Verify that x 7 y 19 and n 6 do not yield a solution to t

Verify that x = 7, y = 19, and n = 6 do not yield a solution to the equation xn + yn = zn where z is a natural number.

Solution

7⁶ + 19⁶ = z⁶

117649 + 47045881 = z⁶

z⁶ = 47163530

z = 47163530^(1/6) = 19.00791 which is not a natural number.

Hence verified.