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1. Cool Number 68886

Just saw "Registered Members: 68886"
The number seemed to be cool.
It reads the same forwards and backwards.
Any mathematician here who can help on what these type of numbers are called?

2. Re: Cool Number 68886

Palindromes.

(You can have them in words or in numbers.) There is a relatively famous "unsolved number problem" called the palindrome conjecture which says that you can start with any number greater than 10, reverse it, and add the two original numbers. If the resulting number is not a palindrome, repeat the procedure with the sum until the resulting number is a palindrome. Yet some numbers do not work, like the famous "196 Problem". (To a lot of people, this simply suggests that the conjecture is false.) The last (recent) years we had that were palindromes were 1991 and 2002.

The number you reference (68886) falls in the sequence that is generated by:
0,22,252,2332,20002,26062,29392,63736,68886,270107 2,2783872

(By the way, you can have palindromic squares, palindromic cubes, palindromic nonagonals, etc. But as far as numbers go these are actually not the most interesting to mathematicians.)

3. Re: Cool Number 68886

Originally posted by Jeff Nyman:
Palindromes.

(You can have them in words or in numbers.)
There is a relatively famous "unsolved number problem" called the palindrome conjecture which says that you can start with any number greater than 10, reverse it, and add the two original numbers. If the resulting number is not a palindrome, repeat the procedure with the sum until the resulting number is a palindrome. Yet some numbers do not work, like the famous "196 Problem". (To a lot of people, this simply suggests that the conjecture is false.) The last (recent) years we had that were palindromes were 1991 and 2002.

The number you reference (68886) falls in the sequence that is generated by:
0,22,252,2332,20002,26062,29392,63736,68886,270107 2,2783872
<font size="2" face="Verdana, Arial, Helvetica">How is that sequence generated, Jeff?
I can see those are all palindromes,
but I don't see an obvious means of calculating the next one...
(By the way, you can have palindromic squares, palindromic cubes, palindromic nonagonals, etc. But as far as numbers go these are actually not the most interesting to mathematicians.)
<font size="2" face="Verdana, Arial, Helvetica">1991 is rather interesting, since by rotating it you get 1661, which (of course) is also a palindrome.
In general, all palindromes made out of digits 0, 1, 6, 8 and 9 behave like this.
(Yep, I'm also a mathematician [img]images/icons/wink.gif[/img] )

Leo

4. Re: Cool Number 68886

Originally posted by Leo_de_Wit:
How is that sequence generated, Jeff?
I can see those are all palindromes,
but I don't see an obvious means of calculating
<font size="2" face="Verdana, Arial, Helvetica">Palindromic quasipronics of the form n(n+x). More specifically, the form n(n+9). A more complete sequence in that sense:

0,22,252,2332,20002,26062,29392,63736,68886,270107 2,2783872,
2884882,29122192,253080352,289050982,25661316652,
237776677732,2393677763932,215331808133512,2187599 69957812,
225588939885522

5. Re: Cool Number 68886

Originally posted by Jeff Nyman:
[SNIP].... But as far as numbers go these are actually not the most interesting to mathematicians.)
<font size="2" face="Verdana, Arial, Helvetica">Particularly since if you convert them to binary or some other base they likely lose their palindromicity (is that a word?). There's a challenge, come up with the largest number you can find that is palidromic in binary, base 6, base 10 and hexadecimal. [img]images/icons/smile.gif[/img]

"Able was I ere I saw Elba."

6. Re: Cool Number 68886

Once you convert, you can certainly get beyond the ability to be palindromic (particularly with hexadecimal). Here is one recent number found (in binary, of course):

1010010001 1101011100 1110010101 0010100001 0100000010 1000010100 1010100111 0011101011 1000100101

That becomes 795280629691202196926082597. (So you cover bases 2 and 10.) With base 6 things are a little tricker, but the common element you can use is 13131 (base 6) is a base 6 palindrome representation of 1999 (base 10) and then factor things like that. Usually people use base 4 as a conversion factor. (So, for example, 91 is the base 10, 231 is base 4 coverted, 363 is the palindrome.) If you consider the notion of consecutive digit sequences, then provided that the highest digit in the consecutive number sequence is even, any resulting palindromes are always divisible by 11. Due to some oddities of what this results in, you can multiply any of the original palindromes of a given sequence and get hexadecimal palindromes as such:

123456787654321 * 11 = 13579ACEECA97531

That is one of the more common examples, admittedly. Yet it is possible to find palindromic convergences between bases. Consider 373 in base 10. This has the form 565 in base 8, 11311 in base 4, and 454 in base 9. A lot of times you want to look for the "limiting cases." For example, the smallest number that never becomes palindromic in binary is 10110. In base 16 I think this is 413. But finding the largest is always a tricky proposition because even if you find one, then how do you know there is not another beyond it? (Well, actually, if you could come up with a mathematical limiting proof, you could do just that.)

By the way, palindromes are of great interest to molecular and cellular biologists such as when looking at DNA methylase or restriction endonuclease because you want to look for sequences that are the same forward and backward. (Restriction enzymes, for example, recognize base 4, base 6, and base 8 sequences.) This is important because you want to look for repeats in any given nucleotide sequence. Basically you will have two general types of palindromes in this context: palindromes on opposite strands of the same section of DNA helix (restriction enzymes) and what are known as inverted repeats (transposons, for example).

wow

8. Re: Cool Number 68886

Thank heavens, Jeff is back. Now i can read all his posts thrice (that's the # of times it takes me to understand) when i am bored.

9. Re: Cool Number 68886

I'd be the #1 Glad person he's back [img]images/icons/smile.gif[/img]

10. Re: Cool Number 68886

A Big WOW. I guess there must a formula that explains everything in this universe [img]images/icons/smile.gif[/img]

Thanks

Rahul

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