Powers in Progression
There cannot be 4 or more nth powers in arithmetic progression [n>1]
Proof:
e.g. n=2, generalizes...
for smallest, hcf (w,x,y,z) = 1
(z+y)(z-y) = (y+x)(y-x) = (x+w)(x-w)
hcf (z+y, z-y) = hcf (y+x, y-x) = hcf (x+w, x-w) = 2
z+y = {2}(y-x)
y-x = {2}(x+w)
x+w = {2}(z-y)
{2}(z-y) = y+x
{2}(y+x) = x-w
{2}(x-w) = z+y
and w = y
=> w = x = y = z