#wap to find Automorphic number..
Q: What is automorphic number ?
Ans:Thus, the automorphic numbers in base 10 are 0, 1, 5, 6, 25, 76, 376, 625, 9376, … (sequence A003226 in the OEIS).
A fixed point of {\displaystyle f(x)}
is a zero of the function {\displaystyle g(x)=f(x)-x}
. In the ring of integers modulo {\displaystyle b}
, there are {\displaystyle 2^{\omega (b)}}
zeroes to {\displaystyle g(x)=x^{2}-x}
, where the prime omega function {\displaystyle \omega (b)}
is the number of distinct prime factors in {\displaystyle b}
. An element {\displaystyle x}
in {\displaystyle \mathbb {Z} /b\mathbb {Z} }
is a zero of {\displaystyle g(x)=x^{2}-x}
if and only if {\displaystyle x\equiv 0{\bmod {p}}^{v_{p}(b)}}
or {\displaystyle x\equiv 1{\bmod {p}}^{v_{p}(b)}}
for all {\displaystyle p|x}
. Since there are two possible values in {\displaystyle \lbrace 0,1\rbrace }
, and there are {\displaystyle \omega (b)}
such {\displaystyle p|x}
, there are {\displaystyle 2^{\omega (b)}}
zeroes of {\displaystyle g(x)=x^{2}-x}
, and thus there are {\displaystyle 2^{\omega (b)}}
fixed points of {\displaystyle f(x)=x^{2}}
. According to Hensel’s lemma, if there are {\displaystyle k}
zeroes or fixed points of a polynomial function modulo {\displaystyle b}
, then there are {\displaystyle k}
corresponding zeroes or fixed points of the same function modulo any power of {\displaystyle b}
, and this remains true in the inverse limit. Thus, in any given base {\displaystyle b}
there are {\displaystyle 2^{\omega (b)}}
{\displaystyle b}
-adic fixed points of {\displaystyle f(x)=x^{2}}
.
As 0 is always a zero divisor, 0 and 1 are always fixed points of {\displaystyle f(x)=x^{2}}
, and 0 and 1 are automorphic numbers in every base. These solutions are called trivial automorphic numbers. If {\displaystyle b}
is a prime power, then the ring of {\displaystyle b}
-adic numbers has no zero divisors other than 0, so the only fixed points of {\displaystyle f(x)=x^{2}}
are 0 and 1. As a result, nontrivial automorphic numbers, those other than 0 and 1, only exist when the base {\displaystyle b}
has at least two distinct prime factors.
