Happy Palindrome Week: What are palindromes?

A palindrome is a word, number, phrase, or other sequence of characters which reads the same backward as forward, such as ‘’taco cat’’ or madam or racecar or the number 10801.

Sentence-length palindromes may be written when allowances are made for adjustments to capital letters, punctuation, and word dividers, such as “A man, a plan, a canal, Panama!”, “Was it a car or a cat I saw?” or “No ‘x’ in Nixon”.

Composing literature in palindromes is an example of constrained writing.
The word palindrome was first published by Henry Peacham in his book, The Truth of Our Times (1638). It is derived from the Greek roots palin (πάλιν; “again”) and dromos (δρóμος; “way, direction”); however, the Greek language uses a different word, i.e. καρκινικός, to refer to letter-by-letter reversible writing.

Sentences and phrases

Palindromes often consist of a sentence or phrase, e.g., “Mr. Owl ate my metal worm”, “Do geese see God?”, “Was it a car or a cat I saw?”, “Murder for a jar of red rum” or “Go hang a salami, I’m a lasagna hog”. Punctuation, capitalization, and spaces are usually ignored. Some, such as “Rats live on no evil star”, “Live on time, emit no evil“, and “Step on no pets”, include the spaces.Palindrome Names
Some names are palindromes, such as the given names Hannah, Ada, Anna, Bob, Nitin and Otto, or the surnames Harrah, Renner, Salas and Nenonen. Lon Nol (1913–1985) was Prime Minister of Cambodia.

Some people have changed their name in order to make of it a palindrome (such as actor Robert Trebor and rock-vocalist Ola Salo), while others were given a palindromic name at birth (such as the philologist Revilo P. Oliver or the flamenco dancer Sara Baras).
There are also palindromic names in fictional media. “Stanley Yelnats” is the name of the main character in Holes, a 1998 novel and 2003 film. Four of the fictional Pokémon species have palindromic names in English (Eevee, Girafarig, Ho-Oh, and Alomomola).
The 1970s pop band ABBA is a palindrome using the starting letter of the first name of each of the four band members.

A palindrome number

A palindromic number is a number whose digits, with decimal representation usually assumed, are the same read backward, for example, 5885. They are studied in recreational mathematics where palindromic numbers with special properties are sought. A palindromic prime is a palindromic number that is a prime number, for example, 191 and 313.
The continued fraction of √n + ⌊√n⌋ is a repeating palindrome when n is an integer, where essentially, for any positive x, ⌊x⌋ denotes the integer part of x.

The question of whether Lychrel numbers exist is an unsolved problem in mathematics about whether all numbers become palindromes when they are continuously reversed and added. For example, 56 is not a Lychrel number as 56 + 65 = 121, and 121 is a palindrome. The number 59 becomes a palindrome after three iterations: 59 + 95 = 154; 154 + 451 = 605; 605 + 506 = 1111, so 59 is not a Lychrel number either. Numbers such as 196 are thought to never become palindromes when this reversal process is carried out and are therefore suspected to be Lychrel numbers. If a number is not a Lychrel number, it is called a “delayed palindrome” (56 has a delay of 1 and 59 has a delay of 3). In January 2017 the number 1,999,291,987,030,606,810 was published in OEIS as A281509, and described as “The Largest Known Most Delayed Palindrome“, with a delay of 261. Several smaller 261-delay palindromes were published separately as A281508.

Remarkably, a 2018 paper has demonstrated that every positive integer can be written as the sum of three palindromic numbers in every number system with base 5 or greater.

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