A272197 -id:A272197 - cp's OEIS frontend

A272198 The p-defect 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 A272197(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

Offset: 1

Wolfdieter Lang, May 02 2016

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, ...

			a(1) = 2 - A272197(1) = 0, and 2 == 2(mod 3).
a(4) = 7 - A272197(4) = 7 - 11 = -4, and 7 = A002476(1) = 2^2 + 3*1^2, 2 = A001479(1+1), 7 = A272201(1), hence a(4) = -2*2 = -4.
a(6) = 13 - A272197(6) = 13 - 11 = 2, and 13 = A002476(2) = 1^2 + 3*2^2; 1 = A001479(2+1), 13 = A272200(1), hence a(6) = +2*1 = +2.

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, A002476, A001479, A001480, A045309, A272197, A272200, A272201.

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.
a(n) = +2*A001479(m+1) if prime(n) == A002476(m) (m is unique) is a prime of A272200 (type I).
a(n) = -2*A001479(m+1) if prime(n) == A002476(m) is from A272201 (type II).
See a comment above for the bisection of the primes 1 (mod 3) into type I and II.

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).

Original entry on oeis.org

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.

A272202 Number of solutions of the congruence y^2 == x^3 - 1 (mod p) as p runs through the primes.

Original entry on oeis.org

2, 3, 5, 3, 11, 11, 17, 27, 23, 29, 27, 47, 41, 51, 47, 53, 59, 47, 51, 71, 83, 75, 83, 89, 83, 101, 123, 107, 107, 113, 147, 131, 137, 123, 149, 147, 143, 171, 167, 173, 179, 155, 191, 191, 197, 171, 195, 195, 227, 251, 233, 239, 227, 251, 257, 263, 269, 243, 251, 281

Offset: 1

Wolfdieter Lang, May 05 2016

In the Martin and Ono reference, in Theorem 2, this elliptic curve appears in the last column, starting with Conductor 144, as a strong Weil curve for the weight 2 newform  eta^{12}(12*z) / (eta^4(6*z) * eta^4(24*z)), symbolically 12^{12} 6^{-4} 24^{-4}, with Im(z) > 0, and the Dedekind eta function. See A187076 which gives the q-expansion (q = exp(2*Pi*i*z)) of exp(-Pi*i*z/3)* eta(2*z)^{12} / (eta^4(z)*eta^4(4*z)). For the q-expansion of 12^{12} 6^{-4} 24^{-4} one has a leading zero and 5 interspersed 0's: 0,1,0,0,0,0,0,4,0,0,0,0,0,2,0,0,0,0,0,-8,...
The discriminant of this elliptic curve is -3^3 = -27.
For the elliptic curve y^2 == x^3 + 1 (mod prime(n)) see A000727, A272197, A272198, A272200 and A272201.

			The first nonnegative complete residue system {0, 1, ..., prime(n)-1} is used. The solutions (x, y) of y^2 == x^3 - 1 (mod prime(n)) begin:
n, prime(n), a(n)\ solutions (x, y)
1,    2,      2:   (0, 1), (1, 0)
2,    3,      3:   (1, 0), (2, 1), (2, 2)
3,    5,      5:   (0, 2), (0, 3), (1, 0),
                   (3, 1), (3, 4)
4,    7,      3:   (1, 0), (2, 0), (4, 0)
5,    11,    11:   (1, 0), (3, 2), (3, 9),
                   (5, 5), (5, 6), (7, 1),
                   (7, 10), (8, 4), (8, 7),
                   (10, 3), (10, 8)
...

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Yves Martin and Ken Ono, Eta-Quotients and Elliptic Curves, Proc. Amer. Math. Soc. 125, No 11 (1997), 3169-3176.

Cf. A000040, A187076, A272203.

a(n) gives the number of solutions of the congruence y^2 == x^3 - 1 (mod prime(n)), n >= 1.

cp's OEIS Frontend

A272198 The p-defect 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 A272197(n).

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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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A272202 Number of solutions of the congruence y^2 == x^3 - 1 (mod p) as p runs through the primes.

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