OFFSET
1,3
COMMENTS
Conjecture: a(n) exists for any n > 0. In general, every rational number r > 1 can be written as m/n with m > n > 0 such that prime(m) - prime(n) is a square.
This conjecture is a supplement to the conjecture in A259712.
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..1250
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.
EXAMPLE
a(3) = 5 since prime(5*3) - prime(5) = 47 - 11 = 6^2.
a(70) = 10413 since prime(10413*70) - prime(10413) = 11039173 - 109537 = 3306^2.
a(1133) = 697092 since prime(697092*1133) - prime(697092) = 17813555143 - 10523959 = 133428^2.
MATHEMATICA
SQ[n_]:=IntegerQ[Sqrt[n]]
Do[k=0; Label[bb]; k=k+1; If[SQ[Prime[n*k]-Prime[k]], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k]; Continue, {n, 1, 70}]
lpi[n_]:=Module[{k=1, sq}, sq=Prime[k*n]-Prime[k]; While[!IntegerQ[ Sqrt[ sq]], k++; sq=Prime[k*n]-Prime[k]]; k]; Array[lpi, 70] (* Harvey P. Dale, Oct 15 2015 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jul 12 2015
STATUS
approved