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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