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Prime Uniqueness

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 minus 1

By factoring the right hand side,
p = (n minus 1)(n2 + n+ 1)

By definition p cannot have any factors, so n minus 1 = 1 implies n = 2.

Hence p = 23 minus 1 = 7.

Investigate this property for other perfect powers.

Problem ID: 48 (May 2001)     Difficulty: 2 Star

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