A279392 -id:A279392 - cp's OEIS frontend

A278720 The p-defect p - N(p) of the congruence y^2 == x^3 + 4*x (mod p) for primes p, where N(p) is the number of solutions given by A276730.

0, 0, -2, 0, 0, 6, 2, 0, 0, -10, 0, -2, 10, 0, 0, 14, 0, -10, 0, 0, -6, 0, 0, 10, 18, -2, 0, 0, 6, -14, 0, 0, -22, 0, 14, 0, 22, 0, 0, -26, 0, -18, 0, -14, -2, 0, 0, 0, 0, 30, 26, 0, -30, 0, 2, 0, -26, 0, -18, 10, 0, -34, 0, 0, 26, 22, 0, 18, 0, -10, 34, 0, 0, 14, 0, 0, -34, 38, 2, -6, 0, 30, 0, 34, 0, 0, -14, 42, 38, 0, 0, 0, 0, 0, 0, 0, -10, -22, 0, -42

Offset: 1

Wolfdieter Lang, Dec 11 2016

This sequence gives also the p-defects for the congruences y^2 == x^3 - x (mod p), y^2 == x^3 - 11*x - 14 (mod p) and y^2 == x^3 - 11*x + 14 (mod p). See the Cremona link, Table 1, N = 32. - Wolfdieter Lang, Dec 22 2016
This elliptic curve y^2 = x^3 + 4*x appears as strong Weil curve for the weight 2 newform (eta(4*tau)*eta(8*tau))^2 of level N=32, with Dedekind's eta function. See the Martin-Ono link, Theorem 2, p. 3173, the row with Conductor 32. See also A002171 for the expansion of this newform in powers of q^4 (but with different offset). The also Nr. 49 of the Martin Table 1.
From this L-series of this elliptic curve one has:
a(n) = 0 if prime(n) == 2 or 3 (mod 4). (see the conjecture by Robert Israel, Sep 28 2016 in A276730).
If prime(n) == 1 (mod 4) = A002144(m) (for a unique m = m(n)) then prime(n) = A(m)^2 + B(m)^2 with the odd A(m) = A002972(m) and the even B(m) = 2*A002973(m). It turns out that 4*A002144(m) - a(m^2) = (2*B(m))^2 for m=m(n), and the sign s(m) of a(m) is + if A(m) + B(m) == 1 (mod 4) and - if A(m) + B(m) == 3 (mod 4). For the primes == 1 (mod 4) leading to sign + or - see A279392 or A279393, respectively. One has thus s(m) = (-1)^((A(m)-1)/2 + B(m)/2). See the Martin-Ono formula for a_{32}(p) in Theorem 3, p. 3175. This leads to the a(n) formula given below.

			a(1) = 0  because prime(1) = 2 == 2 (mod 4).
a(2) = 0 because prime(2) = 3 == 3 (mod 4).
a(3) = -2 because prime(3) = 5 = A002144(1) = A002972(1)^2 + (2*A002973(1))^2 = 1^2 + 2^2. Hence 2*A(1) = 2*A002972(1) = 2, and the sign s(1) = - because A(1) + B(1) = 1 + 2*1 = 3 == 3 (mod 4).
a(6) = +6 because prime(6) = 13 = A002144(2) = A(2)^2 + B(2)^2 = 3^2 + (2*1)^2. Hence 2*A(2) = 6 and the sign is + because A(2) + B(2) = 3 + 2 = 5 == 1 (mod 4).

Seiichi Manyama, Table of n, a(n) for n = 1..10000
George E. Andrews, Number Theory, see S(1) expression p. 135.
J. E. Cremona, Algorithms for Modular Elliptic Curves.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.

Cf. A000040, A002144, A002171, A002972, A002973, A279392, A279393.

a(n) =  my(p=prime(n)); -sum(k=1, p-3, kronecker(k*(k+1)*(k+2), p)); \\ Michel Marcus, Nov 06 2020

a(n) = 0 if prime(n) == 2 or 3 (mod 4) (this is conjecture II from above).
a(n) = s(m)*2*A(m) if prime(n) = A002144(m), with A(m) = A002972(m) and the sign s(m) = (-1)^((A(m)-1)/2 + B(m)/2).
a(n) = - Sum_{k=1..p-3} ((k*(k+1)*(k+2))/p) where (x/y) is the Kronecker symbol. - Michel Marcus, Nov 06 2020

A279393 Bisection of primes congruent to 1 modulo 4 (A002144), depending on the corresponding sum of the A002972 and 2*A002973 entries being congruent to 1 modulo 4 or not. Here we give the second case.

Original entry on oeis.org

5, 29, 37, 61, 73, 101, 113, 137, 173, 181, 193, 197, 241, 269, 277, 293, 349, 389, 409, 449, 509, 521, 541, 593, 613, 617, 653, 661, 673, 677, 701, 757, 761, 773, 821, 877, 929, 937, 941, 977, 1009, 1033, 1069, 1093, 1109, 1117, 1129, 1181, 1217, 1237, 1249, 1289

Offset: 1

Wolfdieter Lang, Dec 11 2016

See A279392 for details of this bisection of the primes of A002144. This sequence gives the part II of primes congruent 1 modulo 4.

			a(1) = 5 = A002144(1) and A002972(1) = 1 and 2*A002973(1) = 2, hence 1 + 2 = 3 == 3 (mod 4), and 5 belongs to part II of this bisection.

Cf. A002144, A002972, A002973, A279392.

A prime A002144(m) = A(m)^2 + B(m)^2 belongs to this sequence iff (-1)^((A(m)-1)/2 + B(m)/2) = -1, where A(m) = A002972(m) and B(m)/2 = A002973(m).

More terms from Jinyuan Wang, Apr 20 2025

cp's OEIS Frontend

A278720 The p-defect p - N(p) of the congruence y^2 == x^3 + 4*x (mod p) for primes p, where N(p) is the number of solutions given by A276730.

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