OFFSET
1,4
COMMENTS
This sequence for an elliptic curve (of the Bachet-Mordell type) is discussed in the Silverman reference. In Exercise 45.5, in the table on p. 405, the p-defects are called a_p, and are shown for primes 2 to 113.
The modularity pattern series is the expansion of the 51st modular cusp form of weight 2 and level N=36, given in the table I of the Martin reference, i.e., eta^4(6*z) in powers of q = exp(2*Pi*i*z), with Im(z) > 0. Here eta is the Dedekind function. See A000727 for the expansion in powers of q^6 (after deleting a factor q^(1/6)). Note that also for the possibly bad prime 2 and the bad prime 3 this expansion gives the correct numbers 0 (the discriminant of this elliptic curve is -3^3).
See also the comment on the Martin-Ono reference in A272197 which implies that eta^4(6*z) provides the modularity sequence for this elliptic curve.
For primes p == 0 and 2 (mod 3) (A045309) a(p) = 0. The proof runs along the same line as the one given in the Silverstein reference on pp. 400 - 402 for 17 replaced by 1. From the expansion of the known modularity function eta^4(6*z) follows that only the coefficients for powers q^n with n == 1 (mod 6) are nonzero, and therefore all a(p) for primes p == 0 and 2 (mod 3) have to vanish.
If prime(n) == 1 (mod 3) = A002476(m) (for a unique m = m(n)) then prime(n) = A(m)^2 + 3*B(m)^2 with A(m) = A001479(m+1) and B(m) = A001480(m+1), m >= 1. In this case (4*prime(n) - a(n)^2)/12, seems to be a square, q(m)^2. In fact is seems that (the positive) q(m) = B(m). This is true at least for the first 80 primes 1 (mod 3), i.e. for such primes <= 997. (In the Silverman reference, in hint c) for Exercise 4.5, on p. 405, a more complicated way is suggested: 4*p is decomposed there non-uniquely instead of p uniquely.) If this conjecture is true then a(n) = 2*(+/-sqrt(prime(n) - 3*B(m)^2)) = +- 2*A(m) for prime(n) = A002476(m). This leads to a bisection of the primes 1 (mod 3) into two types: type I if the + sign applies, and type II for the - sign. Primes of type I are given in A272200: 13, 19, 43, 61, 97, ... and those of type II in A272201: 7, 31, 37, 67, 73, ...
REFERENCES
J. H. Silverman, A Friendly Introduction to Number Theory, 3rd ed., Pearson Education, Inc, 2006, Exercise 45.5, p. 405, Exercise 47.2, p. 415, and pp. 400 - 402 (4th ed., Pearson 2014, Exercise 5, p. 371, Exercise 2, p. 385, and pp. 366 - 368).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
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.
FORMULA
a(n) = prime(n) - N(prime(n)), n = 1, where N(prime(n)) = A272197(n), the number of solutions of the congruence y^2 == x^3 + 1 (mod prime(n)).
a(n) = 0 for prime(n) == 0, 2 (mod 3) (see A045309).
The above given conjecture for primes 1 (mod 3) is true because Mordell proved the Ramanujan conjecture on the expansion coefficients of eta^4(6*z), and with the present a(n) the result of Ramanujan follows. See the references and a comment on A000727.
See a comment above for the bisection of the primes 1 (mod 3) into type I and II.
EXAMPLE
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Wolfdieter Lang, May 02 2016
STATUS
approved