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 102718281 - 5 · 101631138 - 5 · 101087142 - 1 2718281 p423 Aug 2024 Palindrome 2 102000007 - 101127194 - 10872812 - 1 2000007 p423 Jan 2024 Palindrome 3 102000005 - 101051046 - 10948958 - 1 2000005 p423 Jan 2024 Palindrome 4 101888529 - 10944264 - 1 1888529 p423 Oct 2021 Near - repdigit, palindrome 5 101234567 - 20342924302 · 10617278 - 1 1234567 p423 Sep 2021 Palindrome 6 101234567 - 1927633367291 · 10617277 - 1 1234567 p423 Jun 2023 Palindrome 7 101234567 - 3626840486263 · 10617277 - 1 1234567 p423 Sep 2021 Palindrome 8 101234567 - 4708229228074 · 10617277 - 1 1234567 p423 Sep 2021 Palindrome 9 10490000 + 3 · (107383 - 1)/9 · 10241309 + 1 490001 p413 Aug 2021 Palindrome 10 10474500 + 999 · 10237249 + 1 474501 p363 Nov 2014 Palindrome 11 10400000 + 4 · (10102381 - 1)/9 · 10148810 + 1 400001 p413 Jul 2021 Palindrome 12 10390636 + 999 · 10195317 + 1 390637 p363 Nov 2014 Palindrome 13 10362600 + 666 · 10181299 + 1 362601 p363 Nov 2014 Palindrome 14 10320236 + 10160118 + 1 + (137 · 10160119 + 731 · 10159275) · (10843 - 1)/999 320237 p44 Mar 2014 Palindrome 15 10320096 + 10160048 + 1 + (137 · 10160049 + 731 · 10157453) · (102595 - 1)/999 320097 p44 Mar 2014 Palindrome 16 10314727 - 8 · 10157363 - 1 314727 p235 Jan 2013 Near - repdigit, palindrome 17 10300000 + 5 · (1048153 - 1)/9 · 10125924 + 1 300001 p413 Jun 2021 Palindrome 18 10300000 + 10158172 + 11011 · 10149998 + 10141828 + 1 300001 p409 Sep 2024 Palindrome 19 10290253 - 2 · 10145126 - 1 290253 p235 Apr 2012 Near - repdigit, Palindrome 20 10283355 - 737 · 10141676 - 1 283355 p399 May 2020 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.]