The rational numbers can be completed with respect to the standard absolute value and this produces the real numbers. However, there are other absolute values on the rationals besides the standard one. Completing the rationals with respect to one of these produces the p-adic numbers. In this paper, we take some basic number theory concepts and apply them to rational p-adic numbers. Using these concepts, a p-adic division algorithm is developed along with a p-adic Euclidean Algorithm. These algorithms produce a generalized greatest common divisor in the p-adics along with a p-adic simple continued fraction. In Section 2, we describe the p-adic numbers, and in Section 3 we present our p-adic Division Algorithm and Euclidean Algorithm. In Section 4, we show some applications, including a connection to Browkin's p-adic continued fractions, which motivatored our investigations in the first place. Finally, in Section 5, we give some open questions for further study.
Eric Errthum, Department of Mathematics & Statistics, Winona State University EErrthum@winona.edu
"A p-adic Euclidean Algorithm,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 10
, Article 9.
Available at: https://scholar.rose-hulman.edu/rhumj/vol10/iss2/9