We want to solve the equation y2=x3-2 over the integers.

By writing y2+2=x3 we can factor on [-2] as


Using congruencesMathworldPlanetmath modulo 8, one can show that both x,y must be odd, and it can also be shown that (y-i2) and (y+i2) are relatively prime (if it were not the case, any divisorMathworldPlanetmathPlanetmath would have even norm, which is not possible).

Therefore, by unique factorizationMathworldPlanetmath, and using that the only units (http://planetmath.org/UnitsOfQuadraticFields) on [-2] are 1,-1, we have that each factor must be a cube.

So let us write


Then y=a3-6ab2 and 1=3a2b-2b3=b(3a2-2b2). These two equations imply b=±1 and thus a=±1, from where the only possible solutions are x=3,y=±5.


Title y2=x3-2
Canonical name Y2X32
Date of creation 2013-03-22 14:52:05
Last modified on 2013-03-22 14:52:05
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 10
Author CWoo (3771)
Entry type Application
Classification msc 12D05
Classification msc 11R04
Synonym y2+2=x3
Synonym finding integer solutions to y2+2=x3
Related topic UFD