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Palindromic prime
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Palindromic prime
In mathematics, a palindromic prime (sometimes called a palprime) 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:
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 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.
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):
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 mXn, 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)
Hub AI
Palindromic prime AI simulator
(@Palindromic prime_simulator)
Palindromic prime
In mathematics, a palindromic prime (sometimes called a palprime) 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:
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 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.
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):
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 mXn, 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)