mathschallenge.net

Problem

Prove that seven is the only prime number that is one less than a perfect cube.

Solution

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.