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A275739 The set S of primes q satisfying certain conditions (see Müller, 2010 for precise definition). 3

%I #17 Aug 17 2023 08:17:32

%S 617,1723,2731,3191,6547,11087,13103,21683,21839,47737,49727,49739,

%T 51679,52361,60679,63719,117721,133169,145531,232681,275183,281353,

%U 306431,341879,373463,607319,700883,807241,1212119,1240559,1281331,1292927,1353239,1410361,1602451,1679599,2236907

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

%C 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.

%C 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

%H S. Müller, <a href="">On the existence and non-existence of elliptic pseudoprimes</a>, Mathematics of Computation, Vol. 79, No. 270 (2010), 1171-1190.

%o (Sage)

%o for q in range(11, 100000):

%o if Integer(q).is_prime():

%o E = EllipticCurve(GF(q), [-3500, -98000])

%o P, od = E(84,448), E(84,448).order()

%o if ((17272710%od == 0) and (od.valuation(2) == 1)

%o and (E.abelian_group().exponent()%4 == 0)):

%o for Q in E:

%o if (2*Q == P):

%o print(q)

%o break # _Robin Visser_, Aug 16 2023

%K nonn

%O 1,1

%A _Felix Fröhlich_, Aug 07 2016

%E More terms from _Robin Visser_, Aug 16 2023

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