回文数

Palindromic numbers are a creature of notation. 585 mirrors in base ten and also in binary, where it is 1001001001, while 1000 mirrors in neither. Because of that they sit in recreational number theory rather than in any deep part of arithmetic — and recreational number theory is where the most stubborn open problems live.

The best known is reverse-and-add. Take a number, add it to its reversal, repeat: 57 + 75 = 132, then 132 + 231 = 363, and the process stops at a palindrome after two steps. Nearly every starting value stops quickly. 196 does not. The longest-running attack on it, begun by John Walker in 1987 and carried on by others since, has pushed the running sum past hundreds of millions of digits without a palindrome appearing, and there is no proof either way. In 2002 Wade Van Landingham gave numbers suspected of never terminating the name Lychrel — a loose rearrangement of his girlfriend's name, Cheryl — and to date not a single Lychrel number has been proved to be one.

Other questions have been answered. In the early 1970s G. J. Simmons, writing in the Journal of Recreational Mathematics, studied palindromic powers and conjectured that no palindrome is an exact kth power for k of five or more; that remains open. In 2018, by contrast, Javier Cilleruelo, Florian Luca and Lewis Baxter proved something sharp: in every base of five or more, every positive integer is the sum of at most three palindromes.

The primes gave up their secret early. Any palindrome with an even number of digits has an alternating digit sum of zero, so it is divisible by eleven — which makes 11 the only palindromic prime with an even digit count. It is a two-line argument, it removes half of all palindromes from the hunt, and it is the reason this page searches only odd lengths when you ask for primes.

主な性質

• The number of base-10 palindromes with exactly L digits is 9 × 10^(⌈L/2⌉ − 1): nine with one digit, nine with two, ninety with three, ninety with four, nine hundred with five.
• Every palindrome with an even number of digits is divisible by 11, so 11 is the only palindromic prime with an even number of digits.
• More generally, in base b every palindrome with an even number of digits is divisible by b + 1 — the same argument, with b ≡ −1 (mod b+1).
• There are four one-digit palindromic primes (2, 3, 5, 7), exactly one two-digit palindromic prime (11), fifteen with three digits, and none at all with four.
• The only palindromic perfect squares below 1000 are 0, 1, 4, 9, 121, 484 and 676 — the squares of 0, 1, 2, 3, 11, 22 and 26.
• 585 is a palindrome in base 10 and in base 2, where it is written 1001001001.
• Cilleruelo, Luca and Baxter proved in 2018 that in every base b ≥ 5, every positive integer is a sum of at most three palindromes.
• 196 is the smallest number for which repeated reverse-and-add has never been shown to reach a palindrome, and no such number has been proved to exist.

登場する場面

• The bit-reversal permutation at the heart of the Cooley–Tukey FFT reorders data by reversing the bits of each index; the indices it leaves in place are exactly those whose fixed-width bit pattern reads the same backwards.
• The 196 problem has been a fixture of hobbyist computing for decades, with published runs reporting sums hundreds of millions of digits long and still no palindrome.
• Palindromic vehicle registrations and phone numbers are reported to fetch large premiums at official number auctions, notably in India and the Gulf states — a market convention, with no mathematical content.
• Watching an odometer roll to 123,321 and treating palindromic numbers or dates as auspicious are both reactions to notation rather than to quantity: a habit and a belief, not a property of the number.
• The OEIS catalogues base-10 palindromes as A002113, with cross-references to the palindromic primes, squares and cubes that this page can filter for.