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