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Abstract

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.

Author Bio

Cortney Lager is currently a senior at Winona State University majoring in Mathematics and Mathematics Education and will be graduating in fall 2009. During the fall 2008 semester, Cortney and another student took an independent study on p-adic numbers with Dr. Eric Errthum and during this time they discovered some similarities between p-adic numbers and classical number theory. During the following semester, Cortney compiled the discoveries into a paper. In the future, Cortney plans to pursue her desire to become a professor of Mathematics Education.

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