Polynomials and packings: A new proof of de Bruijn's theorem

✍ Scribed by Paul Boisen

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 119 KB
• Volume: 146
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(94)00070-1

✦ Synopsis

In 1969 de Bruijn published a proof of the following fact: An a x ab x abc brick can be used to pack an A x B x C box if, and only if, the integers A, B, C are in some order a multiple of a, a multiple of ab, and a multiple of abc. We give a quick proof of this result based on the following elementary lemma. The polynomial (x a-1)(x ab-1)(x °be-1) divides (x A -1)(x B-1)(x c -1) if, and only if, the integers A, B, C are in some order a multiple of a, a multiple of ab, and a multiple of abc.

Compare the following two results, in which a, b, c, A, B, C denote positive integers.