|
|
A276731
|
|
Number of solutions to y^2 + y == x^3 - 7 (mod p) as p runs through the primes.
|
|
1
|
|
|
2, 3, 5, 8, 11, 8, 17, 26, 23, 29, 35, 26, 41, 35, 47, 53, 59, 62, 62, 71, 80, 62, 83, 89, 116, 101, 116, 107, 107, 113, 107, 131, 137, 116, 149, 170, 143, 188, 167, 173, 179, 188, 191, 170, 197, 188, 224, 251, 227, 251, 233, 239, 224, 251, 257, 263, 269, 242, 251, 281
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
This elliptic curve corresponds to a weight 2 newform which is an eta-quotient, namely, (eta(3t)*eta(9t))^2, see Theorem 2 in Martin & Ono.
a(n) is the number of solutions of the congruence y^2 + y == x^3 - 7 (mod prime(n)), n >= 1.
a(n) is also the number of solutions of the congruence y^2 == x^3 - 432 (mod prime(n)), n >= 1.
|
|
LINKS
|
|
|
EXAMPLE
|
The first nonnegative complete residue system {0, 1, ..., prime(n)-1} is used.
The solutions (x, y) of y^2 + y == x^3 - 7 (mod prime(n)) begin:
n, prime(n), a(n)\ solutions (x, y)
1, 2, 2: (1, 0), (1, 1)
2, 3, 3: (0, 1), (1, 0), (1, 2)
3, 5, 5: (2, 2), (3, 0), (3, 4),
(4, 1), (4, 3)
4, 7, 8: (0, 0), (0, 6), (3, 2),
(3, 4), (5, 2), (5, 4),
(6, 2), (6, 4)
|
|
PROG
|
(Ruby)
require 'prime'
def A(a3, a2, a4, a6, n)
ary = []
Prime.take(n).each{|p|
a = Array.new(p, 0)
(0..p - 1).each{|i| a[(i * i + a3 * i) % p] += 1}
ary << (0..p - 1).inject(0){|s, i| s + a[(i * i * i + a2 * i * i + a4 * i + a6) % p]}
}
ary
end
A(1, 0, 0, -7, n)
end
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|