A243901 Positive integers n such that p_{n+i} is a quadratic residue modulo p_{n+j} for any distinct i and j among 0, 1, ..., 6.

178633, 2355662, 4892172, 5702347, 9256159, 9572343, 13837265, 15147032, 15429648, 15822376, 16603935, 20925043, 22128672, 22462201, 22689295, 27145167, 28031877, 28470899, 29246422, 30772941, 31211796, 32372758

Offset: 1

Author

Zhi-Wei Sun, Jun 14 2014

Keywords

• nonn

Comments

Conjecture: For any integer m > 0, there are infinitely many positive integers n such that p_{n+i} is a quadratic residue modulo p_{n+j} for any distinct i and j among 0, 1, ..., m.

Examples

			a(1) = 178633 since any 6 primes among the 7 integers prime(178633) = 2434589, prime(178634) = 2434609, prime(178635) = 2434613, prime(178636) = 2434657, prime(178637) = 2434669, prime(178638) = 2434673 and prime(178639) = 2434681 are quadratic residues modulo the remaining one of the 7 primes.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..200
• Hao Pan, Z.-W. Sun, Consecutive primes and Legendre symbols, arXiv preprint arXiv:1406.5951 [math.NT], 2014-2018.

Crossrefs

Cf. A000040, A243755, A243837, A243839.

Programs

Mathematica

q[i_,j_]:=JacobiSymbol[Prime[i],Prime[j]]
m=0;Do[Do[If[q[n+i,n+j]==-1,Goto[aa]],{i,0,6},{j,0,6}]; m=m+1;Print[m," ",n];Label[aa];Continue,{n,1,32372758}]
Reap[ Do[ If[ Catch[ Do[ If[ JacobiSymbol[Prime[n + i], Prime[n + j]] != 1, Throw@False], {i, 0, 5}, {j, i + 1, 6}]; True], Sow[n]], {n, 32372758}]][[2, 1]] (* Michael Somos, Jun 15 2014 *)