Example2. Prove that a - b is a factor of aⁿ - bⁿ for all positive integers n.

Proof: Clearly, a - b is a factor of a¹ - b¹; hence the first part of the induction is verified, that is, the statement is true for n = 1. Now we assume that a^k - b^k has a - b as a factor:

Next we must show that a - b is a factor of a^k+1 - b^k+1. But

Now, using the assumption that a^k - b^k = (a - b)M and substituting, we obtain:

We see from this that a - b is a factor of a^k+1 - b^k+1 and hence a - b is a factor of aⁿ - bⁿ for n equal to any positive integer. It is easily seen that the explicit factorization is

Saturday, April 11, 1998