The Top Twenty--a Prime Page Collection

Palindrome

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The Prime Pages

GIMPS has discovered a new largest known prime number: 2^82589933-1 (24,862,048 digits)

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.

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 10⁴⁷⁴⁵⁰⁰ + 999 · 10²³⁷²⁴⁹ + 1 474501 p363 Nov 2014 Palindrome
2 10³⁹⁰⁶³⁶ + 999 · 10¹⁹⁵³¹⁷ + 1 390637 p363 Nov 2014 Palindrome
3 10³⁶²⁶⁰⁰ + 666 · 10¹⁸¹²⁹⁹ + 1 362601 p363 Nov 2014 Palindrome
4 Phi(3, 10¹⁶⁰¹¹⁸) + (137 · 10¹⁶⁰¹¹⁹ + 731 · 10¹⁵⁹²⁷⁵) · (10⁸⁴³ - 1)/999 320237 p44 Mar 2014 Palindrome
5 Phi(3, 10¹⁶⁰⁰⁴⁸) + (137 · 10¹⁶⁰⁰⁴⁹ + 731 · 10¹⁵⁷⁴⁵³) · (10²⁵⁹⁵ - 1)/999 320097 p44 Mar 2014 Palindrome
6 10³¹⁴⁷²⁷ - 8 · 10¹⁵⁷³⁶³ - 1 314727 p235 Jan 2013 Near - repdigit, palindrome
7 10²⁹⁰²⁵³ - 2 · 10¹⁴⁵¹²⁶ - 1 290253 p235 Apr 2012 Near - repdigit, Palindrome
8 Phi(3, 10¹³⁷⁷⁴⁷) + (137 · 10¹³⁷⁷⁴⁸ + 731 · 10¹²⁹²⁹³) · (10⁸⁴⁵⁴ - 1)/999 275495 p44 Jan 2012 Palindrome
9 10²⁶⁹⁴⁷⁹ - 7 · 10¹³⁴⁷³⁹ - 1 269479 p235 Feb 2012 Near - repdigit, Palindrome
10 10²²³⁶⁶³ - 454 · 10¹¹¹⁸³⁰ - 1 223663 p363 Jan 2016 Palindrome
11 10²²⁰²⁸⁵ - 949 · 10¹¹⁰¹⁴¹ - 1 220285 p363 Jan 2016 Palindrome
12 10²¹⁹¹¹³ - 535 · 10¹⁰⁹⁵⁵⁵ - 1 219113 p363 Jan 2016 Palindrome
13 10²¹⁴⁵⁷⁵ - 20002 · 10¹⁰⁷²⁸⁵ - 1 214575 p363 Jan 2016 Palindrome
14 10²¹⁴⁴⁷⁹ - 535 · 10¹⁰⁷²³⁸ - 1 214479 p363 Jan 2016 Palindrome
15 Phi(3, 10¹⁰⁴²⁷⁹) + (137 · 10¹⁰⁴²⁸⁰ + 731 · 10⁹³³⁹⁵) · (10¹⁰⁸⁸⁴ - 1)/999 208559 p44 Feb 2014 Palindrome
16 Phi(3, 10¹⁰⁴²⁷⁶) + (137 · 10¹⁰⁴²⁷⁷ + 731 · 10⁹⁹⁶⁸³) · (10⁴⁵⁹³ - 1)/999 208553 p44 Feb 2014 Palindrome
17 Phi(3, 10¹⁰⁴²⁵⁷) + (137 · 10¹⁰⁴²⁵⁸ + 731 · 10⁹⁹¹⁹³) · (10⁵⁰⁶⁴ - 1)/999 208515 p44 Feb 2014 Palindrome
18 Phi(3, 10¹⁰³²⁸⁹) + (137 · 10¹⁰³²⁹⁰ + 731 · 10⁹⁰⁴⁴⁹) · (10¹²⁸⁴⁰ - 1)/999 206579 p44 Feb 2014 Palindrome
19 Phi(3, 10¹⁰³²⁸²) + (137 · 10¹⁰³²⁸³ + 731 · 10⁸⁵⁰⁰⁹) · (10¹⁸²⁷³ - 1)/999 206565 p44 Feb 2014 Palindrome
20 Phi(3, 10¹⁰³¹⁸²) + (137 · 10¹⁰³¹⁸³ + 731 · 10⁶⁶⁶³⁹) · (10³⁶⁵⁴³ - 1)/999 206365 x29 Jan 2014 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, 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.]

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