Palindromic prime

Last updated
Palindromic prime
Conjectured no. of termsInfinite
First terms 2, 3, 5, 7, 11, 101, 131, 151
Largest known term101888529 - 10944264 - 1
OEIS index
  • A002385
  • Palindromic primes: prime numbers whose decimal expansion is a palindrome

In mathematics, a palindromic prime (sometimes called a palprime [1] ) is a prime number that is also a palindromic number. Palindromicity depends on the base of the number system and its notational conventions, while primality is independent of such concerns. The first few decimal palindromic primes are:

Contents

2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, … (sequence A002385 in the OEIS )

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. The largest known as of October 2021 is

101888529 - 10944264 - 1.

which has 1,888,529 digits, and was found on 18 October 2021 by Ryan Propper and Serge Batalov. [2] On the other hand, it is known that, for any base, almost all palindromic numbers are composite, [3] i.e. the ratio between palindromic composites and all palindromes less than n tends to 1.

Other bases

In binary, the palindromic primes include the Mersenne primes and the Fermat primes. 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 by 3. The sequence of binary palindromic primes begins (in binary):

11, 101, 111, 10001, 11111, 1001001, 1101011, 1111111, 100000001, 100111001, 110111011, ... (sequence A117697 in the OEIS )

The palindromic primes in base 12 are: (using A and B for ten and eleven, respectively)

2, 3, 5, 7, B, 11, 111, 131, 141, 171, 181, 1B1, 535, 545, 565, 575, 585, 5B5, 727, 737, 747, 767, 797, B1B, B2B, B6B, ...

The palindromic prime numbers can also be generated based on Smarandache function (Kempner function) using prime number algorithm. [4]

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. 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. [5]

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. [6] For example, p = 1011310 + 4661664×105652 + 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 [7] that they have calculated 100 Trillion digits of pi using y-cruncher on their cloud platform. The largest Palindromic prime appearing in the known decimal expansion of pi is 9609457639843489367549069. [8]

See also

Related Research Articles

A palindromic number is a number that remains the same when its digits are reversed. In other words, it has reflectional symmetry across a vertical axis. The term palindromic is derived from palindrome, which refers to a word whose spelling is unchanged when its letters are reversed. The first 30 palindromic numbers are:

11 (eleven) is the natural number following 10 and preceding 12. It is the first repdigit. In English, it is the smallest positive integer whose name has three syllables.

In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system. The word is a portmanteau of "repeated" and "digit". Examples are 11, 666, 4444, and 999999. All repdigits are palindromic numbers and are multiples of repunits. Other well-known repdigits include the repunit primes and in particular the Mersenne primes.

27 is the natural number following 26 and preceding 28.

69 (sixty-nine) is the natural number following 68 and preceding 70. An odd number and a composite number, 69 is divisible by 1, 3, 23 and 69. 69 is a semiprime because it is a natural number that is the product of exactly two prime numbers, and an interprime between the numbers of 67 and 71. Because 69 is not divisible by any square number other than 1, it is categorised as a square-free integer. 69 is also a Blum integer since the two factors of 69 are both Gaussian primes. In number theory, 69 is a deficient number, arithmetic number and a congruent number.

73 (seventy-three) is the natural number following 72 and preceding 74. In English, it is the smallest natural number with twelve letters in its spelled out name.

1000 or one thousand is the natural number following 999 and preceding 1001. In most English-speaking countries, it can be written with or without a comma or sometimes a period separating the thousands digit: 1,000.

2000 is a natural number following 1999 and preceding 2001.

10,000 is the natural number following 9,999 and preceding 10,001.

<span class="mw-page-title-main">1,000,000</span> Natural number

1,000,000, or one thousand thousand, is the natural number following 999,999 and preceding 1,000,001. The word is derived from the early Italian millione, from mille, "thousand", plus the augmentative suffix -one.

<span class="mw-page-title-main">1,000,000,000</span> Natural number

1,000,000,000 is the natural number following 999,999,999 and preceding 1,000,000,001. With a number, "billion" can be abbreviated as b, bil or bn.

100,000 (one hundred thousand) is the natural number following 99,999 and preceding 100,001. In scientific notation, it is written as 105.

A cyclic number is an integer for which cyclic permutations of the digits are successive integer multiples of the number. The most widely known is the six-digit number 142857, whose first six integer multiples are

An undulating number is a number that has the digit form ABABAB... when in the base 10 number system. It is sometimes restricted to non-trivial undulating numbers which are required to have at least three digits and A ≠ B. The first few such numbers are:

10,000,000 is the natural number following 9,999,999 and preceding 10,000,001.

100,000,000 is the natural number following 99,999,999 and preceding 100,000,001.

In number theory, a left-truncatable prime is a prime number which, in a given base, contains no 0, and if the leading ("left") digit is successively removed, then all resulting numbers are prime. For example, 9137, since 9137, 137, 37 and 7 are all prime. Decimal representation is often assumed and always used in this article.

20,000 is the natural number that comes after 19,999 and before 20,001.

<span class="mw-page-title-main">Belphegor's prime</span> The prime number 10³⁰+666×10¹⁴+1

Belphegor's prime is the palindromic prime number 1000000000000066600000000000001 (1030 + 666 × 1014 + 1), a number which reads the same both backwards and forwards and is only divisible by itself and one. It was discovered by Harvey Dubner. The name Belphegor refers to one of the Seven Princes of Hell, who was charged with helping people make ingenious inventions and discoveries. "Belphegor's prime" is a name coined by author Clifford A. Pickover. The number itself contains superstitious elements that have given it its name: the number 666 at the heart of Belphegor's Prime is widely associated as being the Number of the Beast, used in symbolism to represent one of the creatures in the Apocalypse or, more commonly, the Devil. This number is surrounded on either side by thirteen zeroes and is 31 digits in length (thirteen reversed), with thirteen itself long regarded superstitiously as an unlucky number in Western culture.

References

  1. ↑ De Geest, Patrick. "World of Palindromic Primes". World!Of Numbers. Retrieved 1 April 2023.
  2. ↑ Chris Caldwell, The Top Twenty: Palindrome
  3. ↑ William D. Banks, Derrick N. Hart, Mayumi Sakata, February 1, 2008 "Almost All Palindromes Are Composite"
  4. ↑ "Palindromes in Some Smarandache-Type Functions". Jurnal Matematika MANTIK Vol. 8, No. 1, 2022, pp.1-9, by Hary Gunarto, S.M.S. Islam and A.A.K. Majumdar. June 2022. Retrieved 8 October 2023.
  5. ↑ See Caldwell, Prime Curios! (CreateSpace, 2009) p. 251, quoted in Wilkinson, Alec (February 2, 2015). "The Pursuit of Beauty". The New Yorker. Retrieved January 29, 2015.
  6. ↑ Paulo Ribenboim, The New Book of Prime Number Records
  7. ↑ "Even more pi in the sky: Calculating 100 trillion digits of pi on Google Cloud". Google Cloud Blog. Retrieved 2022-10-13.
  8. ↑ "Sigma Geek". SigmaGeek. Retrieved 2022-10-13.
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