A275739 The set S of primes q satisfying certain conditions (see Müller, 2010 for precise definition).

617, 1723, 2731, 3191, 6547, 11087, 13103, 21683, 21839, 47737, 49727, 49739, 51679, 52361, 60679, 63719, 117721, 133169, 145531, 232681, 275183, 281353, 306431, 341879, 373463, 607319, 700883, 807241, 1212119, 1240559, 1281331, 1292927, 1353239, 1410361, 1602451, 1679599, 2236907

Offset: 1

Author

Felix Fröhlich, Aug 07 2016

Keywords

• nonn

Comments

Primes q satisfying conditions (18) and (19) on page 1179 of Müller, 2010. The values are given in section 3.2.2 on page 1179.

Let E be the elliptic curve y^2 = x^3 - 3500*x - 98000, and P the point (84, 448) on E. Then these are exactly the primes q satisfying the following four conditions: (i) there exists a point Q in E(F_q) such that 2*Q = P in E(F_q), (ii) the 2-adic valuation of the order of P in E(F_q) equals 1, (iii) there exists a point of order 4 in E(F_q), (iv) the order of P in E(F_q) divides 17272710. Here, E(F_q) denotes the reduction of the elliptic curve E over the finite field of order q. - Robin Visser, Aug 16 2023

Links

• S. Müller, On the existence and non-existence of elliptic pseudoprimes, Mathematics of Computation, Vol. 79, No. 270 (2010), 1171-1190.

Programs

Sage

for q in range(11, 100000):
    if Integer(q).is_prime():
        E = EllipticCurve(GF(q), [-3500, -98000])
        P, od = E(84,448), E(84,448).order()
        if ((17272710%od == 0) and (od.valuation(2) == 1)
            and (E.abelian_group().exponent()%4 == 0)):
            for Q in E:
                if (2*Q == P):
                    print(q)
                    break  # Robin Visser, Aug 16 2023

Extensions

More terms from Robin Visser, Aug 16 2023