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.

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, 22, 78, 10, 98, 36, 84, 11, 119, 63, 75, 52, 93, 40, 108, 27, 123, 12, 142, 74, 104, 13, 167, 88, 102, 61, 137, 47, 157, 14, 194, 80, 128, 32, 178

Offset

1,2

Comments

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.

For N(1) = 1 every integer solves this Diophantine equation, and the representative solution is 0.

For N(2) = 5 there is only one representative solution, namely 2.

For n >= 3 the representative solutions come in nonnegtive power of 2 pairs (x1, x2) with x2 = N - 1 - x1.

See the link in A089270 to the W. Lang paper, section 3, and Table 6.

Links

Table of n, a(n) for n=1..60.

Example

The irregular triangle T(n, k) begins (pairs (x, N - 1 - x) in brackets):
n,    N \ k   1   2     3   4  ...
----------------------------------
1,    1:      0
2,    5:      2
3,   11:     (3   7)
4,   19:     (4  14)
5,   29:     (5  23)
6,   31:    (12  18)
7,   41:     (6  34)
8,   55:     (7  47)
9,   59:    (25  33)
10,  61:    (17  43)
11,  71:     (8  62)
12,  79:    (29  49)
13,  89:     (9  79)
14,  95:    (42  52)
15, 101:    (22  78)
16, 109:    (10  98)
17, 121:    (36  84)
18, 131:    (11 119)
19, 139:    (63  75)
20, 145:    (52  93)
....
29, 209:    (14 194)  (80 128)
...
41, 319:   (139 179) (150 168)
...
43, 341:    (18 322)  (80 260)
...
59, 451:    (47 403) (157 293)
...

Crossrefs

Cf. A089270, A324599 (x^2 - 5 == 0 (mod N)).

Sequence in context: A319863 A320948 A086885 * A381119 A229794 A331318

Adjacent sequences: A324595 A324596 A324597 * A324599 A324600 A324601

Keyword

nonn,tabf

Author

Wolfdieter Lang, Jul 08 2019

Status

approved