cp's OEIS Frontend

An elliptic curve E over a field K is a nonsingular algebraic curve defined by a minimal Weierstrass equation

E/K: y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6

for some coefficients a_1, a_2, a_3, a_4, a_6 in K and discriminant of E not equal to zero.

Associated to E is an L-function L(E,s) = sum_{N} a_N / N^s. The map sending the positive integer N to a_N is a multiplicative function. Moreover, a_p = p + 1 - #E(F_p) with E(F_p) defined below, and a_{p^e} = a_p a_{p^{e-1}} - 1_E(p) p a_{p^{e-2}} for all e >= 2 where 1_E(p) is 1 if p does not divide the discriminant of E and is 0 otherwise.

Let E(Q) be an elliptic curve over Q, the field of rational numbers. We can replace y and x with scalar multiples such that the defining equation of E has integer coefficients.

For an integer N > 2, let E(Z/NZ) be the set of points (x,y) satisfying the defining equation of E in Z/NZ, the ring of integers modulo N, and the "points at infinity" (identity element). If the discriminant of E is coprime to N, then E(Z/NZ) forms an Abelian group.

If the discriminant of E is indivisible by a prime p and if #E(F_p) = p, then p is called an anomalous prime for E. There are no anomalous primes for the curve y^2 = x^3 + 80. This was shown in a paper by H. Qin (2016), see link.

The notions of strong elliptic pseudoprimes and strong elliptic Carmichael numbers are defined in "Anomalous Primes and Extension of the Korselt Criterion", see link. A Korselt criterion for these two notions was proven in "Anomalous Primes and Extension of the Korselt Criterion" based on the Korselt criterion developed in "Elliptic Carmichael Numbers and Elliptic Korselt Criteria" (J. Silverman, 2012).

Let N be a composite number, and P be a point of E(Z/NZ). Suppose that N+1-a_N is even, N has at least two distinct prime factors, and N is coprime to the discriminant of E. Then, N is an Euler elliptic pseudoprime for (E,P) if ((N+1-a_N)/2) P is the identity if P = 2Q for some Q in E(Z/NZ) and is of the form (x,y) where 2y + a_1 x + a_3 = 0 modulo N otherwise. For a prime p, let ord_p(N) be the p-adic order of N. Also let e_{N,p}(E) be the exponent of the group E(Z/p^(ord_p(N))Z). N is an Euler elliptic Carmichael number for E if and only if N has at least two distinct prime factors, N is coprime to the discriminant of E, and, for every prime p dividing N, 2e_{N,p}(E) divides N+1-a_N.

An elliptic curve E over a field K is a nonsingular algebraic curve defined by a minimal Weierstrass equation

E/K: y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6

for some coefficients a_1, a_2, a_3, a_4, a_6 in K and discriminant of E not equal to zero.

Associated to E is an L-function L(E,s) = Sum_{N} a_N / N^s. The map sending the positive integer N to a_N is a multiplicative function. Moreover, a_p = p + 1 - #E(F_p) with E(F_p) defined below, and a_{p^e} = a_p a_{p^{e-1}} - 1_E(p) p a_{p^{e-2}} for all e >= 2 where 1_E(p) is 1 if p does not divide the discriminant of E and is 0 otherwise.

Let E(Q) be an elliptic curve over Q, the field of rational numbers. We can replace y and x with scalar multiples such that the defining equation of E has integer coefficients.

For an integer N > 2, let E(Z/NZ) be the set of points (x,y) satisfying the defining equation of E in Z/NZ, the ring of integers modulo N, and the "points at infinity" (identity element). If the discriminant of E is coprime to N, then E(Z/NZ) forms an Abelian group.

If the discriminant of E is indivisible by a prime p and if #E(F_p) = p, then p is called an anomalous prime for E. There are no anomalous primes for the curve y^2 = x^3 + 80. This was shown in a paper by H. Qin (2016), see link.

The notions of strong elliptic pseudoprimes and strong elliptic Carmichael numbers are defined in "Anomalous Primes and Extension of the Korselt Criterion", see link. A Korselt criterion for these two notions was proven in "Anomalous Primes and Extension of the Korselt Criterion" based on the Korselt criterion developed in "Elliptic Carmichael Numbers and Elliptic Korselt Criteria" (J. Silverman, 2012).

Let N be a composite number, and P be a point of E(Z/NZ). Suppose that N has at least two distinct prime factors and N is coprime to the discriminant of E. Furthermore, write N+1-a_N as N+1-a_N = 2^r * t for integers r and t with t odd. Then, N is a strong elliptic pseudoprime for (E,P) if tP is the identity or (2^s t) P is a point of the form (x,y) where 2y + a_1 x + a_3 = 0 modulo N for some integer s with 0 <= s < r. For a prime p, let ord_p(N) be the p-adic order of N. Also let e_{N,p}(E) be the exponent of the group E(Z/p^(ord_p(N))Z). N is a strong elliptic Carmichael number for E if and only if N has at least two distinct prime factors, N is coprime to the discriminant of E, and, for every prime p dividing N, e_{N,p}(E) divides t.

Let p>3 be a prime and Z/pZ the field of integers modulo p. An elliptic curve E over Z/pZ, denoted by E(Z/pZ), is a set of points (x,y) in Z/pZ x Z/pZ such that y^2 = x^3 + ax + b with discriminant not equal to zero (4*a^3 + 27*b^2 != 0), and an additional point O, called the "point at infinity".

An elliptic curve can be seen as an additive Abelian group with the point at infinity as an identity element. The order of the elliptic curve, the number of points including the point at infinity, is denoted by #E(Z/pZ). There is another equivalent definition of elliptic curve in projective coordinates. Namely, the elliptic curve E(Z/pZ) is a set of points (x:y:z) in P^2(Z/pZ) that satisfy the equation y^2z = x^3 + axz^2 + bz^3. Here, the points (x,y) are mapped to (x:y:1), and O is mapped to (0:1:0). The formulas for computing multiples and adding points can be found in "Elliptic Curves: Number Theory and Cryptography" by L. C. Washington.

For an integer N > 2, let E(Z/NZ) be the set of points (x,y) satisfying the defining equation of E in Z/NZ, the ring of integers modulo N, and the "point at infinity" (identity element). If the discriminant of E is coprime to N, then E(Z/NZ) forms an Abelian group.

Associated to E is an L-function L(E,s) = Sum_{N} a_N / N^s. The map sending the positive integer N to a_N is a multiplicative function. Moreover, a_p = p + 1 - #E(Z/pZ) with E(Z/pZ) defined below, and a_{p^e} = a_p a_{p^{e-1}} - 1_E(p) p a_{p^{e-2}} for all e >= 2 where 1_E(p) is 1 if p does not divide the discriminant of E and is 0 otherwise.

The notions of elliptic pseudoprimes and elliptic Carmichael numbers are defined in "Elliptic Carmichael Numbers and Elliptic Korselt Criteria", see link. Let N be a composite number, and P be a point of E(Z/NZ). Suppose that N has at least two distinct prime factors and N is coprime to the discriminant of E. Then, N is an elliptic pseudoprime for (E,P) if (N+1-a_N)P is the identity. N is a Carmichael number for E if it is a pseudoprime at (E,P) for all point P on E.

A Korselt criterion for the notions of elliptic pseudoprimes and elliptic Carmichael numbers was proved in "Elliptic Carmichael Numbers and Elliptic Korselt Criteria", see link. For a prime p, let ord_p(N) be the p-adic order of N. Also let e_{N,p}(E) be the exponent of the group E(Z/p^(ord_p(N))Z). Then N is an elliptic Carmichael number for E if and only if N has at least two distinct prime factors, N is coprime to the discriminant of E, and, for every prime p dividing N, e_{N,p}(E) divides N+1-a_N.

The resulting sequence is based on work done during the REU program, "Complexity Across Disciplines", supported by the National Science Foundation under the grant DMS -1659872.