A227456 Number of permutations i_0, i_1, ..., i_n of 0, 1, ..., n with i_0 = 0 and i_n = 1 such that all the n+1 numbers i_0^2+i_1, i_1^2+i_2, ..., i_{n-1}^2+i_n, i_n^2+i_0 are of the form (p+1)/4 with p a prime congruent to 3 modulo 4.

1, 1, 1, 1, 1, 2, 4, 11, 15, 15

Offset: 1

Author

Zhi-Wei Sun, Sep 16 2013

Keywords

• nonn
• more
• hard

Comments

Conjecture: a(n) > 0 for all n > 0. Similarly, for any positive integer n, there is a circular permutation i_0, i_1, ..., i_n of 0, 1, ..., n such that all the n+1 numbers i_0^2+i_1, i_1^2+i_2, ..., i_{n-1}^2+i_n, i_n^2+i_0 are of the form (p-1)/4 with p a prime congruent to 1 modulo 4.

Note that if a circular permutation i_0, i_1, ..., i_n of 0, 1, ..., n with i_0 = 0 makes all the n+1 numbers i_0^2+i_1, i_1^2+i_2, ..., i_{n-1}^2+i_n, i_n^2+i_0 of the form (p+1)/4 with p a prime, then we must have i_n = 1. This can be explained as follows: If i_n > 1, then 3 | i_n since 4*(i_n^2+0)-1 is a prime not divisible by 3, and similarly i_{n-1},...,i_1 are also multiples of 3 since 4*(i_{n-1}^2+i_n)-1, ..., 4*(i_1^2+i_2)-1 are primes not divisible by 3. Therefore, i_n > 1 would lead to a contradiction.

Examples

			a(1) = a(2) = a(3) = a(4) = a(5) = 1 due to the permutations (0,1), (0,2,1), (0,3,2,1), (0,3,2,4,1), (0,3,2,4,5,1).
a(6) = 2 due to the permutations
    (0,3,6,2,4,5,1) and (0,3,6,5,2,4,1).
a(7) = 4 due to the permutations
    (0,3,6,2,4,5,7,1), (0,3,6,2,7,4,5,1),
    (0,3,6,5,2,7,4,1), (0,3,6,5,7,4,2,1).
a(8) = 11 due to the permutations
(0,3,6,2,4,5,8,7,1), (0,3,6,2,7,8,4,5,1), (0,3,6,2,8,4,5,7,1),
(0,3,6,2,8,7,4,5,1), (0,3,6,5,2,7,8,4,1), (0,3,6,5,2,8,7,4,1),
(0,3,6,5,7,8,2,4,1), (0,3,6,5,7,8,4,2,1), (0,3,6,5,8,2,7,4,1),
(0,3,6,5,8,4,2,7,1), (0,3,6,5,8,7,4,2,1).
a(9) > 0 due to the permutation (0,3,6,9,2,4,5,8,7,1).
a(10) > 0 due to the permutation (0,3,6,9,2,4,5,10,8,7,1).
		

Links

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

Crossrefs

Cf. A229082, A229141, A229130.

Programs

Mathematica

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