My first experience of problem setting. It feels good when you see people think about your problem and come up with better solutions than yours 😛

This problem is about math and primes. The link is this.

# Category Archives: Number Theory

# 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.