Golden Integers

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The positive root of the golden polynomial is the famous golden ratio and the negative root is . There are two integer solutions and to the congruence , where is a prime and is a positive integer (in this Demonstration, and ). Can either or be called a "golden integer" that is analogous to ?

Contributed by: Robert L Brown (March 2014)
Open content licensed under CC BY-NC-SA


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Kurt Hensel (1861–1941) discovered that for certain prime numbers , there are two roots to the "golden polynomial" modulo . Further, he claimed that for each of those primes and for every natural number , there would also exist two roots congruent modulo .

When you choose the "base " option and vary from 1 to 50, it is clear that the 50 different congruences are related. They are a single infinite sequence written as (base ) that is truncated on the left to leave significant digits . Likewise, the 50 checks are a single infinite integer (base ) that is truncated to digits.

With the "base " option set and , notice the strong analogy between modular arithmetic and real arithmetic as you vary . The infinite sequences (base ) behave like "real" infinitely repeating decimals. And like repeating decimals, the most significant digits are nearer the decimal point.

Writing the 50 congruences as a single truncated sequence (base ) inspired the introduction of -adic numbers by Kurt Hensel in 1897. Powerful applications of -adic numbers have been found in number theory, including the famous proof of Fermat's last theorem by Andrew Wiles.



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