OFFSET
1,1
COMMENTS
Conjecture: Let a,b,c be pairwise relatively prime positive integers with a+b+c even and a not equal to b. Then, for any positive integer n, there are primes p and q such that a*prime(p*n) - b*prime(q*n) = c.
For example, for a = 7, b = 17, c = 20 and n = 30, we have 7*prime(4695851*30) - 17*prime(2020243*30) = 7*2922043519 - 17*1203194389 = 20 with 4695851 and 2020243 both prime.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..200
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2015.
EXAMPLE
a(2) = 157 since 3 + 4*prime(157*2) = 3 + 4*2083 = 8335 = 5*prime(131*2) with 157 and 131 both prime.
MATHEMATICA
f[n_]:=Prime[n]
PQ[p_, n_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]/n]
Do[k=0; Label[bb]; k=k+1; If[PQ[(4*f[n*f[k]]+3)/5, n], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", f[k]]; Continue, {n, 1, 50}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Aug 02 2015
STATUS
approved