 mersenneforum.org > Math P(n+1)<(sqrt(P(n))+1)^2
 Register FAQ Search Today's Posts Mark Forums Read 2005-10-26, 13:32 #1 Crook   Nov 2004 24 Posts P(n+1)<(sqrt(P(n))+1)^2 Let P(n) denote the n-th prime number. Then, does anybody have an idea why P(n+1)<(sqrt(P(n))+1)^2 is true? This would be a lower bound than the Tchebycheff result that there is always a prime between n and 2n. Regards.   2005-10-26, 13:49   #2
R.D. Silverman

"Bob Silverman"
Nov 2003
North of Boston

11101010101002 Posts Quote:
 Originally Posted by Crook Let P(n) denote the n-th prime number. Then, does anybody have an idea why P(n+1)<(sqrt(P(n))+1)^2 is true? This would be a lower bound than the Tchebycheff result that there is always a prime between n and 2n. Regards.
The short answer is no. Noone does. We have no proof that it is
true. The best that has been achieved, when last I looked was that
there is always a prime between x and x + x^(11/20 + epsilon), for epsilon
depending on x as x -->oo. The fraction 11/20 may have been improved.

Note that even R.H does not yield the result you want. R.H. would imply
there is always a prime between x and x + sqrt(x)log x for sufficiently
large x. You want one between x and 2sqrt(x)+1.   2005-10-26, 21:11   #3
maxal

Feb 2005

10416 Posts Quote:
 Originally Posted by Crook Let P(n) denote the n-th prime number. Then, does anybody have an idea why P(n+1)<(sqrt(P(n))+1)^2 is true?
There is a famous Legendre's conjecture AKA the 3rd Landau's problem that there is always a prime between n^2 and (n+1)^2 but I'm not sure if n is required to be integer. If n may be a positive real number then this conjecture directly implies your inequality. Otherwise, it implies a weaker inequality P(n+1)<(ceil(sqrt(P(n))+1)^2.   2005-10-26, 21:29 #4 Citrix   Jun 2003 110010001012 Posts http://www.primepuzzles.net/conjectures/conj_008.htm Is my favorite conjecture related to distribution of primes.  Thread Tools Show Printable Version Email this Page Similar Threads Thread Thread Starter Forum Replies Last Post henryzz Factoring 18 2010-09-26 00:55 tgrdy Msieve 6 2010-08-20 21:51 XYYXF Math 2 2007-12-08 12:31 hallstei Factoring 7 2007-05-01 12:51 R.D. Silverman NFSNET Discussion 11 2006-07-20 17:04

All times are UTC. The time now is 12:27.

Wed Feb 1 12:27:12 UTC 2023 up 167 days, 9:55, 0 users, load averages: 1.32, 1.07, 0.95