A260886 Least prime p such that 3 + 4*prime(p*n) = 5*prime(q*n) for some prime q.

2, 157, 199, 3539, 1973, 9241, 14629, 167, 48281, 2207, 313, 30631, 35993, 33863, 23, 23, 7963, 17077, 11069, 6043, 4931, 3697, 2339, 14153, 35311, 63149, 111143, 491, 247193, 464237, 2293, 12101, 727, 61403, 243437, 40289, 4337, 241, 2719, 13933, 21817, 6803, 52813, 451279, 166409, 45631, 109891, 490969, 153563, 9127

Offset: 1

Author

Zhi-Wei Sun, Aug 02 2015

Keywords

• nonn

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.

This includes the conjectures in A260252 and A260882 as special cases.

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.

Examples

			a(2) = 157 since 3 + 4*prime(157*2) = 3 + 4*2083 = 8335 = 5*prime(131*2) with 157 and 131 both prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..200
• Zhi-Wei Sun, Checking the conjecture for a,b,c = 1..20 and n = 1..30
• Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588 [math.NT], 2012-2015.

Crossrefs

Cf. A000040, A260120, A260252, A260882.

Programs

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}]