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## 14937Re: [PrimeNumbers] Need help-what is Chebyshev's theorem about the distributi...

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• Jun 4, 2004
In a message dated 04/06/04 05:54:15 GMT Daylight Time,
billroscarson@... writes:

> It has been suggested to me that Chebyshev (I am not sure which one-
> apparently there are many)has a theory that says something like
> there is always one prime between n^2 and (n^2+2n). I cannot find
> it, or I don't recognize it when I do find it. Can anyone tell me if
> the theorem is as I stated, and where I can find it documented?
> Thanks for any help.
>

I think you must be thinking of Bertrand's postulate, which
"is that, for every n > 3, there is a prime p satisfying n < p < 2n-2.
Bertrand verified this for n < 3,000,000 and Tchebychef proved it for all n > 3 in
1850."
[Hardy & Wright (1979), p. 373]

If p[n] is the nth prime number, define
d[n] = p[n+1] - p[n]
Then your statement would be equivalent to the assertion that d[n] <=
2*p[n]^0.5.

This is probably true (for n "sufficiently large"), but has never been proved.
In fact, it is widely conjectured that d[n] < p[n]^0.5 except for a finite
number of cases.

If the Riemann hypothesis is assumed, it is (relatively!) easy to prove that,
if t is any fixed number > 0.5, then d[n] < p[n]^t except for a finite number
of cases.

By 1990 the strongest result unconditionally proved (by Mozzochi in 1986) was
that this statement is true with t any fixed number > 1051/1921.
[A..E.Ingham "The Distribution of Prime Numbers" (1990)]

Is this the latest state of play, anyone know?

-Mike Oakes

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