OFFSET
1,1
COMMENTS
Conjecture: (i) If n > 0 and r are relatively prime integers, then there are infinitely many positive integers k such that k*n+r = prime(p) for some prime p.
(ii) Let r be 1 or -1. For any integer n > 0, there is a positive integer k such that k*n+r = prime(p) and k^2*n+1 = prime(q) for some primes p and q.
(iii) For any integer n > 0, there is a positive integer k such that n+k = prime(p) and n+k^2 = prime(q) for some primes p and q.
Note that part (i) is a refinement of Dirichlet's theorem on primes in arithmetic progressions. Also, part (ii) implies that a(n) exists for any n > 0.
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..200
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.
EXAMPLE
a(3) = 286 since 286*3+1 = 859 = prime(149) with 149 prime, and 286^2*3+1 = 245389 = prime(21661) with 21661 prime.
MATHEMATICA
PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
Do[k=0; Label[bb]; k=k+1; If[PQ[k*n+1]&&PQ[k^2*n+1], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k]; Continue, {n, 1, 60}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Aug 18 2015
STATUS
approved