A276731 - OEIS

%I #29 Oct 24 2018 08:08:10

%S 2,3,5,8,11,8,17,26,23,29,35,26,41,35,47,53,59,62,62,71,80,62,83,89,

%T 116,101,116,107,107,113,107,131,137,116,149,170,143,188,167,173,179,

%U 188,191,170,197,188,224,251,227,251,233,239,224,251,257,263,269,242,251,281

%N Number of solutions to y^2 + y == x^3 - 7 (mod p) as p runs through the primes.

%C 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.

%C a(n) is the number of solutions of the congruence y^2 + y == x^3 - 7 (mod prime(n)), n >= 1.

%C a(n) is also the number of solutions of the congruence y^2 == x^3 - 432 (mod prime(n)), n >= 1.

%H Seiichi Manyama, <a href="/A276731/b276731.txt">Table of n, a(n) for n = 1..10000</a>

%H Yves Martin and Ken Ono, <a href="http://dx.doi.org/10.1090/S0002-9939-97-03928-2">Eta-Quotients and Elliptic Curves</a>, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.

%e The first nonnegative complete residue system {0, 1, ..., prime(n)-1} is used.

%e The solutions (x, y) of y^2 + y == x^3 - 7 (mod prime(n)) begin:

%e n, prime(n), a(n)\  solutions (x, y)

%e 1,   2,       2:   (1, 0), (1, 1)

%e 2,   3,       3:   (0, 1), (1, 0), (1, 2)

%e 3,   5,       5:   (2, 2), (3, 0), (3, 4),

%e                    (4, 1), (4, 3)

%e 4,   7,       8:   (0, 0), (0, 6), (3, 2),

%e                    (3, 4), (5, 2), (5, 4),

%e                    (6, 2), (6, 4)

%o (Ruby)

%o require 'prime'

%o def A(a3, a2, a4, a6, n)

%o   ary = []

%o   Prime.take(n).each{|p|

%o     a = Array.new(p, 0)

%o     (0..p - 1).each{|i| a[(i * i + a3 * i) % p] += 1}

%o     ary << (0..p - 1).inject(0){|s, i| s + a[(i * i * i + a2 * i * i + a4 * i + a6) % p]}

%o   }

%o   ary

%o end

%o def A276731(n)

%o   A(1, 0, 0, -7, n)

%o end

%K nonn

%O 1,1

%A _Seiichi Manyama_, Sep 16 2016