cp's OEIS Frontend

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

%I A275739 #17 Aug 17 2023 08:17:32
%S A275739 617,1723,2731,3191,6547,11087,13103,21683,21839,47737,49727,49739,
%T A275739 51679,52361,60679,63719,117721,133169,145531,232681,275183,281353,
%U A275739 306431,341879,373463,607319,700883,807241,1212119,1240559,1281331,1292927,1353239,1410361,1602451,1679599,2236907
%N A275739 The set S of primes q satisfying certain conditions (see Müller, 2010 for precise definition).
%C A275739 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 A275739 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 A275739 S. Müller, <a href="http://dx.doi.org/10.1090/S0025-5718-09-02275-3">On the existence and non-existence of elliptic pseudoprimes</a>, Mathematics of Computation, Vol. 79, No. 270 (2010), 1171-1190.
%o A275739 (Sage)
%o A275739 for q in range(11, 100000):
%o A275739     if Integer(q).is_prime():
%o A275739         E = EllipticCurve(GF(q), [-3500, -98000])
%o A275739         P, od = E(84,448), E(84,448).order()
%o A275739         if ((17272710%od == 0) and (od.valuation(2) == 1)
%o A275739             and (E.abelian_group().exponent()%4 == 0)):
%o A275739             for Q in E:
%o A275739                 if (2*Q == P):
%o A275739                     print(q)
%o A275739                     break  # _Robin Visser_, Aug 16 2023
%K A275739 nonn
%O A275739 1,1
%A A275739 _Felix Fröhlich_, Aug 07 2016
%E A275739 More terms from _Robin Visser_, Aug 16 2023