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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The iteration using Newton’s method for target is .

The iteration using Newton’s method for the golden ratio is .

Column 4 with heading " as -adic" and column 5 with heading "check: as -adic" show very strange looking numbers indeed. These seem to be infinite base numbers. They are in fact numerals, the -adic expansions of and , respectively. In rare cases, suggested by Kurt Hensel in 1897, columns 4 and 5 also converge, in a sense. The "Suggestions" in this Demonstration give the only cases of -adic convergence for .

To appreciate how these -adic numbers can represent rational numbers, observe the following. Columns 4 and 5 are like column 2 in that the digits eventually form an infinitely repeating pattern, but to the left (only 17 digits are displayed). Column 4 is like column 3, entitled " as fraction" (where ), in that in base . Similarly, column 5 behaves like , in that in base .

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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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