A276730 Number of solutions to y^2 == x^3 + 4*x (mod p) as p runs through the primes.

2, 3, 7, 7, 11, 7, 15, 19, 23, 39, 31, 39, 31, 43, 47, 39, 59, 71, 67, 71, 79, 79, 83, 79, 79, 103, 103, 107, 103, 127, 127, 131, 159, 139, 135, 151, 135, 163, 167, 199, 179, 199, 191, 207, 199, 199, 211, 223, 227, 199, 207, 239, 271, 251, 255, 263, 295, 271, 295, 271

Offset: 1

Author

Seiichi Manyama, Sep 16 2016

Keywords

• nonn

Comments

This elliptic curve corresponds to a weight 2 newform which is an eta-quotient, namely, (eta(4t)*eta(8t))^2, see Theorem 2 in Martin & Ono.

It appears that a(n) = prime(n) iff prime(n) == 2 or 3 (mod 4). - Robert Israel, Sep 28 2016 This is true due to the L-function of this elliptic curve. See A278720. - Wolfdieter Lang, Dec 22 2016

The rational solutions of y^2 = x^3 + 4*x are (x,y) = (0,0), (2,4), (2,-4). See the Keith Conrad link, Corollary 3.17., p. 9. - Wolfdieter Lang, Dec 01 2016

For the p-defects p - N(p) see A278720. - Wolfdieter Lang, Dec 22 2016

Examples

			The first nonnegative complete residue system {0, 1, ..., prime(n)-1} is used.
The solutions (x, y) of y^2 == x^3 + 4*x (mod prime(n)) begin:
n, prime(n), a(n)\  solutions (x, y)
1,   2,       2:   (0, 0), (1, 1)
2,   3,       3:   (0, 0), (2, 1), (2, 2)
3,   5,       7:   (0, 0), (1, 0), (2, 1),
                   (2, 4), (3, 2), (3, 3),
                   (4, 0)
4,   7,       7:   (0, 0), (2, 3), (2, 4),
                   (3, 2), (3, 5), (6, 3),
                   (6, 4)
...
The solutions (x, y) of y^2 == x^3 - x (mod prime(n)) begin:
n, prime(n), a(n)\  solutions (x, y)
1,   2,       2:   (0, 0), (1, 0);
2,   3,       3:   (0, 0), (1, 0), (2, 0);
3,   5,       7:   (0, 0), (1, 0), (2, 1),
                   (2, 4), (3, 2), (3, 3),
                   (4, 0);
4,   7,       7:   (0, 0), (1, 0), (4, 2),
                   (4, 5), (5, 1), (5, 6),
                   (6, 0);
... - _Wolfdieter Lang_, Dec 22 2016
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• Keith Conrad, Expository papers, Proofs by Descent.
• Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.

Crossrefs

Cf. A095978, A272207, A278720.

Programs

Maple

seq(nops([msolve(y^2-x^3-4*x, ithprime(n))]),n=1..100); # Robert Israel, Sep 28 2016

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 A276730(n)
  A(0, 0, 4, 0, n)
end

Formula

a(n) is the number of solutions of the congruence y^2 == x^3 + 4*x (mod prime(n)), n >= 1.

a(n) is also the number

of solutions of the congruence y^2 == x^3 - x (mod prime(n)), n >= 1. - Wolfdieter Lang, Dec 22 2016 (See the Cremona link given in A278720).