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REMARKS ON A FACTORIZATION OF Xn −Yn

for each positive integer n and all real x and y. This identity has been shown to have various applications throughout the undergraduate mathematics curriculum and beyond. For example, Johnsonbaugh, using the inequality b − a b − a < (n+ 1)b n for all positive integers n and real a and b with 0 ≤ a < b, an easy consequence of (∗), published an old and relatively simple proof of the monotonicity and boundedness of the sequence {(1 + 1/n)n} in [2]. Evidently, the proof was discovered by Fort in 1864 (see [3]). In [1], there is a nice proof of the existence of nth roots, which is a good deal simpler than other proofs using the Fundamental Axiom of the Reals (see [3]) and proofs using the Intermediate Value Theorem (see [5]). The following characterization of Cn,k is an interesting by-product obtained by relating (∗) to the Binomial Theorem: If n and k are positive integers with k ≤ n, then