

A275739


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


3



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
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OFFSET

1,1


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 2adic 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



PROG

(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)


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



