“Shanghai tap water comes from the sea” is a classic sentence that everyone knows, and it reads the same forwards and backwards; such sentences are called “palindromes.”
In natural numbers, there are also some numbers with this characteristic, which are called palindromic numbers.
What is a palindromic number?
For any natural number n, if we reverse the order of its digits to get a new natural number m, and if m equals n, then n is a palindromic number.
For example:
For the natural number n=123454321, reversing the digits gives m=123454321, and since m=n, n is a palindromic number.
On the other hand, for the natural number n1=123456789, reversing the digits gives m=987654321, and since m does not equal n1, n1 is not a palindromic number.
The smallest palindromic number in natural numbers is 0. The digits 0, 1, …, 8, 9 are all palindromic numbers, and two-digit numbers like 11, 22, 33, …, 88, 99 are also palindromic numbers. There are even more palindromic numbers in three digits like 111, 121, …, 181, 191, 212, 222, …, 292, …, 919, 929, …, 989, 999.
Some natural numbers have additional interesting features aside from being palindromic. So what are some interesting palindromic numbers?
Square Palindromic Numbers
A natural number that is a palindromic number and also the square of another natural number is called a square palindromic number.
For example: 11 squared is 121, 22 squared is 484, and 26 squared is 676; thus, 121, 484, and 676 are all square palindromic numbers.
Additionally, there are some numbers whose squares are palindromic numbers:
For example:
1 squared is 1
11 squared is 121
111 squared is 12321
1111 squared is 1234321
…………
With the advancement of science and technology, aided by computers, it has been discovered that the proportion of palindromic numbers among perfect squares and perfect cubes is much higher than that among general natural numbers.
For instance: 11 squared is 121, 22 squared is 484, 7 cubed is 343, 11 cubed is 1331, and 11 to the power of 4 is 14641…… and so on; these numbers are all palindromic numbers.
So how do we find palindromic numbers among natural numbers?
There is an interesting algorithm for finding palindromic numbers:
Generally, for a natural number n, repeat the following steps until a palindromic number is obtained:
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Reverse the digits of the natural number n to get the reverse number m, then calculate m+n.
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If m+n is not a palindromic number, assign the value of m+n to n and repeat step 1.
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Repeat the above steps until m+n is a palindromic number. If we have repeated the steps several times, we say we have obtained a palindromic number after a few calculations.
Let’s see which of these natural numbers can obtain a palindromic number after a few calculations:
1. 28+82=110 110+011=121 28 gets a palindromic number after 2 steps.
2. 96+69=165 165+561=726 726+627=1353 1353+3531=4884; 96 gets a palindromic number after 4 steps.
Some natural numbers cannot obtain a palindromic number even after many calculations, but we cannot prove that they will never obtain a palindromic number after further calculations. Numbers that seem unable to form a palindromic number but we cannot determine whether they eventually can are called Lychrel numbers. If further calculations yield a palindromic number, then that number is not a Lychrel number. The first discovered Lychrel number is 196.
Some natural numbers need many calculations to finally obtain a palindromic number.
A 19-digit number 1,186,060,307,891,929,990 needs 261 steps to obtain a palindromic number; 261 steps is currently the world record for the number of calculations to obtain a palindromic number.
We will stop our introduction to palindromic numbers here; how to program to find palindromic numbers will be discussed in future articles. There are also some palindromic equations that you can check out if you are interested.
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