A272204 -id:A272204 - cp's OEIS frontend

A272203 P-defects p - N(p) of the congruence y^2 == x^3 - 1 (mod p) for primes p, where N(p) is the number of solutions given by A272202(n).

0, 0, 0, 4, 0, 2, 0, -8, 0, 0, 4, -10, 0, -8, 0, 0, 0, 14, 16, 0, -10, 4, 0, 0, 14, 0, -20, 0, 2, 0, -20, 0, 0, 16, 0, 4, 14, -8, 0, 0, 0, 26, 0, 2, 0, 28, 16, 28, 0, -22, 0, 0, 14, 0, 0, 0, 0, 28, 26, 0, -32, 0, 16, 0, -22, 0, -32, -34, 0, 14, 0, 0, 4, 38, -8, 0, 0, -34, 0, 38, 0, -22, 0, 2, 28, 0, 0, -10, 0, -20, 0, 0, -44, 0, -32, 0, 0, 0, -8, -46, 40, 0, 0, 0, 16, -46

Offset: 1

Wolfdieter Lang, May 05 2016

The analysis of this elliptic curve runs along the same lines as in A000727, A272197 and A272198, and it is inspired by the Silverman reference where the curve y^2 = x^3 + 1 modulo primes is treated.
The series showing the modularity pattern is the expansion of the 67th modular cusp form of weight 2 and level N=144, given in the table I of the Martin reference, i.e., eta^{12}(12*z)/( eta^4(6*z)*eta^4(24*z)), symbolically 12^{12} 6^(-4) 24^{-4}, in powers of q = exp(2*Pi*i*z), with Im(z) > 0. Here eta is the Dedekind function. See A187076 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 A272202 which implies that 12^{12} 6^(-4) 24^{-4} provides the modularity sequence for this elliptic curve.
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). 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 A272204: 7,13,31,61,67, ... and those of type II in A272205: 19,37,43,73,103, ...

			a(1) = 2 - A272202(1) = 0, and 2 == 2 (mod 3).
a(4) = 7 - A272202(4) = 7 - 3 = +4, and 7 = A002476(1) = 2^2 + 3*1^2, 2 = A001479(1+1), 7 = A272204(1), hence a(4) = +2*2 = +4.
a(8) = 19 - A272202(8) = 19 - 27 = -8, and 19 = A002476(3) = 4^2 + 3*1^2; 4=A001479(3+1), 19 = A272205(1), hence a(8) = 2*(-4) = -8.

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).

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.

Cf. A000040, A187076, A272202, A272204, A272205.

a(n) = prime(n) - N(prime(n)), n = 1, where N(prime(n)) = A272202(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 expected to be true by analogy to the case A272198 where only the signs differ.
a(n) = +2*A001479(m+1) if prime(n) == A002476(m) (m is unique) is a prime of A272204 (type I).
a(n) = -2*A001479(m+1) if prime(n) == A002476(m) is from A272205 (type II).
See a comment above for this bisection of the primes 1 (mod 3) into type I and II.

A272205 A bisection of the primes congruent to 1 modulo 3 (A002476). This is the part depending on the corresponding A001479 entry being congruent to 4 or 5 modulo 6.

Original entry on oeis.org

19, 37, 43, 73, 103, 127, 163, 229, 283, 313, 331, 337, 379, 397, 421, 457, 463, 487, 499, 523, 541, 577, 607, 613, 619, 631, 691, 709, 727, 787, 811, 829, 853, 859, 877, 883, 967, 991, 997

Offset: 1

Wolfdieter Lang, May 05 2016

The other part of this bisection appears in A272204.
Each prime == 1 (mod 3) has a unique representation A002476(m) = A(m)^2 + 3*B(m)^2 with positive A(m) = A001479(m+1) and B(m) = A001480(m+1), m >= 1. The present sequence gives such primes corresponding to A(m) == 4, 5 (mod 6). The ones corresponding to A(m) == 1, 2 (mod 6) (the complement) are given in A272205.
The corresponding A001479 entries are 4, 5, 4, 5, 10, 10, 4, 11, 16, 11, 16, 17, 4, 17, 11, 5, 10, 22, 16, 4, 23, 23, 10, 5, 16, 22, 4, 11, 22, 28, 28, 23, 29, 28, 17, 4, 10, 22, 5, ...
See  A272204 for a comment on the relevance of this bisection in connection with the signs of the q-expansion coefficients of the modular cusp form eta^{12}(12*z) / (eta^4(6*z)*eta^4(24*z)).

Cf. A001479, A001480, A002476, A047239, A187076, A272203, A272204 (complement relative to A002476).

This sequence collects the 1 (mod 3) primes p(m) = A002476(m) = A(m)^2 + 3*B(m)^2 with positive A(m) == 4, 5 (mod 6), for m >= 1. A(m) = A001479(m+1).

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A272203 P-defects p - N(p) of the congruence y^2 == x^3 - 1 (mod p) for primes p, where N(p) is the number of solutions given by A272202(n).

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