In mathematics, a palindromic prime (sometimes called a palprime) is a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

that is also a

palindromic number A palindromic number (also known as a numeral palindrome or a numeric palindrome) is a number (such as 16461) that remains the same when its digits are reversed. In other words, it has reflectional symmetry across a vertical axis. The term ''palin ...

. Palindromicity depends on the base of the number system and its notational conventions, while primality is independent of such concerns. The first few

decimal The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...

palindromic primes are: : 2, 3, 5, 7, 11, 101,

131 131 may refer to: *131 (number) *AD 131 *131 BC *131 (album), the album by Emarosa *131 (MBTA bus), the Massachusetts Bay Transportation Authority bus. For the MBTA bus, see 131 (MBTA bus). *131 (New Jersey bus) 131 may refer to: *131 (number) *AD ...

151 Year 151 ( CLI) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Condianus and Valerius (or, less frequently, year 904 ''Ab urbe con ...

, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, … Except for 11, all palindromic primes have an

odd Odd means unpaired, occasional, strange or unusual, or a person who is viewed as eccentric. Odd may also refer to: Acronym * ODD (Text Encoding Initiative) ("One Document Does it all"), an abstracted literate-programming format for describing X ...

number of digits, because the divisibility test for 11 tells us that every palindromic number with an

even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East **Even language, a language spoken by the Evens * Odd and Even, a solitaire game wh ...

number of digits is a multiple of 11. It is not known if there are infinitely many palindromic primes in base 10. The largest known is :10^1888529 - 10⁹⁴⁴²⁶⁴ - 1. which has 1,888,529 digits, and was found on 18 October 2021 by Ryan Propper and Serge Batalov. On the other hand, it is known that, for any base,

almost all In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the mathema ...

palindromic numbers are

composite Composite or compositing may refer to: Materials * Composite material, a material that is made from several different substances ** Metal matrix composite, composed of metal and other parts ** Cermet, a composite of ceramic and metallic materials ...

, i.e. the ratio between palindromic composites and all palindromes below ''n'' tends to 1.

Other bases

In binary, the palindromic primes include the

Mersenne prime In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17 ...

s and the

Fermat prime In mathematics, a Fermat number, named after Pierre de Fermat, who first studied them, is a positive integer of the form :F_ = 2^ + 1, where ''n'' is a non-negative integer. The first few Fermat numbers are: : 3, 5, 17, 257, 65537, 429496 ...

s. All binary palindromic primes except binary 11 (decimal 3) have an odd number of digits; those palindromes with an even number of digits are

divisible In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

by 3. The

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...

of binary palindromic primes begins (in binary): :11, 101, 111, 10001, 11111, 1001001, 1101011, 1111111, 100000001, 100111001, 110111011, ... The palindromic primes in

base 12 The duodecimal system (also known as base 12, dozenal, or, rarely, uncial) is a positional notation numeral system using twelve as its base. The number twelve (that is, the number written as "12" in the decimal numerical system) is instead wr ...

are: (using reversed two and three for ten and eleven, respectively) :2, 3, 5, 7, Ɛ, 11, 111, 131, 141, 171, 181, 1Ɛ1, 535, 545, 565, 575, 585, 5Ɛ5, 727, 737, 747, 767, 797, Ɛ1Ɛ, Ɛ2Ɛ, Ɛ6Ɛ, ...

Property

Due to the

superstitious A superstition is any belief or practice considered by non-practitioners to be irrational or supernatural, attributed to fate or magic, perceived supernatural influence, or fear of that which is unknown. It is commonly applied to beliefs and ...

significance of the numbers it contains, the palindromic prime 1000000000000066600000000000001 is known as Belphegor's Prime, named after

Belphegor In Christian demonology, Belphegor (or Beelphegor, he, בַּעַל-פְּעוֹר ''Báʿal-pəʿór'' - Lord of the Gap) is a demon. In later Kabbalah Belphegor is a demon who helps people make discoveries. He seduces people by suggesting to ...

, one of the seven princes of

Hell In religion and folklore, hell is a location in the afterlife in which evil souls are subjected to punitive suffering, most often through torture, as eternal punishment after death. Religions with a linear divine history often depict hell ...

. Belphegor's Prime consists of the number

666 666 may refer to: * 666 (number) * 666 BC, a year * AD 666, a year * The number of the beast, a reference in the Book of Revelation in the New Testament Places * 666 Desdemona, a minor planet in the asteroid belt * U.S. Route 666, an America ...

, on either side enclosed by

thirteen Thirteen or 13 may refer to: * 13 (number), the natural number following 12 and preceding 14 * One of the years 13 BC, AD 13, 1913, 2013 Music * 13AD (band), an Indian classic and hard rock band Albums * ''13'' (Black Sabbath album), 2013 * ...

zeroes and a one. Belphegor's Prime is an example of a beastly palindromic prime in which a prime ''p'' is palindromic with 666 in the center. Another beastly palindromic prime is 700666007. Ribenboim defines a triply palindromic prime as a prime ''p'' for which: ''p'' is a palindromic prime with ''q'' digits, where ''q'' is a palindromic prime with ''r'' digits, where ''r'' is also a palindromic prime. For example, ''p'' = 10¹¹³¹⁰ + 4661664 + 1, which has ''q'' = 11311 digits, and 11311 has ''r'' = 5 digits. The first (base-10) triply palindromic prime is the 11-digit number 10000500001. It is possible that a triply palindromic prime in base 10 may also be palindromic in another base, such as base 2, but it would be highly remarkable if it were also a triply palindromic prime in that base as well.

Palindromic Prime in Decimal Expansion of Pi

On June 8, 2022 Google cloud announced that they have calculated 100 Trillion digits of pi using

y-cruncher The Chudnovsky algorithm is a fast method for calculating the digits of , based on Ramanujan’s formulae. It was published by the Chudnovsky brothers in 1988. It was used in the world record calculations of 2.7 trillion digits of in December ...

on their cloud platform. The largest Palindromic prime appearing in the known decimal expansion of pi is 9609457639843489367549069.

References

{{Prime number classes Base-dependent integer sequences Classes of prime numbers Unsolved problems in mathematics Palindromes

Other bases

Property

Palindromic Prime in Decimal Expansion of Pi

See also

References