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A269596 Irregular triangle giving in row n the smaller of the two roots x1 of x^2 + b modulo prime(n) from {0, 1, ..., prime(n)-1} corresponding to b from row n of A269595.

%I A269596 #10 Feb 13 2018 00:19:47
%S A269596 1,1,2,1,2,3,1,3,4,2,5,1,5,6,3,2,4,1,4,7,8,3,5,2,6,1,6,4,7,3,8,5,9,2,
%T A269596 1,8,4,6,9,3,10,11,2,7,5,1,12,5,13,9,14,7,4,10,3,6,8,11,2,1
%N A269596 Irregular triangle giving in row n the smaller of the two roots x1 of x^2 + b modulo prime(n) from {0, 1, ..., prime(n)-1} corresponding to b from row n of A269595.
%C A269596 The length of row n >= 2 is (prime(n)-1)/2 =  A005097(n-1), and for row n = 1 it is 1.
%C A269596 The other roots of x^2 + b modulo prime(n) are given in A269597.
%C A269596 See A269595 for the irregular triangle with the quadratic residues -b modulo prime(n) = A000040(n), for n >= 1.  For n=1 (prime 2) there is a double root x1 = x2 = 1 of x^2 + 1 (mod 2).
%C A269596 Each row n >= 2 consists of a certain permutation of  1, 2, ..., (prime(n)-1)/2.
%C A269596 For a(n), n >= 2, see column x_1 of the table in the Wolfdieter Lang link.
%H A269596 Wolfdieter Lang, <a href="/A268922/a268922.pdf">Note on a Recurrence for Approximation Sequences of p-adic Square Roots</a>
%F A269596 T(n, k) gives the smaller zero of x^2 + A269595(n, k) == 0 (mod prime(n)), n >= 1, for k=1 if n=1 and k = 1, 2, ..., (prime(n)-1)/2 = A005097(n-1) for n >= 2. Representatives are taken from the complete residue class {0, 1 ,..., prime(n)-1}.
%F A269596 T(n, k) = prime(n) - A269597(n, k).
%e A269596 The irregular triangle T(n, k) begins (P(n) stands here for prime(n)):
%e A269596 n, P(n)\k 1  2  3  4  5  6  7  8  9 10 11 12 13 14
%e A269596 1,   2:   1
%e A269596 2,   3:   1
%e A269596 3,   5:   2  1
%e A269596 4,   7:   2  3  1
%e A269596 5,  11:   3  4  2  5  1
%e A269596 6:  13:   5  6  3  2  4  1
%e A269596 7,  17:   4  7  8  3  5  2  6  1
%e A269596 8,  19:   6  4  7  3  8  5  9  2  1
%e A269596 9,  23:   8  4  6  9  3 10 11  2  7  5  1
%e A269596 10, 29:  12  5 13  9 14  7  4 10  3  6  8 11  2  1
%e A269596 ...
%e A269596 Row n=7 (prime 17) is the permutation (in cycle notation) (1,4,3,8)(2,7,6) of {1, 2, ..., 8}.
%t A269596 nn = 12; s = Table[Select[Range[Prime@ n - 1], JacobiSymbol[#, Prime@ n] == 1 &], {n, nn}]; t = Table[Prime@ n - s[[n, (Prime@ n - 1)/2 - k + 1]], {n, Length@ s}, {k, (Prime@ n - 1)/2}] /. {} -> {1};
%t A269596 Prepend[Table[SelectFirst[Range@ #, Function[x, Mod[x^2 + t[[n, k]], #] == 0]] &@ Prime@ n, {n, 2, Length@ t}, {k, (Prime@ n - 1)/2}], {1}] // Flatten (* _Michael De Vlieger_, Apr 04 2016, Version 10 *)
%Y A269596 Cf. A000040, A005097, A269595, A269597 (x2).
%K A269596 nonn,tabf,easy
%O A269596 1,3
%A A269596 _Wolfdieter Lang_, Apr 03 2016