A271229 Number of solutions of the congruence y^2 == x^3 + x^2 + x (mod p) as p runs through the primes.

2, 2, 7, 7, 15, 15, 15, 15, 15, 23, 39, 31, 47, 47, 47, 55, 63, 63, 63, 79, 63, 71, 79, 95, 95, 119, 119, 95, 111, 95, 119, 127, 143, 127, 135, 135, 159, 175, 191, 167, 191, 175, 191, 191, 215, 215, 191, 215, 239, 207, 223, 223, 223, 271, 255, 255, 279, 279, 303, 255

Offset: 1

Author

Wolfdieter Lang, Apr 18 2016

Keywords

• nonn
• easy

Comments

The discriminant of the elliptic curve y^2 = x^3 + x^2 + x is -3.

Examples

			Here P(n) stands for prime(n).
n,  P(n), a(n)\ Solutions (x, y) modulo P(n)
1,   2,    2:  (0, 0), (1, 1)
2,   3:    2:  (0, 0), (1, 0)
3,   5,    7:  (0, 0), (2, 2), (2, 3), (3, 2), (3, 3),
               (4, 2), (4, 3)
4,   7,    7:  (0, 0), (2, 0), (3, 2), (3, 5), (4, 0),
               (5, 1), (5, 6)
5,  11,   15:  (0, 0), (1, 5), (1, 6), (2, 5), (2, 6),
               (5, 1), (5, 10), (6, 4), (6, 7), (7, 5),
               (7, 6), (8, 1), (8, 10), (9, 4), (9, 7)
...
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Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A159819. A271230. A271231.

Formula

a(n) is the number of solutions of the congruence y^2 == x^3 + x^2 + x (mod prime(n)), n >= 1.

a(n) = prime(n) - A271230(n), n >= 1.