A259764 Least prime p such that prime(p*n)-1 is a square, or 0 if no such p exists.

3, 13, 41, 3, 11, 2, 241, 181, 5, 2927, 5, 523, 2, 4967, 3, 421, 33053, 8447, 17107, 20747, 1811, 5743, 20407, 99643, 165443, 769, 21269, 46099, 3121, 9883, 16301, 523, 10771, 41603, 17, 7, 48383, 455353, 711317, 1637, 3, 105397, 43, 12071, 186113, 56437, 303157, 211, 25951, 178817

Offset: 1

Author

Zhi-Wei Sun, Jul 04 2015

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 0.

This is stronger than the conjecture in A259731. It implies the well-known conjecture that there are infinitely many primes of the form x^2-1 with x an integer.

I also conjecture that for any positive integer n there exists a prime p such that prime(p*n)+2 is a square.

Examples

			a(1) = 3 since 3 is prime and prime(3*1)-1 = 2^2 is a square.
a(2) = 13 since 13 is prime and prime(13*2)-1 = 10^2 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 = 1..500
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000040, A000290, A002496, A028871, A259731.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[k=0;Label[bb];k=k+1;If[SQ[Prime[Prime[k]*n]-1],Goto[aa],Goto[bb]];Label[aa];Print[n," ",Prime[k]];Continue,{n,1,50}]