Show that 3^w+3^x+3^y+3^z is a perfect cube for infinitely many integers

we know

(a+1)^3 = a^3 + 3a^2 + 3a + 1

put a = 3^p to get

(3^p+1)^2 = 3^(3p) + 3^(2p+1) + 3^(p+1) + 1

the above is true for any p and by multiplying by (3^3)^q we get more results

one solution set is

w = 3p + 3q
x = 2p+1 + 3q
y = p + 1 + 3q
z = 3q   ( for p and q over integers)