A229141 Number of circular permutations i_1, ..., i_n of 1, ..., n such that all the n sums i_1^2+i_2, ..., i_{n-1}^2+i_n, i_n^2+i_1 are among those integers m with the Jacobi symbol (m/(2n+1)) equal to 1.

1, 0, 0, 2, 0, 1, 0, 5, 35, 0

Offset: 1

Author

Zhi-Wei Sun, Sep 15 2013

Keywords

• nonn
• more
• hard

Comments

Conjecture: a(n) > 0 if 2*n+1 is a prime greater than 11.

Zhi-Wei Sun also made the following conjectures:

(1) For any prime p = 2*n+1 > 11, there is a circular permutation i_1, ..., i_n of 1, ..., n such that all the n numbers i_1^2-i_2, i_2^2-i_3, ..., i_{n-1}^2-i_n, i_n^2-i_1 are quadratic residues modulo p.

(2) Let p = 2*n+1 be an odd prime. If p > 13 (resp., p > 11), then there is a circular permutation i_1, ..., i_n of 1, ..., n such that all the n numbers i_1^2+i_2, i_2^2+i_3, ..., i_{n-1}^2+i_n, i_n^2+i_1 (resp., the n numbers i_1^2-i_2, i_2^2-i_3, ..., i_{n-1}^2-i_n, i_n^2-i_1) are primitive roots modulo p.

(3) Let p = 2*n+1 be an odd prime. If p > 19 (resp. p > 13), then there is a circular permutation i_1, ..., i_n of 1, ..., n such that all the n numbers i_1^2+i_2^2, i_2^2+i_3^2, ..., i_{n-1}^2+i_n^2, i_n^2+i_1^2 (resp., the n numbers i_1^2-i_2^2, i_2^2-i_3^2, ..., i_{n-1}^2-i_n^2, i_n^2-i_1^2) are primitive roots modulo p.

See also the linked arXiv paper of Sun for more conjectures involving primitive roots modulo primes.

Examples

			a(4) = 2 due to the permutations (1,3,2,4) and (1,4,3,2).
a(6) = 1 due to the permutation (1,3,5,2,6,4).
a(8) = 5 due to the permutations
   (1,3,4,2,5,8,6,7), (1,8,3,6,2,4,5,7), (1,8,3,6,7,4,2,5),
   (1,8,3,7,6,2,4,5), (1,8,6,7,3,4,2,5).
a(9) > 0 due to the permutation (1,3,7,6,8,4,9,2,5).
		

Links

• Zhi-Wei Sun, Some new problems in additive combinatorics, preprint, arXiv:1309.1679 [math.NT], 2013-2014.

Crossrefs

Cf. A229038, A229082, A229130, A229005.

Programs

Mathematica

(* A program to compute the desired circular permutations for n = 8. *)
f[i_,j_,p_]:=f[i,j,p]=JacobiSymbol[i^2+j,p]==1
V[i_]:=Part[Permutations[{2,3,4,5,6,7,8}],i]
m=0
Do[Do[If[f[If[j==0,1,Part[V[i],j]],If[j<7,Part[V[i],j+1],1],17]==False,Goto[aa]],{j,0,7}];
m=m+1;Print[m,":"," ",1," ",Part[V[i],1]," ",Part[V[i],2]," ",Part[V[i],3]," ",Part[V[i],4]," ",Part[V[i],5]," ",Part[V[i],6]," ",Part[V[i],7]];Label[aa];Continue,{i,1,7!}]

Extensions

a(10) = 0 from R. J. Mathar, Sep 15 2013