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

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.

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).

EXAMPLE

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

MAPLE

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

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 A276730(n)

  A(0, 0, 4, 0, n)

end

CROSSREFS

Cf. A095978, A272207, A278720.

Sequence in context: A011161 A305420 A171464 * A179894 A060215 A309666

Adjacent sequences:  A276727 A276728 A276729 * A276731 A276732 A276733

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Sep 16 2016

STATUS

approved

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Last modified October 21 13:44 EDT 2021. Contains 348155 sequences. (Running on oeis4.)