02 Sep '03 17:02>
S(d*2^a) = S(d)/2, of course, provided d is odd. Maybe I mentioned, maybe not. If d is even, say d=2, then S(d) = N/2. But S(4) = N/2 as well. Just to clarify.
Originally posted by royalchickenSorry, but I'm still not convinced.
THe answer is 1/3. Good going iamatiger!
Here's my proof. Let S(d) be the number of rationals in (0,N] with denominator d for some integer N.
Clearly, we are counting pairs (m,d) such that gcd(m,d)=1. If d is prime, then:
S(d) = N(1-1/d)
If d has k distinct prime factors, then:
S(d) = N(1-1/p1 - 1/p2 - ...-1/pk + 1/p1p2+...+1/p1p2p3p4 ...[text shortened]... t being even guarantees the presence of a prime factor, 2, while being odd guarantees nothing.
Originally posted by FiathahelNo, Fiathahel. All I proved is that IN ANY FINITE INTERVAL (for any finite maximum denominator), S(d) = 2S(2d) for odd d. S(d) counts the number of rationals of denominator d with a numerator less than or equal to N. For example, let d = 3 and N = 5. Then we can use the set of rationals {1/3, 2/3, 4/3, 5/3}. Thus S(3) = 4 in this case. Now for 2d = 6, we can have {1/6, 5/6}, so S(6) = S(3)/2. In general, as was shown, for any integer N and odd denominator d, there are twice as many rationals with numerator at most N with denom. d as there are for denom. 2d. Your argument doesn't apply here, since the second set allows larger numerators than the first.
Sorry, but I'm still not convinced.
You said:
'Thus if d is odd, then for all a, S(d*2^a) = S(d)/2.'
This is true of course, but then you compare an odd k with 2k. and thus also {1,3,5,7,9,11} with {2,6,10,14,18,22} and conclude that the second set 'makes' half as many rationals as the first. While you want to look at {1,2,3,...,11,12}. Or eve ...[text shortened]... with an infinite mumber of rationals which each 'makes' half as many rationals as k.
Originally posted by royalchickenWhat was wrong with my proof anyway, if we are quibbling about proofs? I can restate it a bit more clearly if you like...
No, Fiathahel. All I proved is that IN ANY FINITE INTERVAL (for any finite maximum denominator), S(d) = 2S(2d) for odd d. S(d) counts the number of rationals of denominator d with a numerator less than or equal to N. For example, let d = 3 and N = 5. Then we can use the set of rationals {1/3, 2/3, 4/3, 5/3}. Thus S(3) = 4 in this case. Now for 2d ...[text shortened]... com/pdp10/hbaker/hakmem/number.html
it does not give a proof, but does give the same result.
Originally posted by royalchickenWhat your prove says is that lim S(1)+S(3)+S(5)+S(7)+...+S(2n+1) = lim 2*[S(2)+S(6)+S(10)+S(14)+...+ S(4n+2)]. And not lim S(1)+S(3)+S(5)+S(7)+... S(2+1n) = 2*[S(2)+S(4)+S(6)+S(8)+S(10)+S(14)+...+S(2n+2)]. Or I'm missing something?
No, Fiathahel. All I proved is that IN ANY FINITE INTERVAL (for any finite maximum denominator), S(d) = 2S(2d) for odd d. S(d) counts the number of rationals of denominator d with a numerator less than or equal to N. For example, let d = 3 and N = 5. Then we can use the set of rationals {1/3, 2/3, 4/3, 5/3}. Thus S(3) = 4 in this case. Now for 2d ...[text shortened]... com/pdp10/hbaker/hakmem/number.html
it does not give a proof, but does give the same result.
Originally posted by Fiathahellim S(1)+S(3)+S(5)+S(7)+...+S(2n+1) does not converge. I think that you're forgetting that S is functionally related to TWO numbers, namely the denominator d and the maximum numerator N. My proof merely says that in any interval for which the upper (integral) bound is the maximum numerator in a set of rationals, there are twice as many odd denominator rationals as even denominator rationals. The ratio in question is:
What your prove says is that lim S(1)+S(3)+S(5)+S(7)+...+S(2n+1) = lim 2*[S(2)+S(6)+S(10)+S(14)+...+ S(4n+2)]. And not lim S(1)+S(3)+S(5)+S(7)+... S(2+1n) = 2*[S(2)+S(4)+S(6)+S(8)+S(10)+S(14)+...+S(2n+2)]. Or I'm missing something?
Originally posted by royalchickenThanks for the encouragement!
I'd like to see a restatement, because I think it's good.
Originally posted by royalchickenNow I'm confused. Why not count the number of d-denominator rationals in (0,N]? Also, why not set N=1, if it saves confusion?
Ok, I made a typo, but not exactly the one Fiathahel meant. If S(d) referred to the number of d-denominator rationals in (0,N] as I said, then indeed
S(d) = dN(1-1/p1)....(1-1/pk) as Fiathehel said. However, I meant to say that S(d) counts rationals with NUMERATORS IN (0,N], ie those rationals in (0,N/d] so that the rest is as I said.
S(d) = N(1- ...[text shortened]... h makes S(d) pretty meaningless since we can't make comparisons among different denominators.
Originally posted by iamatigerBy making m/n this way, you allow them to be like 4/2 as well as 6/3 while they are the same. But I don't know if that's makes any difference, cause you count all rationals infinite times this way.
Thanks for the encouragement!
......
Therefore the probability that a random rational number will have an even divisor is also 1/3
Originally posted by FiathahelI thought about that, it would change the odds I think but since m and n are mutually prime each fraction is only counted once. Fractions that are not in their lowest form can by identified by m & n having a common factor that can be divided out - which due to the way m&n are constructed, cannot be the case here.
By making m/n this way, you allow them to be like 4/2 as well as 6/3 while they are the same. But I don't know if that's makes any difference, cause you count all rationals infinite times this way.
You can't pick an integer X random ...[text shortened]... of for X in [0,Q] for all Q. So it is also true for the Q --> inf.
Originally posted by iamatigersorry, my fault. The construction methode works a bit different then I thought.
I thought about that, it would change the odds I think but since m and n are mutually prime each fraction is only counted once. Fractions that are not in their lowest form can by identified by m & n having a common factor that can be divided out - which due to the way m&n are constructed, cannot be the case here.