CONTINUOUS FUNCTIONS AND THE GAUSS LEMMA

Article Type

Research Article

Publication Title

Mathematics Student

Abstract

In Article 42 of his celebrated book ‘Disquisitiones Arith-meticae’, Gauss proved the following result: If the coefficients A, B, C, · · ·, N; a, b, c, · · · n of two functions of the form (formula presented) are all rational and not all integers, and if the product of (P) and (Q) = xm+µ + Axm+µ−1 + Bxm+µ−2 + etc. + Z then not all the coefficients A, B, · · ·, Z can be integers. This is the famous Gauss lemma which has been rephrased and gen-eralized in several ways over 150 years. Some of the statements have only existential proofs while some have surprisingly explicit proofs. We discuss these aspects of the Gauss lemma and its generalizations.

First Page

205

Last Page

215

Publication Date

7-1-2022

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