Palindromic prime

Palindromic prime
Conjectured no. of terms	Infinite
First terms	7, 11, 101, 131, 151
Largest known term	101888529 - 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:

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

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

10^1888529 - 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

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

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]

References

^ De Geest, Patrick. "World of Palindromic Primes". World!Of Numbers. Retrieved 1 April 2023.
^ Chris Caldwell, The Top Twenty: Palindrome
^ William D. Banks, Derrick N. Hart, Mayumi Sakata, February 1, 2008 "Almost All Palindromes Are Composite"
^ "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.
^ 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.
^ Paulo Ribenboim, The New Book of Prime Number Records
^ "Even more pi in the sky: Calculating 100 trillion digits of pi on Google Cloud". Google Cloud Blog. Retrieved 2022-10-13.
^ "Sigma Geek". SigmaGeek. Retrieved 2022-10-13.

[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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v t e Prime number classes
By formula	Fermat ( $2 2 n + 1$ ) Mersenne ( $2 p - 1$ ) Double Mersenne ( $2 2 p -1 - 1$ ) Wagstaff $(2 p + 1)/3$ Proth ( $k \cdot2 n + 1$ ) Factorial ( $n! \pm 1$ ) Primorial ( $p n # \pm 1$ ) Euclid ( $p n # + 1$ ) Pythagorean ( $4 n + 1$ ) Pierpont ( $2 m \cdot3 n + 1$ ) Quartan ( $x 4 + y 4$ ) Solinas ( $2 m \pm 2 n \pm 1$ ) Cullen ( $n \cdot2 n + 1$ ) Woodall ( $n \cdot2 n - 1$ ) Cuban ( $x 3 - y 3)/(x - y$ ) Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin
By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient Unique
Base-dependent	Palindromic Emirp Repunit $(10 n - 1)/9$ Permutable Circular Truncatable Minimal Delicate Primeval Full reptend Unique Happy Self Smarandache–Wellin Strobogrammatic Dihedral Tetradic
Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n \pm 1, 2 n \pm 1, 4 n \pm 1, \dots$ ) Triplet ( $p, p + 2 or p + 4, p + 6$ ) Quadruplet ( $p, p + 2, p + 6, p + 8$ ) k-tuple Cousin ( $p, p + 4$ ) Sexy ( $p, p + 6$ ) Chen Sophie Germain/Safe ( $p, 2 p + 1$ ) Cunningham ( $p, 2 p \pm 1, 4 p \pm 3, 8 p \pm 7, ...$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3, ...$ ) Balanced ( $consecutive p - n, p, p + n$ )
By size	Mega (1,000,000+ digits) Largest known list
Complex numbers	Eisenstein prime Gaussian prime
Composite numbers	Pseudoprime Catalan Elliptic Euler Euler–Jacobi Fermat Frobenius Lucas Perrin Somer–Lucas Strong Carmichael number Almost prime Semiprime Sphenic number Interprime Pernicious
Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap
First 60 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
List of prime numbers

Other bases

Property

Palindromic Prime in Decimal Expansion of Pi

See also

References