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 2006-01-24, 16:56 #1 mfgoode Bronze Medalist     Jan 2004 Mumbai,India 22·33·19 Posts Random numbers and proper factors In my current reading of Maths I have come across the following problem which I am at a loss to ascertain. Take three positive whole numbers at random. What is the chance they have no proper factor in common? Answer around 83%- to be precise, 0.83190737258070746868......:surprised Can anyone elucidate? Mally
2006-01-24, 17:11   #2
R.D. Silverman

Nov 2003

164448 Posts

Quote:
 Originally Posted by mfgoode In my current reading of Maths I have come across the following problem which I am at a loss to ascertain. Take three positive whole numbers at random. What is the chance they have no proper factor in common? Answer around 83%- to be precise, 0.83190737258070746868......:surprised Can anyone elucidate? Mally
Yes. I can.

 2006-01-24, 17:37 #3 akruppa     "Nancy" Aug 2002 Alexandria 2,467 Posts I suppose the analysis will be similar to that for two randomly chosen integers. We had that one before in this forum. The result for two random integers was 1/zeta(2), the results for 3 random integers should be 1/zeta(3) which matches your constant. Alex
2006-01-24, 17:59   #4
mfgoode
Bronze Medalist

Jan 2004
Mumbai,India

22·33·19 Posts
Random numbers and proper factors.

Quote:
 Originally Posted by akruppa I suppose the analysis will be similar to that for two randomly chosen integers. We had that one before in this forum. The result for two random integers was 1/zeta(2), the results for 3 random integers should be 1/zeta(3) which matches your constant. Alex
Okay Alex well and good.
Could you link this number with another well known number in electro dynamics ? Its amazing !
Mally

Last fiddled with by mfgoode on 2006-01-24 at 17:59

 2006-01-24, 18:18 #5 alpertron     Aug 2002 Buenos Aires, Argentina 5×269 Posts The quotient $\zeta {(n)}/\pi^n$ for n even is always rational, but for odd values of n, such as in this case there is no known explicit finite expression using elementary functions and constants. Last fiddled with by alpertron on 2006-01-24 at 18:19
2006-01-25, 15:39   #6
mfgoode
Bronze Medalist

Jan 2004
Mumbai,India

22×33×19 Posts

Quote:
 Originally Posted by R.D. Silverman Yes. I can.
Kindly explain the derivation as I have requested.
Thanking you,
Mally

2006-01-25, 15:47   #7
mfgoode
Bronze Medalist

Jan 2004
Mumbai,India

22·33·19 Posts

Quote:
 Originally Posted by alpertron The quotient $\zeta {(n)}/\pi^n$ for n even is always rational, but for odd values of n, such as in this case there is no known explicit finite expression using elementary functions and constants.

We know that Z(2) = (pi^2)/6. Would you call this a rational number?
Similarly Z(4) = (pi^4)/90.
Mally

2006-01-25, 17:23   #8

"Richard B. Woods"
Aug 2002
Wisconsin USA

22·3·641 Posts

Quote:
 Originally Posted by mfgoode We know that Z(2) = (pi^2)/6. Would you call this a rational number?
Note that Alpertron was talking about the quotient (Z(n)/pi^n) being rational, not Z(n) itself being rational.

 2006-01-25, 23:49 #9 John Renze     Nov 2005 608 Posts What does "two randomly chosen integers" mean? You can't put a uniform distribution on the entire set of integers, so there is some interpretation involved. Specifically, you seem to be using the following heuristic: Let $n$ be an random integer and $p$ be a prime. Then $Pr(p | n) = 1/p$. This is perfectly sensible and number theorists use this kind of reasoning all the time. I was wondering, though, how you formulate this precisely,.
 2006-01-26, 00:17 #10 John Renze     Nov 2005 24·3 Posts Thinking about this some more... you are making some statement over the interval (0, n) and then take a limit as n ->\infty. Is there somewhere easily accessible these details are written down? I'm still grappling with whether this really corresponds to "random integers," but I'll do the philosophical musings on my own time. Simpler question: How do I put this as a postscript to my previous post, rather than a new posting?
2006-01-26, 16:28   #11
mfgoode
Bronze Medalist

Jan 2004
Mumbai,India

80416 Posts
Random numbers and proper factors.

Quote:
 Originally Posted by cheesehead Note that Alpertron was talking about the quotient (Z(n)/pi^n) being rational, not Z(n) itself being rational.
Thank you cheesehead that makes it much clearer and precise.
These rational numbers I take it are of even no.s and no odd has been found to describe the quotient rationally.
Mally

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