2,4

Conjecture: (i) a(p) > 0 for any prime p > 3. Moreover, for any finite field F(q) with q elements and a polynomial P(x) over F(q) of degree smaller than q-1, if P(x) is not of the form c-x, then there is a circular permutation a_1, ..., a_q of all the elements of F(q) with

{P(a_1)+a_2, P(a_2)+a_3, ..., P(a_{q-1})+a_q, P(a_q)+a_1}

equal to F(q).

(ii) Let F be any field with |F| > 7, and let A be a finite subset of F with |A| = n > 2. Let P(x) be a polynomial over F whose degree is smaller than p-1 if F is of prime characteristic p. If P(x) is not of the form c-x, then there is a circular permutation a_1, ..., a_n of all the elements of A such that the n sums P(a_1)+a_2, P(a_2)+a_3, ..., P(a_{n-1})+a_n, P(a_n)+a_1 are pairwise distinct.

Clearly, if a(n) > 0, then we must have 1^2+2^2+...+n^2 == 0 (mod n), i.e., (n+1)*(2n+1) == 0 (mod 6). Thus, when n is divisible by 2 or 3, we must have a(n) = 0.

It is well known that for any finite field F(q) with q elements we have sum_{x in F(q)} x^k = 0 for every k = 0, ..., q-2.

Verified for n < 256: a(n) > 0 iff n is not divisible by 2 or 3. - Bert Dobbelaere, Apr 23 2021

Zhi-Wei Sun, A combinatorial conjecture on finite fields, a message to Number Theory List, Sept. 30, 2013.

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

a(5) = 4 due to the circular permutations

(1,3,4,5,2), (1,4,2,3,5), (1,5,2,4,3), (1,5,4,2,3).

a(7) > 0 due to the circular permutation (1,2,3,7,4,6,5).

a(11) > 0 due to the circular permutation

(1,2,3,4,6,5,9,11,10,8,7).

(* A program to compute desired circular permutations for n = 7. *)

V[i_]:=Part[Permutations[{2, 3, 4, 5, 6, 7}], i]

m=0

Do[If[Length[Union[Table[Mod[If[j==0, 1, Part[V[i], j]]^2+If[j<6, Part[V[i], j+1], 1], 7], {j, 0, 6}]]]<7, Goto[aa]];

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]]; Label[aa]; Continue, {i, 1, 6!}]

Cf. A227456, A229082, A229141.

Sequence in context: A076021 A179270 A246132 * A295839 A243270 A242027

Adjacent sequences: A229824 A229825 A229826 * A229828 A229829 A229830

nonn,more,hard

Zhi-Wei Sun, Sep 30 2013

a(11)-a(22) from Bert Dobbelaere, Apr 23 2021

approved