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A259764 Least prime p such that prime(p*n)-1 is a square, or 0 if no such p exists. 2


%S 3,13,41,3,11,2,241,181,5,2927,5,523,2,4967,3,421,33053,8447,17107,

%T 20747,1811,5743,20407,99643,165443,769,21269,46099,3121,9883,16301,

%U 523,10771,41603,17,7,48383,455353,711317,1637,3,105397,43,12071,186113,56437,303157,211,25951,178817

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

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

%C 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.

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

%D 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.

%H Zhi-Wei Sun, <a href="/A259764/b259764.txt">Table of n, a(n) for n = 1..500</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.

%e a(1) = 3 since 3 is prime and prime(3*1)-1 = 2^2 is a square.

%e a(2) = 13 since 13 is prime and prime(13*2)-1 = 10^2 is a square.

%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]

%t 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}]

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

%K nonn

%O 1,1

%A _Zhi-Wei Sun_, Jul 04 2015

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