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 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.
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)
break # Robin Visser, Aug 16 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Felix Fröhlich, Aug 07 2016
EXTENSIONS
More terms from Robin Visser, Aug 16 2023
STATUS
approved