A165242 The larger member of the n-th twin prime pair, modulo 8.

5, 7, 5, 3, 7, 3, 5, 1, 7, 5, 3, 7, 5, 1, 7, 5, 1, 7, 3, 1, 5, 5, 1, 7, 3, 3, 1, 3, 3, 5, 3, 7, 5, 3, 3, 5, 1, 3, 7, 5, 1, 7, 7, 3, 7, 1, 5, 5, 3, 1, 1, 5, 5, 3, 3, 5, 1, 7, 5, 7, 7, 5, 3, 1, 1, 3, 7, 7, 5, 7, 5, 7, 7, 1, 3, 1, 1, 3, 7, 3, 3, 1, 1, 1, 5, 3, 5, 3, 1, 5, 7, 7, 5, 1, 5, 7, 7, 1, 1, 7, 5, 7, 3, 3, 5

Offset: 1

Author

Jonathan Vos Post, Sep 09 2009

Keywords

• easy
• nonn

Comments

Related to the rank of some elliptic curves by the conjecture on page 2 of [Hatley]:

Let E_p be the elliptic curve defined by y^2 = x(x-p)(x-2) where p and p-2 are twin primes.

Then Rank(E_p) = 0 if p == 7 (mod 8), 1 if p == 3,5 (mod 8), 2 if p == 1 (mod 8).

References

• Joseph H. Silverman, The Arithmetic of Elliptic Curves, Springer-Verlag, 1986.

Links

• Jeffrey Hatley, On the Rank of the Elliptic Curve y^2=x(x-p)(x-2), arXiv:0909.1614 [math.NT], 2009.

Crossrefs

Cf. A000040, A001359, A010877.

Programs

Maple

A006512 := proc(n) if n = 1 then 5; else for a from procname(n-1)+2 by 2 do if isprime(a) and isprime(a-2) then RETURN(a) ; fi; od: fi; end:
A165242 := proc(n) A006512(n) mod 8 ; end: seq(A165242(n),n=1..120) ; # R. J. Mathar, Sep 16 2009

Mathematica

Mod[#,8]&/@(Select[Partition[Prime[Range[800]],2,1],#[[2]]-#[[1]]==2&][[All,2]]) (* Harvey P. Dale, Sep 26 2016 *)

Formula

a(n) = A010877(A006512(n)).

Extensions

Redefined for the larger member of twin primes by R. J. Mathar, Sep 16 2009