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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
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
Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.
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
def A276731(n)
A(1, 0, 0, -7, n)
end
CROSSREFS
Sequence in context: A023176 A063178 A280205 * A028891 A028890 A189716
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Sep 16 2016
STATUS
approved