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A206549 Nontrivial solution of x^2 == 1 (Modd p), where p is the n-th prime of the form 4*k+1, for odd restricted residue classes Modd p.

%I A206549 #56 Oct 23 2024 01:33:36
%S A206549 3,5,13,17,31,9,23,11,27,55,75,91,33,15,37,105,129,93,19,81,183,107,
%T A206549 89,177,241,187,217,53,155,25,203,189,213,311,269,115,63,381,143,29,
%U A206549 179,67,109,413,301,235,489,439,483,553
%N A206549 Nontrivial solution of x^2 == 1 (Modd p), where p is the n-th prime of the form 4*k+1, for odd restricted residue classes Modd p.
%C A206549 For multiplication Modd n (not to be confused with multiplication mod n) see a comment on A203571.
%C A206549 The trivial solution of x^2 == 1 (Modd n) is x = 1 (Modd n). Note that x = -1 (Modd n) == +1 (Modd n). In the ordinary mod n case the trivial solution is 1 (mod 2) for n=2 (-1 == +1 (mod 2)) and if n>2 the two trivial solutions are 1 (mod n) and the noncongruent -1 (mod n) == n-1 (mod n).
%C A206549 Here multiplication on the reduced residue system Modd p, p an odd prime, with only odd numbers is considered (which is possible, contrary to mod p). In order to have inverses one has to exclude all reduced residue classes Modd p with even numbers. The (p-1)/2 residue classes are then [1],[3],...,[p-2]. For m=1,3,...,p-2, the class [m] Modd p is the union of the ordinary reduced residue classes mod 2p: [m] and [-m]=[2p-m]. Besides the trivial solution x=+1 (Modd p) (note that -1 == +1 (Modd p)) there is a further nontrivial solution if and only if (p-1)/2 is even, i.e. p=A002144(n) (primes of the form 4*k+1), n>=1. The present sequence entry a(n) gives the smallest positive representative for this nontrivial solution Modd A002144(n).
%C A206549 This result uses the fact that every finite group of prime order p is the cyclic group Z_p (Corollary to Lagrange's or also Cauchy's theorem on finite groups or see A000688 for abelian groups). Here for the multiplicative group Modd p (on the odd residue classes) which has order (p-1)/2, for p an odd prime. This turns out to be the Galois group for the minimal polynomial C(p,x), whose coefficients are found in A187360. The sequence {a(n)} arises if one asks for the smallest positive members of the odd restricted residue system Modd p, namely [1],[3],...,[p-2], which are their own inverses besides the trivial element 1 (Modd p).
%C A206549 The row a(n) has in the table for the multiplicative group Modd A002144(n) a 1 on the diagonal, if the table is written for the odd representatives 1,3,...,p-2. The only other diagonal entry 1 appears for the element 1. Note that in the ordinary mod p case, p an odd prime, one always has for the multiplicative group mod p (which has p-1 residue classes) in the multiplication table the diagonal entry 1 only for the representatives 1 and p-1, but p-1 == -1 (mod p) is a trivial solution of x^2 == 1 (mod p).
%H A206549 Rémi Guillaume, <a href="/A206549/a206549_1.txt">Calculations supporting my examples & showing that multiplication can be done class-wise</a>
%F A206549 a(n)^2 == 1 (Modd A002144(n)), n>=1, a(n) the smallest positive solution not 1. For Modd p, p an odd prime, see the comment section and the examples.
%e A206549 a(6)=9 because the corresponding prime is A002144(6)=41, 9^2 = 81, and 81 Modd 41 is per definition -81 mod 2*41 = +1 (the definition uses the parity of floor(81/41) = 1 being odd, hence the - sign), thus 9^2 == 1 (Modd 41), and 9 is not congruent to 1 (Modd 41) (or -1 (Modd 41)), hence a nontrivial solution.
%e A206549 A002144(n):  5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, ...
%e A206549 a(n):        3,  5, 13, 17, 31,  9, 23, 11, 27, 55, 75, ...
%e A206549 3^2 = 9, 9 Modd 5 := -9 mod 10 = 1, the smallest positive representative of the class 1 (Modd 5) = {+-1,+-9,+-11,+-19,...}.
%e A206549 5^2 = 25, 25 Modd 13 := -25 mod 26 = 1.
%e A206549 13^2 = 169, 169 Modd 17 := -169 mod 34 = 1.
%e A206549 17^2 = 289, 289 Modd 29 := -289 mod 2*29 = 1.
%e A206549 ...
%e A206549 E.g., for the odd prime 7, not in A002144, there are no self-inverse elements in the multiplicative group Modd 7 (on the odd numbers) except the trivial 1. The inverse of 3 is 5 (Modd 7) and vice versa, since 3*5 = 15 and 15 Modd 7 := 15 mod 14 = 1. (*)
%e A206549 From _Rémi Guillaume_, Sep 08 2024: (Start)
%e A206549 (*) The finite multiplicative group Modd 7 (on the odd residue classes) is of odd order: (7-1)/2 = 3, and is isomorphic to the additive cyclic group Z_3. Moreover, Z_3 has two generating elements: [1] and [2] (mod 3), and no nontrivial self-opposite elements -- since [1]+[2] = [0] (mod 3); likewise, {[1],[3],[5]} (Modd 7) has two generating elements: [3] and [5] (Modd 7), and no nontrivial self-inverse elements -- since [3]*[5] = [1] (Modd 7).
%e A206549 13 is an odd prime in A002144; the finite multiplicative group Modd 13 (on the odd residue classes) is of even order: (13-1)/2 = 6 = 2*3, and is isomorphic to the additive cyclic group Z_6. Moreover, Z_6 has two generating elements: [1] and [5] (mod 6), and one nontrivial self-opposite element: 3*[1] = 3*[5] = [3] (mod 6) -- since [1]+[5] = [2]+[4] = 2*[3] = [0] (mod 6); likewise, {[1],[3],[5],[7],[9],[11]} (Modd 13) has two generating elements: [7] and [11] (Modd 13), and one nontrivial self-inverse element: [7]^3 = [11]^3 = [5] (Modd 13) -- since [3]*[9] = [5]^2 = [7]*[11] = [1] (Modd 13).
%e A206549 17 is an odd prime in A002144; the finite multiplicative group Modd 17 (on the odd residue classes) is of even order: (17-1)/2 = 8 = 2*4, and is isomorphic to the additive cyclic group Z_8. Moreover, Z_8 has four generating elements: [1], [3], [5], [7] (mod 8), and one nontrivial self-opposite element: 4*[1] = 4*[3] = 4*[5] = 4*[7] = [4] (mod 8) -- since [1]+[7] = [2]+[6] = [3]+[5] = 2*[4] = [0] (mod 8); likewise, {[1],[3],[5],[7],[9],[11],[13],[15]} (Modd 17) has four generating elements: [3], [5], [7], [11] (Modd 17), and one nontrivial self-inverse element: [3]^4 = [5]^4 = [7]^4 = [11]^4 = [13] (Modd 17) -- since [3]*[11] = [5]*[7] = [9]*[15] = [13]^2 = [1] (Modd 17).
%e A206549 (End)
%Y A206549 See A203571 for Modd n, A002144 for corresponding primes.
%K A206549 nonn
%O A206549 1,1
%A A206549 _Wolfdieter Lang_, Feb 13 2012