Chebyshev Said It and Ill Say It Again
Number of Primes Between n and 2n
- Thread starter 1MileCrash
- Beginning date
But, as n grows larger and larger, the range of numbers between n and 2n grows larger advertizement larger.
Do these two counteract each other? Does this crusade the number of primes between n and 2n to stay relatively consistant, increase, or just shrink regardless?
Answers and Replies
π(n) ~ north/ln(n), and then π(2n) - π(north) ~ 2n/ln(2n) - n/ln(n) which is approximately n/ln(n).
Using the prime theorem, this can be estimated.
?(northward) ~ n/ln(n), so ?(2n) - ?(n) ~ 2n/ln(2n) - northward/ln(n) which is approximately northward/ln(n).
So as north goes to infinity, the number of primes between n and 2n goes to infinity?
Yes, in the sense that n/logn -> infinity equally n -> infinity. (Simply utilise L'Infirmary'southward to see this.)
Aye, that'south what i did.
I don't really like the quote past chebyshev and erdos considering it is scrap misleading.
But what is misleading nearly it?
Aye, that's what i did.I don't really similar the quote by chebyshev and erdos because it is scrap misleading.
I tin can empathise that the quote is entirely inappropriate to the thread, only calling it misleading seems far-fetched to me.
Well, the quote is by Erdos. Come across, Chebyshev proved that there was e'er a prime between n an 2n using a rather technical argument. Erdos proved it (I think when he was xix or so) using a combinatorial argument, then he said "Chebyshev said it I'll say it again, there's always a prime between due north and 2n."Just what is misleading about information technology?
Well, we agreed above that the number of primes between n and 2n goes to infinity as n goes to infinlty. Thats a lot of primes. The Erdos quote says there is always a prime number, that does not sound like a lot, so IMHO the quote focuses on the triviality of pocket-sized n and disregards the greater truth for big due north.
I just posted the quote as a joke in the start place.
Well sure, what we said was a sharper upshot than Bertrand's Postulate. I yet don't meet how this is misleading (though I will agree information technology really doesn't have much to do with the the thread.) The fact that at that place is ALWAYS a prime between n and 2n for all n is interesting on its ain, espicialy when you consider that I can give you an arbirarily long list of consecutive blended integers.I just posted the quote equally a joke in the start place.
Yep simply that list of r consecutive composite integers must first at some integer due north and my gut tells me that north is much larger than r for big r. For instance if r is 100 and then northward is going to be much larger than 100 and of course 2n-due north is the number of integers in this interval. what would be 100/n? my gut tells me information technology's about 0
so in a simillar fashion r/due north goes to zero as r goes to infinity. Remember that r is the number of consecutive composite integers and north is the integer where this sequence starts.
Is my gut mistaken?
It's all good, you don't take to see it my mode, my opinions are not 'etched in stone'
But that'due south not really the bespeak. The fact that PNT, in a fashion, implies Chebyshev'south theorem does non, in whatsoever way, hateful that Chebyshev's Theorem is "misleading" or unimportant. And information technology certainly does not take away from the fact that Erdos came upward with a magnificent proof of information technology.
PNT as well implies that in that location are infinite primes, but this does non take anything away from Eucilid's proof of the aforementioned.
The sequence begins at (due north+1)!.
Not necessarily.
OK....
I was speaking of constructing a sequence like this:
(n+1)! + ii, (n+1)! + 3, ..., (north+1)! + n + 1
This is a listing of northward consecutive composites. When y'all say "not necessarily" do you mean that y'all know of another manner to construct an arbitrarily long list of composite numbers? If then, I'd exist interested. Or, do you mean that not every list of m consecutive composites has to be synthetic the fashion I describe? If this is the case, then I don't really care equally I never claimed that the Merely way to construct a list of n composites was the construction I mentioned in a higher place. Perhaps what I wrote in my last post was disruptive, but that'due south only because I thought nosotros understood that I was referring to creating a listing of size northward for a general northward, not a specific n. Certainly for "most" n, there are much lower sequences of n consecutive composites; but this doesn't tell us annihilation about a general n.
In that location are occasions that (northward+i)! -ane or (north+1)! +1 or both are composite likewise, then the sequence begins much sooner at (due north+1)! - (north+one), just I don't know if that was what agentredlum had in heed. Also, Yous must consider agentredlum's prior post where he said the sequence of r blended integers begins at some n. I think he had in mind so the lowest possible northward for r consecutive blended integers. Unfortunately your post was for due north consecutive composites rather than for r consecutive composites, but n consecutive composites is more than germane to the original posts.OK....I was speaking of constructing a sequence like this:
(n+ane)! + 2, (north+ane)! + three, ..., (n+1)! + n + ane
This is a listing of n sequent composites. When y'all say "not necessarily" exercise you mean that you lot know of some other mode to construct an arbitrarily long list of blended numbers? If and so, I'd be interested. Or, exercise you lot mean that not every listing of thousand consecutive composites has to be constructed the way I draw? If this is the case, and so I don't really care as I never claimed that the Just mode to construct a list of n composites was the construction I mentioned above. Maybe what I wrote in my last post was disruptive, simply that's only because I thought we understood that I was referring to creating a listing of size north for a general n, not a specific north. Certainly for "most" n, there are much lower sequences of north consecutive composites; but this doesn't tell u.s. annihilation about a general northward.
100 < #21 > 200
...
1000 < #135 > 2000
...
10000 < #1033 > 20000
...
100000 < #8392 > 200000
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