Prove that seven is the only prime number that is one less than a perfect cube.
Let p be a prime number one less than a perfect cube, p = n3 1
By factoring the right hand side,
p = (n 1)(n2 + n+ 1)
By definition p cannot have any factors, so n 1 = 1 n = 2.
Hence p = 23 1 = 7.
Investigate this property for other perfect powers.