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%I #48 Apr 23 2021 01:55:45
%S 0,0,0,4,0,24,0,0,0,5308,0,123884,0,0,0,147288372,0,7238567052,0,0,0
%N Number of circular permutations i_1,...,i_n of 1,...,n such that the n numbers i_1^2+i_2, i_2^2+i_3, ..., i_{n-1}^2+i_n, i_n^2+i_1 form a complete system of residues modulo n.
%C 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
%C {P(a_1)+a_2, P(a_2)+a_3, ..., P(a_{q-1})+a_q, P(a_q)+a_1}
%C equal to F(q).
%C (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.
%C 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.
%C 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.
%C Verified for n < 256: a(n) > 0 iff n is not divisible by 2 or 3. - _Bert Dobbelaere_, Apr 23 2021
%H Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;c27cfe4c.1309">A combinatorial conjecture on finite fields</a>, a message to Number Theory List, Sept. 30, 2013.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1309.1679">Some new problems in additive combinatorics</a> , preprint, arXiv:1309.1679 [math.NT], 2013-2014.
%e a(5) = 4 due to the circular permutations
%e (1,3,4,5,2), (1,4,2,3,5), (1,5,2,4,3), (1,5,4,2,3).
%e a(7) > 0 due to the circular permutation (1,2,3,7,4,6,5).
%e a(11) > 0 due to the circular permutation
%e (1,2,3,4,6,5,9,11,10,8,7).
%t (* A program to compute desired circular permutations for n = 7. *)
%t V[i_]:=Part[Permutations[{2,3,4,5,6,7}],i]
%t m=0
%t 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]];
%t 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!}]
%Y Cf. A227456, A229082, A229141.
%K nonn,more,hard
%O 2,4
%A _Zhi-Wei Sun_, Sep 30 2013
%E a(11)-a(22) from _Bert Dobbelaere_, Apr 23 2021
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