OFFSET
1,1
COMMENTS
Conjecture: Let n be any positive integer, and let s and t belong to the set {1,-1}. Then each positive rational number r can be written as (prime(p*n)+s)/(prime(q*n)+t) with p and q both prime, unless n > r = 1 and {s,t} = {1,-1}.
This extends the conjecture in A258803.
For example, for n = 8, s = t = -1 and r = 16/11, we have (prime(407249*8)-1) /(prime(286411*8)-1) = 54568320/37515720 = r with 407249 and 286411 both prime. Also, for n = 10, s = -1, t = 1, and r = 23/17, we have (prime(1923029*10)-1)/(prime(1444903*10)+1) = 358404768/264907872 = r with 1923029 and 1444903 both prime.
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..2000
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641.
EXAMPLE
a(1) = 2 since 1 = (prime(6*2)-1)/(prime(6*2)-1) with 2 prime.
a(2) = 18253 since 2 = 2868672/1434336 = (prime(6*34673)-1)/(prime(6*18253)-1) with 18253 and 34673 both prime.
MATHEMATICA
PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]/6]
Do[k=0; Label[aa]; k=k+1; If[PQ[(Prime[6*Prime[k]]-1)*n+1], Goto[bb], Goto[aa]]; Label[bb]; Print[n, " ", Prime[k]]; Continue, {n, 1, 50}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jul 20 2015
STATUS
approved