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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