A261339 Least positive integer k such that both k and k*n belong to the set {m>0: m+1, m^2+1 and m^2+prime(m)^2 are all prime}.

1, 1, 47500, 20440, 2, 124560, 17850, 2730, 185550, 1, 518910, 429180, 10, 687480, 81030, 36, 1568340, 2, 1165750, 7410, 10, 6780, 481140, 10, 10, 5430, 240, 2730, 72660, 2080, 18700, 291720, 295080, 52860, 5430, 1, 81030, 56400, 12490, 43590, 124560, 40030, 5170, 278700, 2091850, 131320, 184110, 11206510, 12910, 1245780

Offset: 1

Author

Zhi-Wei Sun, Aug 15 2015

Keywords

• nonn

Comments

Conjecture: a(n) exists for any n > 0. In general, any positive rational number r can be written as m/n with m and n in the set {k>0: k+1, k^2+1 and k^2+prime(k)^2 are all prime}.

For example, 5/8 = 3567600/5708160 with 3567600+1, 3567600^2+1 = 12727769760001, 3567600^2 + prime(3567600)^2 = 3567600^2 + 60098671^2 = 3624578025726241, 5708160+1, 5708160^2+1 = 32583090585601, and 5708160^2 + prime(5708160)^2 = 5708160^2 + 99018553^2 = 9837256928799409 all prime.

The conjecture implies that there are infinitely many primes p with (p-1)^2+1 and (p-1)^2+prime(p-1)^2 both prime.

We also guess that any positive rational number can be written as m/n, where m and n are positive integers with m^2+prime(m)^2, m^2+prime(n)^2, n^2+prime(m)^2 and n^2+prime(n)^2 all prime.

Examples

			a(3) = 47500 since 47501, 47500^2 + 1 = 2256250001, 47500^2 + prime(47500)^2 = 47500^2 + 578827^2 = 337296945929, 47500*3 + 1 = 142501, (47500*3)^2 + 1 = 20306250001, and (47500*3)^2 + prime(47500*3)^2 = 142500^2 + 1907023^2 = 3657042972529 are all prime.
		

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..1000
• Zhi-Wei Sun, Checking the conjecture for r = a/b with a,b = 1..60
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000040, A005574, A236193, A259492, A259540, A260120.

Programs

Mathematica

PQ[n_]:=PrimeQ[n+1]&&PrimeQ[n^2+1]&&PrimeQ[n^2+Prime[n]^2]
Do[k=0;Label[bb];k=k+1;If[PQ[k]&&PQ[k*n],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,50}]