Tool to check, generate, and list palindromic numbers online. Discover their properties, examples, and applications in mathematics.
Palindromic Numbers - dCode
Tag(s) : Arithmetics
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Palindromic Numbers
Palindromic Number Checker
Palindromic Number Search
Lychrel Number Tester
A Lychrel number is a natural number that cannot form a palindrome through the iterative process of reversing and adding its digits (also called algorithm 196)
Answers to Questions (FAQ)
What is a palindromic number? (Definition)
A palindromic number (or digital palindrome) is a mirror number that remains the same when the order of its digits is reversed.
Example: 121 or 12300321 are palindromic numbers
How can you check if a number is palindromic?
To check if a number is palindromic, compare the first digit from the beginning with the first digit from the end of the number.
If the two digits are different, then the number is not a palindrome; otherwise, repeat the process with the second digit from the beginning and the last.
Once half the number has been tested, if all the digits are identical, then the number is palindromic.
Negative numbers are generally excluded, or the minus sign is ignored.
What are the variations of palindromic numbers?
Palindromic numbers can be classified according to their mathematical properties:
— Palindromes in different bases
A number can be palindromic in one base (such as base 10, the decimal system) but not in another.
Example: The number 121 (base 10) is palindromic in base 10, but also in bases 3, 7, and 8.
— Prime palindromes
A palindromic prime number is a prime number that is also a palindrome: 2, 3, 5, 7, 11, 101, 131, 151, etc. See OEIS A002385 here
A Lychrel number is a number that never becomes palindromic, even after a large number of iterations of the 196 algorithm (adding a number to its inverse).
How to generate palindromic numbers?
To obtain a palindrome with an even number of digits: take a number, duplicate it, reverse it, and then concatenate the two digits.
Example: 123 becomes 321 reversed, and concatenation gives 123321.
For palindromes with an odd number of digits: take a number, duplicate it but without its last digit, reverse the resulting number, and then concatenate the two digits.
Example: 123 becomes 12321.
What are the largest known palindromic numbers?
There is no largest palindromic number, because it is always possible to construct a larger palindromic number from a given palindromic number.
What are the first few palindromic numbers?
Here is the list of palindromic numbers up to 1000:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, 242, 252, 262, 272, 282, 292, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 404, 414, 424, 434, 444, 454, 464, 474, 484, 494, 505, 515, 525, 535, 545, 555, 565, 575, 585, 595, 606, 616, 626, 636, 646, 656, 666, 676, 686, 696, 707, 717, 727, 737, 747, 757, 767, 777, 787, 797, 808, 818, 828, 838, 848, 858, 868, 878, 888, 898, 909, 919, 929, 939, 949, 959, 969, 979, 989, 999
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- Palindromic Number Checker
- Palindromic Number Search
- Lychrel Number Tester
- What is a palindromic number? (Definition)
- How can you check if a number is palindromic?
- What are the variations of palindromic numbers?
- How to generate palindromic numbers?
- What are the largest known palindromic numbers?
- What are the first few palindromic numbers?
