cp's OEIS Frontend

Jud McCranie reports that he was able to find a solution for each n <= 225 (2n <= 450) in just a few seconds. - Jul 05 2002

Is there a proof that this can always be done?

The Mathematica program for this sequence uses backtracking to find all solutions for a given n. To verify that at least one solution exists for a given n, the backtracking function be made to stop when the first solution is found. Solutions have been found for n <= 48. - T. D. Noe, Jun 19 2002

This sequence is from the prime circle problem. There is no known proof that a(n) > 0 for all n. However, for many n (see A072618 and A072676), we can prove that a(n) > 0. Also, the sequence A072616 seems to imply that there are always solutions in which the odd (or even) numbers are in order around the circle. - T. D. Noe, Jul 01 2002

Prime circles can apparently be generated for any n using the Mathematica programs given in A072676 and A072184. - T. D. Noe, Jul 08 2002

The following seems to always produce a solution: Work around the circle starting with 1 but after that always choosing the largest remaining number that fits. For example, if n = 4 this gives 1, 6, 7, 4, 3, 8, 5, 2. See A088643 for a sequence on a related idea. - Paul Boddington, Oct 30 2007

See A228917 for a similar conjecture on twin primes. - Zhi-Wei Sun, Sep 08 2013

See A242527 for a similar problem on the set of numbers {0 through (n-1)}. - Stanislav Sykora, May 30 2014

James Tilley and Stan Wagon report that all terms up to n = 10^6 are nonzero. Charles R Greathouse IV, Feb 05 2016

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

Note that if i-j = (p-1)/2 and i+j = (q-1)/2 for some odd primes p and q then 4*i+2 is the sum of the two primes p and q. So the conjecture is related to Goldbach's conjecture.

Zhi-Wei Sun also made the following similar conjecture: For any integer n > 5, there exists a circular permutation i_0, i_1, ..., i_n of 0, 1, ..., n such that all the 2*n+2 numbers 2*|i_k-i_{k+1}|+1 and 2*(i_k+i_{k+1})-1 (k = 0,...,n) (with i_{n+1} = i_0) are primes.

Conjecture: a(n) > 0 except for n = 2, 4.

Note that this conjecture is stronger than the one stated in the comments in A228860. Also, there is no permutation i_0, ..., i_7 of 0, ..., 7 with i_0+i_1, i_1+i_2, ..., i_6+i_7, i_7+i_0 all coprime to 7*13 - 1 = 90.

For n > 1 let S(n) be an undirected simple graph with vertices 0, 1, ..., n which has an edge connecting two distinct vertices i and j if and only if i + j is relatively prime to both n-1 and n+1. Then a(n) is just the number of those Hamiltonian cycles in S(n) on which the vertices 0 and n are adjacent.

We have some other similar conjectures. For example, for any positive integer n there is a permutation i_0,i_1,...,i_n of 0,1,...,n with i_0 = 0 and i_n = n such that i_0 + i_1, i_1 + i_2, ..., i_{n-1} + i_n, i_n + i_0 are all relatively prime to both 2*n-1 and 2*n+1.

On Sep 10 2013, Zhi-Wei Sun proved the conjecture for any positive odd integer n. In fact, if n == 1 or 3 (mod 6) then we may take the permutation 0,n-2,2,n-4,4,...,1,n-1,n which meets the requirement; if n == 3 or 5 (mod 6) then the permutation 0,1,n-1,3,n-3,...,n-2,2,n suffices for our purpose.

Conjecture: a(n) > 0 except for n = 2, 4.

Conjecture: a(n) > 0 except for n = 4.

Note that if a circular permutation i_0, i_1, ..., i_n of 0, 1, ..., n with i_0 = 0 meets the requirement then we must have i_n = 1. This can be explained as follows: If i_n > 1, then 3 | i_n since 2*(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 2*(i_{n-1}^2+i_n)+1, ..., 2*(i_1^2+i_2)+1 are primes not divisible by 3. Therefore, i_n > 1 would lead to a contradiction.

Conjecture: a(n) > 0 except for n = 2, 5, 11. Similarly, for any positive integer n not equal to 4, there is a permutation i_0, i_1, ..., i_n of 0, 1, ..., n with i_0 = 0 and i_n = n such that 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 all coprime to both n-1 and n+1.

Zhi-Wei Sun also made the following general conjecture:

For any positive integer k, define E(k) to be the set of those positive integers n for which there is no permutation i_0, i_1, ..., i_n of 0, 1, ..., n with i_0 = 0 and i_n = n such that all the n+1 numbers i_0^k+i_1, i_1^k+i_2, ..., i_{n-1}^k+i_n, i_n^k+i_0 are coprime to both n-1 and n+1. Then E(k) is always finite; in particular, E(1) = {2,4}, E(2) = {2,5,11} and E(3) = {2,4}.

Conjecture: a(n) > 0 for all n > 5 with n not equal to 13.

Zhi-Wei Sun also made the following conjectures:

(1) For any integer n > 1, there is a permutation pi(1), ..., pi(n) of 1, ..., n such that the n numbers 2*pi(1)*pi(2)-1, ..., 2*pi(n-1)*pi(n)-1, 2*pi(n)*pi(1)-1 are all prime. Also, for any positive integer n not equal to 4, there is a permutation pi(1), ..., pi(n) of 1, ..., n such that the n numbers 2*pi(1)*pi(2)+1, ..., 2*pi(n-1)*pi(n)+1, 2*pi(n)*pi(1)+1 are all prime.

(2) Let F be a finite field with q > 7 elements. Then, there is a circular permutation a_1,...,a_{q-1} of the q-1 nonzero elements of F such that all the q-1 elements a_1*a_2-1, a_2*a_3-1, ..., a_{q-2}*a_{q-1}-1, a_{q-1}*a_1-1 are primitive elements of the field F (i.e., generators of the multiplicative group F\{0}). Also, there is a circular permutation b_1,...,b_{q-1} of the q-1 nonzero elements of F such that all the q-1 elements b_1*b_2+1, b_2*b_3+1, ..., b_{q-2}*b_{q-1}+1, b_{q-1}*b_1+1 are primitive elements of the field F.