Palindrome Numbers

A palindrome number is one that reads the same in both directions. For example, 121, 1331, and 12321 are palindromes. The term comes from the Greek "palin" (again) and "dromos" (path), literally meaning "to travel the path again".

Mathematical properties of palindromes

Palindrome numbers follow fascinating patterns in their distribution:

• All single-digit numbers (1-9) are palindromes: there are 9 in total.
• There are 9 two-digit palindromes: 11, 22, 33, 44, 55, 66, 77, 88, 99.
• There are 90 three-digit palindromes: from 101 to 999 (the first digit determines the last, and the middle one can be anything: 9 × 10).
• There are 90 four-digit palindromes: from 1001 to 9999 (the first two digits determine the last two: 9 × 10).
• There are 900 five-digit palindromes and 900 six-digit palindromes.

In general, the number of n-digit palindromes follows the pattern: 9 × 10^{⌊(n-1)/2⌋}. This means palindromes become proportionally rarer as numbers grow, but they remain infinite.

Palindromic primes

Some palindrome numbers are also prime, making them doubly special. The only even palindromic prime is 11 (since any other even palindrome would end in an even digit, making it divisible by 2).

The first palindromic primes are:

It is unknown whether there are infinitely many palindromic primes, although it is conjectured that there are. The largest known palindromic primes have hundreds of thousands of digits and are found using advanced computational techniques.

The 196 problem

One of the most famous open problems in recreational mathematics is the 196 problem (also called the "Lychrel problem"). The procedure is simple: take a number, reverse its digits, and add the two together. Repeat until you get a palindrome.

Most numbers reach a palindrome in just a few steps:

The number 196 has been tested up to more than one billion digits without ever reaching a palindrome. It is believed it never will, but no one has been able to prove it formally. This is one of the easiest-to-state yet hardest-to-solve open problems in all of mathematics.

Palindromes from 1 to 500

All palindrome numbers between 1 and 500. Click any of them to explore their mathematical properties: