{110141}- 1220285 p363 Jan 2016 Palindrome 18 10^{219113}- 535 · 10^{109555}- 1219113 p363 Jan 2016 Palindrome 19 10^{216091}- 7 · (10^{37627}- 1)/9 · 10^{89232}- 1216091 p413 Oct 2020 Palindrome 20 10^{214575}- 20002 · 10^{107285}- 1214575 p363 Jan 2016 Palindrome

- selected palidromic primes with more than 1000 digits

- DO94
H. DubnerandR. Ondrejka, "A PRIMEr on palindromes,"J. Recreational Math.,26:4 (1994) 256--267.- GC1969
H. GabaiandD. Coogan, "On palindromes and palindromic primes,"Math. Mag.,42(1969) 252--254.MR0253979- HC2000
G. L. Honaker, Jr.andC. Caldwell, "Palindromic prime pyramids,"J. Recreational Math.,30:3 (1999-2000) 169--176.- Iseki1988
Iséki, Kiyoshi, "Palindromic prime numbers from experimental number theory,"Math. Japon.,33:5 (1988) 715--720.MR 972382- Iseki1988b
Iséki, Kiyoshi, "Palindromic prime numbers,"Math. Japon.,33:6 (1988) 861--862.MR 975864- Iseki1988c
Iséki, Kiyoshi, "Palindromic prime numbers from experimental number theory. II,"Math. Japon.,33:6 (1988) 863--872.MR 975865- McDaniel87b
W. McDaniel, "Palindromic Smith numbers,"J. Recreational Math.,19:1 (1987) 34--37.- Ribenboim95
P. Ribenboim,The new book of prime number records, 3rd edition, Springer-Verlag, 1995. New York, NY, pp. xxiv+541, ISBN 0-387-94457-5.MR 96k:11112[An excellent resource for those with some college mathematics. Basically a Guinness Book of World Records for primes with much of the relevant mathematics. The extensive bibliography is seventy-five pages.]

Chris K. Caldwell
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