OFFSET
1,1
COMMENTS
Conjecture: a(n) exists for any n > 0. Moreover, for any integers s and t with |s| = |t| = 1, each positive rational number r can be written as (prime(p) + s)/(prime(q) + t) with p and q both prime. - Sun
I have verified the conjecture for all those rational numbers r = a/b with a, b = 1, ..., 500. - Zhi-Wei Sun, Jun 13 2015
REFERENCES
Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28-Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641.
EXAMPLE
a(1) = 2 since 1 = (prime(2) - 1)/(prime(2) - 1) with 2 prime.
a(2) = 2 since 2 = (prime(3) - 1)/(prime(2) - 1) with 2 and 3 both prime.
a(14) = 397 since 14 = (prime(4021) - 1)/(prime(397) - 1) = (38053 - 1)/(2719 - 1) with 379 and 4021 both prime.
a(23) = 1039 since 23 = (prime(17209) - 1)/(prime(1039) - 1) = (190579 - 1)/(8287 - 1) with 1039 and 17209 both prime.
MATHEMATICA
PQ[n_]:=PrimeQ[n] && PrimeQ[PrimePi[n]];
Do[k = 0; Label[bb]; k = k + 1; If[PQ[n * (Prime[Prime[k]] - 1) + 1], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", Prime[k]]; Continue, {n, 1, 100}] (* Sun *)
pq[n_] := PrimeQ@n && PrimeQ@ PrimePi@ n; a[n_] := Block[{k = 1}, While[!pq[1 + n*(Prime@ Prime@ k - 1)], k++]; Prime@k]; Array[a, 100] (* Giovanni Resta, Jun 11 2015 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jun 10 2015
STATUS
approved