# Newton's Method in -adic Form

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This Demonstration applies Newton's method to three polynomials with rational coefficients to approximate the irrational numbers , , and (with these decimal approximations to 10 places: , , and ). If the iteration begins with a rational number , each successive iteration is also rational as shown in column 3, with heading " as fraction".

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Contributed by: Robert L. Brown (January 2015)

Open content licensed under CC BY-NC-SA

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## Details

The sense in which " as -adic" converges to the target irrational number is called a Cauchy sequence using a -adic norm metric. Hensel's lifting lemma [1] ensures that given a polynomial , if there is an integer that satisfies , it can be used for as -adic.

Powerful applications of -adic numbers have been found in number theory, including the famous proof of Fermat’s last theorem by Andrew Wiles.

Reference

[1] K. Conrad. "Hensel's Lemma." (Nov 13, 2014) www.math.uconn.edu/~kconrad/blurbs/gradnumthy/hensel.pdf.

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