FIVE SMARANDACHE CONJECTURES ON PRIMES
edited by M.L.Perez
x x
1) The equation p - p = 1,
n+1 n
where p is the n-th prime, has a unique solution between 0.5 and 1;
n
- the maximum solution occurs for n = 1, i.e.
x x
3 - 2 = 1 when x = 1;
- the minimum solution occurs for n = 31, i.e.
x x
127 - 113 = 1 when x = 0.567148... = a .
0
Thus, Andrica's conjecture
1/2 1/2
A = p - p < 1
n n+1 n
is generalized to:
a a
2) B = p - p < 1, where a < a .
n n+1 n 0
1/k 1/k
3) C = p - p < 2/k, where k >= 2 .
n n+1 n
a a
4) D = p - p < 1/n, where a < a and n big enough, n = n(a), holds
n n+1 n 0
for infinitely many consecutive primes.
a) Is this still available for a < a < 1 ?
0
b) Is there any rank n depending on a and n such that (4) is verified
0
for all n >= n ?
0
5) p / p <= 5/3, and the maximum occurs at n = 2.
n+1 n
(This last conjecture has been proved to be true by Jozsef Sandor,
Babes-Bolyai University of Cluj, "On A Conjecture of Smarandache on
Prime Numbers", , Vol. 10, 2000, to appear.)
References:
[1] Sloane, N.J.A., Sequence A001223/M0296 ("Andrica's Conjecture") in
.
[2] Sloane, N.J.A., Sequence A038458 ("Smarandache Constant" =
0.56714813020201771464684687553348256458679024938863820684028522182...)
in ,
http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/
eishis.cgi.
[3] Smarandache, Florentin, "Conjectures which Generalize Andrica's
Conjecture", Arizona State University, Hayden Library, Special
Collections, Tempe, AZ, USA.
[4] Weisstein, Eric W., "CRC Concise Encyclopedia of Mathematics", CRC
Press, Boca Raton, FL, USA, p. 44, 1998.