A267859 The p-defect p - N(p) of the elliptic curve y^2 = x^3 + x for primes p congruent to 1 modulo 4 (A002144).

2, -6, 2, 10, 2, 10, -14, 10, -6, 10, 18, 2, -6, -14, -22, -14, -22, 26, 18, -14, 2, -30, 26, -30, 2, 26, 18, 10, 34, 26, -22, 18, 10, 34, -14, 34, -38, 2, -6, -30, 34, -14, 42, -38, 10, -22, 42, -38, 26, 2, -46, 10, 34, -38, 50, 26, 50, -46, 2, 10, -30, -54, 18, -38, 50, 34, -22, 10, 50, -54

Offset: 1

Author

Wolfdieter Lang, Feb 06 2016

Keywords

• sign

Comments

See A002172 for a differently signed sequence.

The number N(p) of solutions modulo a prime p of the elliptic curve y^2 = x^3 + x (of discriminant -4) is given for all p in A095978.

The p-defect a_p = p - N(p) for prime 2 and primes congruent to 3 modulo 4 vanishes.

A002144(n) - (a(n)/2)^2 = (2*A002973(n))^2, n >= 1. See the formula for A095978 for primes 1 (mod 4).

This sequence gives also the non-vanishing p-defects of the elliptic curve y^2 = x^3 - 4*x. See a comment on A138515 with the Martin and Ono link for the modularity series for these two elliptic curves. - Wolfdieter Lang, May 26 2016

Examples

			n = 2: p = A002144(2) = 13 = A000040(6), m = 6, a(2) = 13 - A095978(6) = 13 - 19  = -6.
n = 2:  -6 = A138515((A002144(2) - 1)/4) =
A138515(3) = -6. - _Wolfdieter Lang_, May 26 2016
		

References

• J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, p. 398. In the 4th ed., 2014, p. 371.

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000040, A002144, A002172, A095978, A138515.

Programs

Mathematica

terms = 100; A002144 = Select[Range[5, 20*terms, 4], PrimeQ]; A095978[n_] := Module[{p, xy, x}, p = Prime[n]; If[n==1 || Mod[p, 4]==3, Return[p]]; xy = {Re[#], Im[#]}& @ FactorInteger[p, GaussianIntegers -> True][[2, 1]]; x = SelectFirst[xy, OddQ]; If[Mod[x, 4]==1, p - 2*x, p + 2*x]]; a[n_] := (p = A002144[[n]]; m = PrimePi[p]; p - A095978[m]); Array[a, terms] (* Jean-François Alcover, Feb 26 2016, after Robert Israel (A095978) *)

Formula

a(n) = A002144(n) - A095978(m) with A002144(n) = A000040(m), n >= 1.

a(n) = A138515((A002144(n) - 1)/4), n >= 1. - Wolfdieter Lang, May 26 2016