Check if a number is a palindromic prime or generate palindromic primes in a range
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Range To
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Number Verdict
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Palindromic Primes in Range
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Detailed Derivation
Palindromic Prime Properties
Prime that reads same forwards and backwards
Examples: 2,3,5,7,11,101,131,151,181,191
No even-digit palindromic primes (except 11)
Only even palindromic prime: 2
Palindromic primes are prime numbers that are also palindromes - they read the same forwards and backwards. They combine two fascinating properties: primality and symmetry. These numbers have been studied by mathematicians throughout history.
⚠Palindromic primes with an even number of digits (except 11) are always divisible by 11. All palindromic primes beyond 11 have an odd number of digits.
What Is a Palindromic Prime?
A palindromic prime is a prime number that remains the same when its digits are reversed. These numbers are a fascinating intersection of primality and numerical symmetry. They have been studied since ancient times and continue to interest mathematicians.
Palindrome Check
A number is palindromic if it reads the same backwards: 101 reversed is 101. 123 reversed is 321 (not palindromic).
Divisibility by 11
Every even-digit palindrome (except 11) is divisible by 11. Thus no palindromic prime with even digits exists beyond 11. This is a well-known divisibility rule.
Emirp vs Palindromic
Emirps become a different prime when reversed (13->31). Palindromic primes are the same when reversed. The two sets are disjoint (except single digit primes).
Known Counts
Up to 10^3: 20. Up to 10^4: 36. Up to 10^5: 58. Up to 10^6: 109. Up to 10^7: 198. Up to 10^8: 595. Density decreases rapidly.
Teaching Example: Test 101. Is it prime? 101 has no divisors other than 1 and 101. Is it palindromic? 101 reversed is 101. Yes! 101 is a palindromic prime. Palindromic primes below 200: 2,3,5,7,11,101,131,151,181,191. Total: 10.
A prime that reads the same forwards and backwards: 2,3,5,7,11,101,131,151. Both prime and palindrome.
Why no even-digit palindromic primes?▼
Every even-digit palindrome is divisible by 11 (a known divisibility rule). The only exception is 11 itself (prime). So all palindromic primes beyond 11 have odd digits.
What is the largest palindromic prime?▼
The largest known palindromic prime has 10^5+ digits. It was discovered in 2021 as part of distributed computing projects. Most large palindromic primes are repunit-like (all 1s).
What is an emirp?▼
An emirp is a prime that gives a different prime when reversed. 13 (emirp because 31 is also prime). 101 is NOT an emirp (reads same). Emirps and palindromic primes are disjoint for 3+ digits.
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