Hello and a question

Hi all, thanks for allowing me to join this group.

I have a question: I am relatively uneducated in math, but fascinated by it and always have been. I want to know if there is a name for a field of math in which numbers are studied so that one can, 'reduce,' the sum of the product of the number when it has been raised to a certain power.

As an example, start with the number 8, then multiply by 2, resulting in 16. 1 and 6 make 7.

I have found that each number has a series of repeating patterns when studying the solutions. As an example, 8 has a repeating pattern of 7, 6, 5, 4, 3, 2, 1, 9, 8, 7, 6, 5, 4, 3, 2, 1...

I haven't studied these far enough yet to find if the pattern breaks off after a certain power. (I just got into this over the last couple of days.)

Years ago I studied these for many numbers a lot. I think I threw most of my old work out because I have a messy desk and it felt petty. It was fun though, and I'd do it again if I had time.

Well.. multiplying by a number is not raising it to any power, unless you multiply it by itself, in which case it is raised to the power of 2.

I guess you would study this sort of thing in number theory, but it is not exactly my main line of work..

What you are doing is taking a number and then multiplying it by 2,3,4...n and then taking the sum of the digits?? And you want to now if all numbers have repeating patterns in the sum of the digits?

Thanks for the correction, and I'm just looking for the name for a field of study regarding this phenomenon.

I don't think that it has a name on its own.. It would just be number theory.. but I guess you can look for "number theory + sum of  the digits" on blackle (the energy saving version of google) ..

Have you considered that there may some kind of inductive method of proving that the repeating patterns will persist?

My guess is also that it would be Number Theory, but I really don't know.

Thanks. I have a bit, but I'm not knowlegable enough I guess right now to prove or disprove. It fascinates me though, as some have alternating sequences, like 5 for example. The pattern goes: 1,6,2,7,3,8,4,9,5,1...Some of the sequences go up, some down. Some stay the same, like 9 - every result is 9.

I understand what strangequark felt, though. It doesn't hold much practical implication, I suppose. But I'm still studying it, and just wanted to know how I can find out more about it.

Do you have a way of proving/disproving if they would persist?

Hehe.. I am trying to look into the problem right now.. couldn't you have postponed your question till after I had done all my matlab simulations!? =)

Sorry, and thanks!

But take 18, and add 1 and 8 = 9...this has to be done with many of the solutions to stay within the pattern, but the pattern still holds...

Some I've stopped at 10, as with 9...but others I pursued further, as with 7, so far I've gone to 19, and found the pattern to be 5, 3, (10)1, 8, 6, (13)4, (11)2, 9, 7...

No, no, I've worked this a lot. Remember in 18, 1+8 still =9, it can always be broken down further. Btw, I have found that 9 is definitely the most common number. Oh, it will be practical if it's not already.

Several weeks ago I pursued this with, (I think) 9 and took it to 9 to the 21st and found the pattern stopped...(don't have my notes with me right now, but I think that's right.)

No, I think it was 21 as an exponent

See my message. I can post it here I suppose.

Nine times anything will always have a sum of the digits that is 9... I think you learn that in 4 th grade elementary...

 Yes, yes, of course. But there are other numbers too that relate to 9 with these net sums all the time.

(Like I said, I'm fairly uneducated with regard to this)

since ... 9x2 = 9 + 9 = 18, 1=8 =9

9x3 = 9 + 9 + 9 = 18 + 9, 1+8 =9 => 9 +9 = 18, 1+8=9

nx9 = 9 + ... + 9 = (n-1)/2 x 18 + 9 if n is odd, and n/2 x 9 if n is even ...

Ups sorry ... it is of course n/2 x 18 on that last part...

You don't have to be educated in mathematics to have fun with it .. =) .. I spent much of my time in elementary school playing around with numbers instead of paying attention.. at one point I discovered another way of doing multiplikation with two number that are bigger than 9 .. Later I found out that Russian merchants had been doing it the same way for hundreds of years... #) .. 

Thanks for the comments.