20041031, 03:03  #1 
"Mike"
Aug 2002
2^{3}·3·331 Posts 
Smooth?
In several threads I have come across a term that I do not understand:
"smooth group order" What does this mean? Not only do I not know what "smooth" means, I have no idea what a "group order" is... Thanks! 
20041031, 08:03  #2  
Aug 2004
2·5·13 Posts 
Quote:
A group is an abstract mathematical object consisting of a set and an operation that produces a member of the set from pairs of members of the set, and which satisfies 4 axioms (I won't give all the details here). The set could be finite or infinite, but in the case that it is finite, then the number of elements in the set is called the "order" of the group, or the "group order". A simple example of a group is the set of integers {1, ..., p1}, p a prime number, with the operation being multiplication modulo p, and in this case the group order is p1. So, "smooth group order" refers to a finite group whose order is smooth. One place this comes up is in the P1 factorisation method, where the success of the method depends on the group in the example described above having smooth group order. HTH Chris 

20041031, 08:07  #3  
Bamboozled!
"πΊππ·π·π"
May 2003
Down not across
3^{3}·389 Posts 
Quote:
The order of a group is the number of elements in it. It's just an integer The term "smooth" when applied to an integer means that the integer may be factored entirely into small primes. This, of course, begs the question of what is meant by "small". Frequently it can be deduced from context. Where greater precision is needed, the term "Bsmooth" is generally used. A Bsmooth integer has all its prime factors less than or equal to B. So, for example, 128 is 5smooth (indeed, it's 2smooth), as are 125 and 120, but 121 is not 5smooth, though it is 11smooth. Paul 

20041031, 14:33  #4 
"Mike"
Aug 2002
2^{3}×3×331 Posts 
Many thanks!
I actually took "finite mathematics" which dealt mostly with groups, sets and stuff like that, but I totally forgot about it! So if I understand right, "smooth" works like this: 128 = 2Γ2Γ2Γ2Γ2Γ2Γ2 < 2 smooth 125 = 5Γ5Γ5 < 5 smooth 120 = 2Γ2Γ2Γ3Γ5 < 5 smooth 121 = 11Γ11 < 11 smooth Is it safe to say that you must have the absolute factorization before you can assign a "smooth" value? I'm not too sure where the "b" is coming from when you mention "bsmooth"... Finally, what is the correct way to write out a factorization? I've seen it done several ways but I imagine there is an accepted proper way... 
20041104, 16:21  #5  
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}×3×641 Posts 
Quote:
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20041104, 18:20  #6 
Sep 2002
2×131 Posts 
Many Mersenne numbers have a factor that is Msmooth
since M is the largest factor of p1 ex: 2^291 has factor 233 = 2.2.2.29+1 Joss Last fiddled with by jocelynl on 20041104 at 18:24 
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