OFFSET
2,1
COMMENTS
Conjecture: a(n) exists for any n > 1. Also, for any n > 0 there is a number k > 0 such that prime(k*n) + prime(k+n) is a square.
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 = 2..600
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.
EXAMPLE
a(2) = 2 since prime(2*2) - prime(2+2) = 7 - 7 = 0^2.
a(3) = 24 since prime(24*3) - prime(24+3) = 359 - 103 = 16^2.
MATHEMATICA
SQ[n_]:=IntegerQ[Sqrt[n]]
Do[k=0; Label[bb]; k=k+1; If[SQ[Prime[k*n]-Prime[k+n]], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k]; Continue, {n, 2, 60}]
lpi[n_]:=Module[{k=1}, While[!IntegerQ[Sqrt[Prime[k*n]-Prime[k+n]]], k++]; k]; Array[lpi, 60, 2] (* Harvey P. Dale, Mar 12 2019 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jul 12 2015
STATUS
approved