OFFSET
1,1
COMMENTS
k has a set of general solutions for any integers (a,b,c): k=a^4 (b^4-12 c^4)(3 b^4-4 c^4);
y^2==x^3+k^3 has a solution: {x,y} = {a^4 (-3 b^8+48 b^6 c^2-152 b^4 c^4+192 b^2 c^6-48 c^8),12 a^6 (3 b^11 c-30 b^9 c^3+136 b^7 c^5-272 b^5 c^7+240 b^3 c^9-96 b c^11)};
y^2==x^3-k^3 has a solution: {x,y} = {a^4 (3 b^8+48 b^6 c^2+152 b^4 c^4+192 b^2 c^6+48 c^8),12 a^6 (3 b^11 c+30 b^9 c^3+136 b^7 c^5+272 b^5 c^7+240 b^3 c^9+96 b c^11)}.
LINKS
Zhining Yang, Table of n, a(n) for n = 1..1202
XiaoYang Zhang, How many prime numbers P are currently known to satisfy the equation y^2 = x^3 +- P^2? (in Chinese).
EXAMPLE
Integer 11 is in the sequence because both Mordell equations y^2 = x^3 + 11^3 and y^2 = x^3 - 11^3 have integer solutions as {{37,228}} and {{443,9324}}.
PROG
(SageMath)
def f(start):
for k in range(start, start + 201):
try:
E1 = EllipticCurve([0, 0, 0, 0, -k^3])
E2 = EllipticCurve([0, 0, 0, 0, k^3])
pts1 = E1.integral_points()
pts2 = E2.integral_points()
if any(pt[0] != k for pt in pts1) and any(pt[0] != -k for pt in pts2):
print(k)
except:
pass
f(1)
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhining Yang, Jun 19 2026
STATUS
approved