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 16361) 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 (''decimal fractions'') of th ...

palindromic primes are: : 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, … Except for 11, all palindromic primes have an odd number of digits, because the divisibility test for 11 tells us that every palindromic number with an even number of digits is a multiple of 11. It is not known if there are infinitely many palindromic primes in base 10. For any base,

almost all In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if X is a set (mathematics), set, "almost all elements of X" means "all elements of X but those in a negligible set, negligible subset of X". The meaning o ...

palindromic numbers are composite, i.e. the ratio between palindromic composites and all palindromes less than ''n'' tends to 1. A few decorative examples do however exist; in base 10 the following are primes:
11, 122333221, and 1223334444555554444333221. So are: 13331, and 12233355555333221. For a large example, consider:
:, which has 1,888,529 digits. It was found on 18 October 2021 by Ryan Propper and Serge Batalov.

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 1 ...

s and the

Fermat prime In mathematics, a Fermat number, named after Pierre de Fermat (1601–1665), the first known to have 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, ...

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 divisibl ...

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 cal ...

of binary palindromic primes begins (in binary): :11, 101, 111, 10001, 11111, 1001001, 1101011, 1111111, 100000001, 100111001, 110111011, ... Any number that can be expressed as a repetition of just one digit ''d'' in some base must trivially be palindromic in that base and must be a multiple of ''d'' in every base. Accordingly, no number that consists only of a string of repetitions of the same digit in at least one base, can be a prime unless it is a string of 1s in that base. Furthermore, the string must be of prime length, otherwise, if the string is of length ''m''X''n'', it is divisible by strings of lengths ''m'' and ''n'' in that base. For example 111111111111111 (15 digits) is divisible by 111 and 11111 in that base. If a number ''m'' can be expressed as a string of prime length to some base, such a number may or may not be prime, but commonly is not; for example, to base 10, there are only three such numbers of length less than 100 (1 is by definition, not prime). The three are:
11 (length 2), 1111111111111111111 (length 19), and 11111111111111111111111 (length 23)

Property

Due to the superstitious significance of the numbers it contains, the palindromic prime 1000000000000066600000000000001 is known as Belphegor's Prime, named after Belphegor, one of the seven princes of

Hell In religion and folklore, hell is a location or state in the afterlife in which souls are subjected to punishment after death. Religions with a linear divine history sometimes depict hells as eternal destinations, such as Christianity and I ...

. Belphegor's Prime consists of the number 666, on either side enclosed by thirteen 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. Paulo Ribenboim, ''The New Book of Prime Number Records'' 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.

References

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

Other bases

Property

See also

References