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A324598 Irregular triangle with the representative solutions of the Diophantine equation x^2 + x - 1 congruent to 0 modulo N(n), with N(n) = A089270(n), for n >= 1.

%I A324598 #10 Jul 13 2019 14:13:06
%S A324598 0,2,3,7,4,14,5,23,12,18,6,34,7,47,25,33,17,43,8,62,29,49,9,79,42,52,
%T A324598 22,78,10,98,36,84,11,119,63,75,52,93,40,108,27,123,12,142,74,104,13,
%U A324598 167,88,102,61,137,47,157,14,194,80,128,32,178
%N A324598 Irregular triangle with the representative solutions of the Diophantine equation x^2 + x - 1 congruent to  0 modulo N(n), with N(n) = A089270(n), for n >= 1.
%C A324598 The length of row n is 1 for n = 1 and n = 2, and for n >= 3 it is 2^{r1 + r4} with the number r1 and r4 of distinct primes congruent to 1 and 4 modulo 5, respectively, in the prime number factorization of N(n). E.g., n = 29, N = 209 = 11*19, has r1 = 1 and r4 = 1, with four solutions. The next rows with four solutions are n = 41, 43, 59,..., with N = 319, 341, 451, ... ;  for n = 643, 688, 896, ..., with N  = 6061, 6479, 8569, ..., there are eight solutions.
%C A324598 For N(1) = 1 every integer solves this Diophantine equation, and the representative solution is 0.
%C A324598 For N(2) = 5 there is only one representative solution, namely 2.
%C A324598 For n >= 3 the representative solutions come in nonnegtive power of 2 pairs (x1, x2) with x2 = N - 1 - x1.
%C A324598 See the link in A089270 to the W. Lang paper, section 3, and Table 6.
%e A324598 The irregular triangle T(n, k) begins (pairs (x, N - 1 - x) in brackets):
%e A324598 n,    N \ k   1   2     3   4  ...
%e A324598 ----------------------------------
%e A324598 1,    1:      0
%e A324598 2,    5:      2
%e A324598 3,   11:     (3   7)
%e A324598 4,   19:     (4  14)
%e A324598 5,   29:     (5  23)
%e A324598 6,   31:    (12  18)
%e A324598 7,   41:     (6  34)
%e A324598 8,   55:     (7  47)
%e A324598 9,   59:    (25  33)
%e A324598 10,  61:    (17  43)
%e A324598 11,  71:     (8  62)
%e A324598 12,  79:    (29  49)
%e A324598 13,  89:     (9  79)
%e A324598 14,  95:    (42  52)
%e A324598 15, 101:    (22  78)
%e A324598 16, 109:    (10  98)
%e A324598 17, 121:    (36  84)
%e A324598 18, 131:    (11 119)
%e A324598 19, 139:    (63  75)
%e A324598 20, 145:    (52  93)
%e A324598 ....
%e A324598 29, 209:    (14 194)  (80 128)
%e A324598 ...
%e A324598 41, 319:   (139 179) (150 168)
%e A324598 ...
%e A324598 43, 341:    (18 322)  (80 260)
%e A324598 ...
%e A324598 59, 451:    (47 403) (157 293)
%e A324598 ...
%Y A324598 Cf. A089270, A324599 (x^2 - 5 == 0 (mod N)).
%K A324598 nonn,tabf
%O A324598 1,2
%A A324598 _Wolfdieter Lang_, Jul 08 2019