A396946 Primes p such that both Mordell equations y^2 = x^3 + p^2 and y^2 = x^3 - p^2 have nonzero integer solutions.

11, 17, 47, 101, 197, 431, 2027, 2351, 24337, 27647, 65537, 90001, 156817, 184511, 284591, 314927, 414737, 720299, 746497, 1726597, 5532491

Offset

1,1

Comments

a(22) > 10^8.

Conjecture: All terms can be expressed in the form of 4*m^2 + 1 or 12*m^2 - 1.

Links

Table of n, a(n) for n=1..21.

XiaoYang Zhang, How many prime numbers P are currently known to satisfy the equation y^2 = x^3 +- P^2? (in Chinese).

Example

Prime 101 is in the sequence because both Mordell equations y^2 = x^3 + 101^2 and y^2 = x^3 - 101^2 have integer solutions as {{24, 155}} and {{101, 1010}}.

Prog

(SageMath)
def f(p):
    for k in primes(p, p + 1000):
        try:
            E1 = EllipticCurve([0, 0, 0, 0, -k^2])
            E2 = EllipticCurve([0, 0, 0, 0,  k^2])
            pts1 = E1.integral_points()
            pts2 = E2.integral_points()
            if any(pt[0] != 0 for pt in pts1) and any(pt[0] != 0 for pt in pts2):
                print(k)
        except:
            pass
f(1)

Crossrefs

Cf. A081120, A081119, A396853.

Sequence in context: A267772 A046122 A217064 * A377856 A242244 A391321

Adjacent sequences: A396942 A396943 A396944 * A396948 A396949 A396950

Keyword

nonn,hard,more

Author

Zhining Yang, Jun 11 2026

Status

approved