Palindromes

Standard
What is a Palindrome? 
A palindrome is a word, which reads backwards the same as it does forwards. Well known examples are Anna or radar.
You can apply this principle to numbers. For instance 1001 or 69896 are palindromes.

Counting the Palindromes 
All the digits are palindromes (1,2,3,…,9).

There are also 9 palindromes with two digits (11,22,33, …,99).

You can find to every two-digit number one, and only one number with three digits and with four digits.
For example: For the number 34 there are 343 and 3443.
You can conclude that there are 90 palindromes with three and also 90 palindromes with four digits.

You can find to every three-digit number one, and only one number with five digits and with six digits.
For example: To the number 562 there are 56265 and 562265.
You can conclude that there are 900 palindromes with five and 900 palindromes with six digits.

You have 9+9+90+90+900+900 = 1998 palindromes up to one million. That’s 0,1998 %. About every 500th number is a palindrome. 

Position of the Palindromes
But they are not spread over all numbers regularly. This shows the picture below, which includes the first 1000 numbers.

Products with the Digit 1
11×11 = 121
111×111 = 12321
1111×1111 = 1234321 

111 111 111 x 111 111 111=12345678987654321

again…

11×111 = 1221
111×1111 = 123321
1 111×11111 = 12344321

111 111 111 x 1 111 111 111=123456789987654321

I suppose that all products with the digit 1 are palindromes, if one.factor has at the most 9 digits. 
All palindromes have the shape 123…..321.

 

Prime Numbers among the Palindromes
All palindromic primes with 3 digits:

101
131
151
181
191
313
353
373
383
.
727
757
787
797
.
919
929
.
.
.

There are no primes with 4 digits. They all have the factor 11. (Example:4554=4004+550=4×1001+550=4x91x11+11×50=11x(4×91+50)
There are 93 primes with 5 digits.
There are no primes with 6 digits. They all have the factor 11.
There are 668 primes with 7 digits.

 

196-Problem
Pick a number. Add the number, which you must read from the right to the left (mirror number), to the original number. Maybe the sum is a palindrome. If the sum isn’t a palindrome, add the mirror number of the sum to the sum. Maybe you have a palindrome now, otherwise repeat the process. Nearly all numbers have a palindrome in the end. 
Example: 49       49+94=143       143+341= 484 !
There are some numbers, which have no palindromes. The lowest one is 196. But the proof is still missing.

6 thoughts on “Palindromes

  1. Great post. One thing I found when thinking about palindromes is, if u reverse any number and find the difference between the two you always get a number that is dividable by 9.

    21 – 12 = 9
    321 – 123 = 198

    Ofcourse it’s not true for same palindromes that are same.
    Just a thought, does it have any significance with palindromes?

    • At first I’m really sorry for this late reply🙂 btw I am curious about what you said for palindromes. Palindromes remain same if you reverse them. Then the difference would always be 0, isn’t it?

      • Yes, that’s true, I just didn’t look at it right then. But what is really amazing is that Number ~ Reverse(Number) = Always divisible by 9 or more specifically by the prime 3. Does that hold for all cases?

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