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".
Contributed by: Robert L. Brown (January 2015)
Open content licensed under CC BY-NC-SA
Snapshots
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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