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.

Definitions and Notes

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.

Record Primes of this Type

rank prime digits who when comment 1 101888529 - 10944264 - 1 1888529 p423 Oct 2021 Near - repdigit, palindrome 2 101234567 - 20342924302 · 10617278 - 1 1234567 p423 Sep 2021 Palindrome 3 101234567 - 3626840486263 · 10617277 - 1 1234567 p423 Sep 2021 Palindrome 4 101234567 - 4708229228074 · 10617277 - 1 1234567 p423 Sep 2021 Palindrome 5 10490000 + 3 · (107383 - 1)/9 · 10241309 + 1 490001 p413 Aug 2021 Palindrome 6 10474500 + 999 · 10237249 + 1 474501 p363 Nov 2014 Palindrome 7 10400000 + 4 · (10102381 - 1)/9 · 10148810 + 1 400001 p413 Jul 2021 Palindrome 8 10390636 + 999 · 10195317 + 1 390637 p363 Nov 2014 Palindrome 9 10362600 + 666 · 10181299 + 1 362601 p363 Nov 2014 Palindrome 10 Phi(3, 10160118) + (137 · 10160119 + 731 · 10159275) · (10843 - 1)/999 320237 p44 Mar 2014 Palindrome 11 Phi(3, 10160048) + (137 · 10160049 + 731 · 10157453) · (102595 - 1)/999 320097 p44 Mar 2014 Palindrome 12 10314727 - 8 · 10157363 - 1 314727 p235 Jan 2013 Near - repdigit, palindrome 13 10300000 + 5 · (1048153 - 1)/9 · 10125924 + 1 300001 p413 Jun 2021 Palindrome 14 10290253 - 2 · 10145126 - 1 290253 p235 Apr 2012 Near - repdigit, Palindrome 15 10283355 - 737 · 10141676 - 1 283355 p399 May 2020 Palindrome 16 Phi(3, 10137747) + (137 · 10137748 + 731 · 10129293) · (108454 - 1)/999 275495 p44 Jan 2012 Palindrome 17 10269479 - 7 · 10134739 - 1 269479 p235 Feb 2012 Near - repdigit, Palindrome 18 10262144 + 7 · (105193 - 1)/9 · 10128476 + 1 262145 p413 Jun 2021 Palindrome 19 10223663 - 454 · 10111830 - 1 223663 p363 Jan 2016 Palindrome 20 10220285 - 949 · 10110141 - 1 220285 p363 Jan 2016 Palindrome

Related Pages

• selected palidromic primes with more than 1000 digits

References

DO94

H. Dubner and R. Ondrejka, "A PRIMEr on palindromes," J. Recreational Math., 26:4 (1994) 256--267.

GC1969

H. Gabai and D. Coogan, "On palindromes and palindromic primes," Math. Mag., 42 (1969) 252--254.  MR0253979

HC2000

G. L. Honaker, Jr. and C. 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, New York, NY, 1995.  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.]