Automorphic Numbers

In this post we discuss automorphic numbers. Note that all numbers involved in this post are non negative integers.

Definition: Let x have n digits in its base 10 representation. Then x is said to be automorphic if x^2\equiv x\pmod {10^n}.

In our base 10 representation this means that the last n digits of x^2 are precisely x. In other words, the digits of x are appended to x^2. Note that since the digits of x are appended to x^2 they are also appended to x^m for any m\ge 1.

A list of some automorphic numbers can be found here. Our aim here is to characterize all such numbers in base 10.

Theorem: The automorphic numbers in base 10 are given by \{0,1\}\cup\{5^{2^m}\pmod {10^m}:m\ge 1\}\cup\{16^{5^m}\pmod {10^m}:m\ge 1\}.

Proof: Suppose x is an automorphic number of n digits. Then,

x(x-1)\equiv 0\pmod {10^n}

which is equivalent to

\Big(x\equiv 0\text{ or }1\pmod{2^n}\Big) \text{ and } \Big(x\equiv 0\text{ or }1\pmod{5^n}\Big)

which is equivalent to the occurrence of exactly one of the following:

Case 1: x\equiv 0\pmod {2^n} \text{ and } x\equiv 0\pmod {5^n}

Case 2: x\equiv 1\pmod {2^n} \text{ and } x\equiv 1\pmod {5^n}

Case 3: x\equiv 1\pmod {2^n} \text{ and } x\equiv 0\pmod {5^n}

Case 4: x\equiv 0\pmod {2^n} \text{ and } x\equiv 1\pmod {5^n}

Now we consider all these cases one by one. Case 1 is equivalent to x\equiv 0\pmod {10^n}. Since x had n digits so this means x=0. Similarly x=1 in case 2.

Now suppose case 3 holds. By the Chinese Remainder theorem there exists a unique solution to these two congruences modulo 10^n. We show that 5^{2^n} is a solution to them. By uniqueness it will then follow that case 3 is equivalent to saying that x\equiv 5^{2^n}\pmod {10^n}. Since x has n digits and 5^{2^n}\pmod {10^n} has at most n digits so we will therefore have x equal to 5^{2^n}\pmod {10^n}. Hence x\in\{5^{2^m}\pmod {10^m}:m\ge 1\}.

We first show that 5^{2^n}\equiv 1\pmod {2^n}. Proceed by induction on n. For n=1 note that 5^{2}\equiv 1\pmod 2.

Now assume the result for k. Then 5^{2^k}=q2^k+1 and squaring yields 5^{2^{k+1}}=q^2 2^{2k}+q2^{k+1}+1=(q^2 2^{k-1}+q)2^{k+1}+1 and so 5^{2^{k+1}}\equiv 1\pmod {2^{k+1}} thereby completing the induction step. Next observe that as n<2^n so 5^{2^n}\equiv 0\pmod {5^n} follows immediately, thereby completing case 3.

Note that our argument also implies that any number of the form 5^{2^m}\pmod{10^m} is automorphic. Indeed, if a=5^{2^m}\pmod{10^m} had t\le m digits then we have shown that a^2\equiv a\pmod {10^m}. Now as 10^t divides 10^m so a^2\equiv a\pmod {10^t}, i.e. a is automorphic.

To finish the proof we now consider case 4. We show that 16^{5^n} is a solution to the congruences of case 4. Clearly 16^{5^n}\equiv 0\pmod {2^n} as n<4.5^n. To show 16^{5^n}\equiv 1\pmod {5^n} we proceed by induction as before. The base case may be explicitly worked out or be shown to hold by invoking Fermat’s theorem: m^p\equiv m\pmod p, for any prime p and integer m. Next assume the result for k, so that 16^{5^k}=q5^k+1. Raising both sides to the fifth power yields the result for k+1. The rest of the argument is similar.\Box


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Filed under Miscellaneous, Number theory, Recreational Mathematics

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