Since every 6th power is also a cube, write (n*a)^4+(a^2)^3=c^2.
Then LHS = a^4(a^2+n^4). Let (a^2+n^4)=x^2, since then we have a^4x^2, which is square.
We can also write this as n^4=(x^2-a^2), a difference of squares, so that that at least one solution is guaranteed for every n.
Say n=3, then a^2+81=x^2 has a solution a=40, x=41, with b=120 and c=65600.
Say n=5, then a^2+625=x^2 has a solution a=312, x=313, with b=1560 and c=30468672.
etc.
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Posted by broll
on 2019-10-31 09:28:36 |
As many as you like
