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Old 2004-04-17, 10:42   #1
juergen
 
Mar 2004

2910 Posts
Question least common multiple of numbers of the form a^x-1

Hello,

what is the form of the least common multiple of numbers of the form

v^(a*b*c) - 1, v^(a*b*d) - 1, v^(a*c*d) - 1 and v^(b*c*d) - 1 ?

I know that it is not just the product of these numbers.

Sorry unfortunately I can't use math symbols here. As I wrote this article I recognized that without mathematical symbols like the sum sing (sigma) it looks pretty much complicated. But I guess my question it is not that complicated in reality. So If you try to understand the following (which may be not necessary to answer the question anyway) then I suggest you to write it down on paper in the mathematical form using the sum sign. Sorry for this inconvenience. Why do they have dozens of different smiley simbols here, but no mathematical symbols in a forum which is about math?

I use Si(a,b,f(i)) for the sum sign with the variable i running from a to b. So Si(a,b,f(i)) would build the sum of all the results of f(i) for all the values of i which are between a and b => so Si(....) is just my textual representation of the Singma-Sign :o).

Using this notation for sigma the least common multiple

[v^(a*b*c) - 1; v^(a*b*d) - 1; v^(a*c*d) - 1; v^(b*c*d) - 1]

could be reduced by using polinom division (v^(ab)-1 is a factor of (v^(abc) - 1) as well als of (v^(abd) - 1) to the following:

[v^(a*b*c) - 1; Si(0 ,d-1, v^(i*a*b)); Si(0 ,d-1, v^(i*a*c)); Si(0 ,d-1, v^(i*b*c))]

but it seems that also

Si(0 ,d-1, v^(i*a*b)) and Si(0 ,d-1, v^(i*a*c)) and Si(0 ,d-1, v^(i*b*c))

have common divisors. Is there a formula for this common divisor?

Thank you very much in advance

Juergen Bullinger

Last fiddled with by juergen on 2004-04-17 at 10:45
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Old 2004-04-17, 10:53   #2
cyrix
 
Jul 2003
Thuringia; Germany

2·29 Posts
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Quote:
Originally Posted by juergen
[...] Sorry unfortunately I can't use math symbols here. As I wrote this article I recognized that without mathematical symbols like the sum sing (sigma) it looks pretty much complicated. [...]
Juergen Bullinger
You could use LATeX-Code, or attach a file, were you could use the symbols you need.

Cyrix
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Old 2004-04-17, 12:19   #3
juergen
 
Mar 2004

29 Posts
Default

Quote:
Originally Posted by cyrix
You could use LATeX-Code, or attach a file, were you could use the symbols you need.

Cyrix
Thank you for your suggestion. I don't know how to add files here, so I put a pdf file on a web server you can download it on

informatik.gmxhome.de/lcm.pdf

Kind regards

Juergen
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