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.