20130202, 03:46  #1  
"Frank <^>"
Dec 2004
CDP Janesville
2·1,061 Posts 
Is a search for aliquot 3cycles feasible?
From Perfect, Amicable and Sociable Numbers by Song Y. Yan comes this, from the chapter Algrebraic Rules for Aliquot kCycles:
Quote:
Quote:
The only potential problem I see is a little further down the page where he says Quote:


20130203, 01:47  #2 
"Robert Gerbicz"
Oct 2005
Hungary
5×17^{2} Posts 
"Is a search for aliquot 3cycles feasible?"
Heuristics gives O(1/u) probability for each p1,p2,p3 number to be prime for fixed v and O(1/v) for p. So the probability that you will find a chain is (assume that these p,p1,p2,p3 are primes "almost" indepedently) O(1/(u^3*v)) summing this for all (u,v) pairs (use that 2u+1==0 mod v) gives a finite (small) value. This also shows that if there is no chain for very small u,v pairs then it is a hopeless search. Last fiddled with by R. Gerbicz on 20130203 at 01:49 
20130203, 17:15  #3 
Sep 2009
5×401 Posts 
Will all aliquot 3cycles have to meet those criteria? If not what criteria would they meet?
Chris 
20130207, 11:57  #4  
"Frank <^>"
Dec 2004
CDP Janesville
84A_{16} Posts 
Quote:
Aliquot kcycles are defined earlier: Quote:
Quote:
Quote:


20130207, 13:25  #5  
"Forget I exist"
Jul 2009
Dumbassville
2^{6}·131 Posts 
Quote:
so is there anything that can be said of v in this situation ? n ? ( and yes I know n could simplify it more, or at least make it down to one variable) Last fiddled with by science_man_88 on 20130207 at 13:36 

20130207, 17:01  #6 
"Forget I exist"
Jul 2009
Dumbassville
2^{6}·131 Posts 
is what I get reducing to just n in all equations. 
20130208, 01:27  #7  
"Frank <^>"
Dec 2004
CDP Janesville
2×1,061 Posts 
Quote:
Hint: If [i]p[/i] is prime we [u]know[/u] that [i]v[/i] is [u]one of only [STRIKE]47[/STRIKE] 48 values![/u] 

20130208, 01:33  #8 
"Forget I exist"
Jul 2009
Dumbassville
2^{6}·131 Posts 

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