Prime palindrome

A prime number palindrome is a prime number whose digits, read from the front and back, produce the same number, analogous to the palindrome , which reads from the front and back produces the same word. The prime number palindrome is a special number palindrome .

The property of a number to be a prime number has nothing to do with representation and depends only on the number itself. In contrast, the property of being a palindrome depends very much on the representation of the number. In fact, every prime number is prime palindrome for a suitably chosen base of the number system .

It is not known whether there are infinitely many prime number palindromes on a fixed basis.

Explanation

If is the prime number and the digit of the prime number is in the position , then: ${\displaystyle p}$ ${\displaystyle n_{x}}$ ${\displaystyle x}$

${\displaystyle p=(n_{x}n_{x-1}\ldots n_{1}n_{0})=(n_{0}n_{1}\ldots n_{x-1}n_{x})}$

There are no decimal prime number palindromes with an even number of digits other than 11, because all number palindromes with an even number of digits have the divisor 11 (the alternating checksum is always 0). In general, in every adic number system , if there is a prime number palindrome with an even number of digits, it can only be 11 of the corresponding number system.

Examples in number systems

Decimal system

• 2, 3, 5, 7, 11, 101 , 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 10301 ... (sequence A002385 in OEIS )
• The largest known prime number palindrome in decimal notation was once with 180,005 decimal places, found in 2007 by Harvey Dubner. ${\displaystyle 10^{180004}[phone]\cdot 10^{89998}+1}$
• In the meantime, 10 ³²⁰²³⁶ + 10 ¹⁶⁰¹¹⁸ + (137 × 10 ¹⁶⁰¹¹⁹ + 731 × 10 ¹⁵⁹²⁷⁵ ) × (10 ⁸⁴³ - 1) / 999 + 1, a larger prime number palindrome on base 10 (320,237 digits).
• In November 2014, the largest known prime number palindrome was with 474,501 digits. ${\displaystyle 10^{474\,500}+999\cdot 10^{237\,249}+1}$
• Belphegor's prime number 1000000000000066600000000000001 is a palindrome and named after the demon Belphegor .

Dual system

• The largest known prime number (as of January 3, 2018) is the Mersenne prime number 2 ^77,232,917 -1. In binary representation , this is a column of 77,232,917 ones and thus - like every Mersenne number - a palindrome of numbers in the form of a column of binary ones.
• All Fermat prime numbers are, in binary terms, number palindromes. These are numbers in which an odd number of zeros are framed by a one. As with the Mersenne prime numbers, the number palindrome property of Fermat's prime numbers is not tied to the prim property, but applies to all Fermat numbers .

Strictly non-palindromic numbers

Every natural number is palindrome to the base . There it has the representation 11. Furthermore, every natural number is a palindrome for every base , because here the representation of is single-digit. The only interesting question is whether a given natural number has a multi-digit palindrome representation other than 11. ${\displaystyle n>2}$ ${\displaystyle n-1}$ ${\displaystyle n}$ ${\displaystyle b>n}$ ${\displaystyle n}$ ${\displaystyle n}$

Numbers that cannot be written as a number palindrome> 11 in any adic number system are referred to as strictly non-palindromic numbers . All numbers of this kind that are> 6 are prime numbers. (Follow A016038 in OEIS )

Web links

• The largest known Primzahlpalindrome (English)

Individual evidence

1. ↑ Eric W. Weisstein : Palindromic Prime . In: MathWorld (English).
2. ↑ Eric W. Weisstein : Belphegor Prime . In: MathWorld (English).
3. ^ Great Internet Mersenne Prime Search - PrimeNet. Accessed January 5, 2018 .