A272198 -id:A272198 - cp's OEIS frontend

A272197 Number of solutions of the congruence y^2 == x^3 + 1 (mod p) as p runs through the primes.

2, 3, 5, 11, 11, 11, 17, 11, 23, 29, 35, 47, 41, 35, 47, 53, 59, 47, 83, 71, 83, 83, 83, 89, 83, 101, 83, 107, 107, 113, 107, 131, 137, 155, 149, 155, 143, 155, 167, 173, 179, 155, 191, 191, 197, 227, 227, 251, 227, 251, 233, 239, 227, 251, 257, 263, 269, 299, 251, 281

Offset: 1

Wolfdieter Lang, May 02 2016

This elliptic curve is discussed in the Silverman reference. In the table the p-defects prime(n) - a(n) are shown for primes 2 to 113.
In the Martin and Ono reference, in Theorem 2, this elliptic curve appears in the eighth row, starting with Conductor 36, as a strong Weil curve for the weight 2 newform eta(6*z)^4, with Im(z) > 0, and the Dedekind eta function. See A000727 which gives the q-expansion (q = exp(2*Pi*i*z)) of exp(-Pi*i*z/3)*eta(z)^4. For the q-expansion of eta(6*z)^4 one has 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.

			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:  (0, 1), (0, 2), (2, 0)
3,   5,       5:  (0, 1), (0, 4), (2, 2),
                  (2, 3), (4, 0)
4,   7,      11:  (0, 1), (0, 6), (1, 3),
                  (1, 4), (2, 3), (2, 4),
                  (3, 0), (4, 3), (4, 4),
                  (5, 0), (6, 0)
5,  11,      11:  (0, 1), (0, 10), (2, 3),
                  (2, 8), (5, 4), (5, 7),
                  (7, 5), (7, 6), (9, 2),
                  (9, 9), (10, 0)

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. (4th ed., Pearson 2014, Exercise 5, p. 371, Exercise 2, p. 385).

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, A000727, A272198.

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

A272200 Bisection of primes congruent to 1 modulo 3 (A002476), depending on the corresponding A001479 entry being congruent to 1 modulo 3 or not. Here the first case.

Original entry on oeis.org

13, 19, 43, 61, 97, 103, 109, 127, 157, 163, 181, 193, 241, 277, 283, 331, 349, 373, 379, 409, 433, 463, 487, 499, 523, 601, 607, 619, 631, 661, 673, 691, 727, 733, 757, 769, 787, 811, 859, 883, 937, 967, 991

Offset: 1

Wolfdieter Lang, Apr 28 2016

The other primes congruent to 1 modulo 3 are given in A272201.
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 (see also A001479). The present sequence gives such primes corresponding to A(m+1) == 1 (mod 3). The ones corresponding to A(m+1) not == 1 (mod 3) (the complement) are given in A272201.
This bisection of the primes from A002476 is needed in the formula for the coefficients of the q-expansion (q = exp(2*Pi*i*z), Im(z) > 0) of the modular cusp form (eta(6*z))^4|A000727%20which%20gives%20the%20coefficients%20of%20the%20q-expansion%20of%20F(q)%20=%20Eta64(q%5E(1/6))/q%5E(1/6)%20=%20(Product">{z=z(q)} = Eta64(q) with Dedekind's eta function. See A000727 which gives the coefficients of the q-expansion of F(q) = Eta64(q^(1/6))/q^(1/6) = (Product{m>=0} (1 - q^m))^4. The coefficients F(q) = Sum_{n>=0} f(6*n+1)*q^n are given in the Finch link on p. 5, using multiplicativity. For primes congruent to 1 modulo 6 the formula involves x_p and y_p which are the present A and B for prime p == 1 (mod 3).
See also the p-defects of the elliptic curve y^2 = x^3 + 1, related to (eta(6*z))^4, given in A272198 with another (equivalent) way to find the coefficients of the Eta64(q) expansion, hence those of F(q).

Robert Israel, Table of n, a(n) for n = 1..10000
Steven R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.

Cf. A000727, A001479, A002476, A001480, A272198, A272201 (complement relative to A002476).

filter:= proc(n) local S,x,y;
    if not isprime(n) then return false fi;
    S:= remove(hastype,[isolve(x^2+3*y^2=n)],negative);
    subs(S[1],x) mod 3 = 1
end proc:
select(filter, [seq(i,i=7..1000,6)]); # Robert Israel, Apr 29 2019

filterQ[n_] := Module[{S, x, y}, If[!PrimeQ[n], Return[False]]; S = Solve[x > 0 && y > 0 && x^2 + 3 y^2 == n, Integers]; Mod[x /. S[[1]], 3] == 1];
Select[Range[7, 1000, 6], filterQ] (* Jean-François Alcover, Apr 21 2020, after Robert Israel *)

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

A272201 Bisection of primes congruent to 1 modulo 3 (A002476), depending on the corresponding A001479 entry being congruent to 1 modulo 3 or not. Here the second case.

Original entry on oeis.org

7, 31, 37, 67, 73, 79, 139, 151, 199, 211, 223, 229, 271, 307, 313, 337, 367, 397, 421, 439, 457, 541, 547, 571, 577, 613, 643, 709, 739, 751, 823, 829, 853, 877, 907, 919, 997

Offset: 1

Wolfdieter Lang, Apr 28 2016

The other primes congruent to 1 modulo 3 are given in A272200, where also more details are given.

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 (see also A001479). The present sequence gives these primes corresponding to A(m+1) not congruent 1 modulo 3. The ones corresponding to A(m+1) == 1 (mod 3) (the complement) are given in A272200.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000727, A001479, A002476, A001480, A272198, A272200 (complement relative to A002476).

filter:= proc(n) local S,x,y;
    if not isprime(n) then return false fi;
    S:= remove(hastype,[isolve(x^2+3*y^2=n)],negative);
    subs(S[1],x) mod 3 <> 1
end proc:
select(filter, [seq(i,i=7..1000,6)]); # Robert Israel, Apr 29 2019

filterQ[n_] := Module[{S, x, y}, If[!PrimeQ[n], Return[False]]; S = Solve[x > 0 && y > 0 && x^2 + 3 y^2 == n, Integers]; Mod[x /. S[[1]], 3] != 1];
Select[Range[7, 1000, 6], filterQ] (* Jean-François Alcover, Apr 21 2020, after Robert Israel *)

This sequence collects the 1 (mod 3) primes p(m) = A002476(m) = A(m)^2 + 3*B(m)^2 with positive A(m) not == 1 (mod 3), for m >= 1. A(m) = A001479(m+1).

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

A272197 Number of solutions of the congruence y^2 == x^3 + 1 (mod p) as p runs through the primes.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

A272200 Bisection of primes congruent to 1 modulo 3 (A002476), depending on the corresponding A001479 entry being congruent to 1 modulo 3 or not. Here the first case.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A272201 Bisection of primes congruent to 1 modulo 3 (A002476), depending on the corresponding A001479 entry being congruent to 1 modulo 3 or not. Here the second case.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula