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1991 USAMO Problems/Problem 3

Revision as of 18:05, 5 June 2023 by Nedsanger (talk | contribs) (→‎Solution)

Problem

Show that, for any fixed integer $\,n \geq 1,\,$ the sequence \[2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots \pmod{n}\] is eventually constant.

[The tower of exponents is defined by $a_1 = 2, \; a_{i+1} = 2^{a_i}$. Also $a_i \pmod{n}$ means the remainder which results from dividing $\,a_i\,$ by $\,n$.]

Solution 1

Suppose that the problem statement is false for some integer $n \ge 1$. Then there is a least $n$, which we call $b$, for which the statement is false.

Since all integers are equivalent mod 1, $b\neq 1$.

Note that for all integers $b$, the sequence $2^0, 2^1, 2^2, \dotsc$ eventually becomes cyclic mod $b$. Let $k$ be the period of this cycle. Since there are $k-1$ nonzero residues mod $b$. $1 \le k\le b-1 < b$. Since \[2, 2^2, 2^{2^2}, 2^{2^{2^2}}, \dotsc\] does not become constant mod $b$, it follows the sequence of exponents of these terms, i.e., the sequence \[1, 2, 2^2, 2^{2^{2}}, \dotsc\] does not become constant mod $k$. Then the problem statement is false for $n=k$. Since $k<b$, this is a contradiction. Therefore the problem statement is true. $\blacksquare$

Note that we may replace 2 with any other positive integer, and both the problem and this solution remain valid.

Solution 2

We’ll prove by strong induction that for every natural number $n$, the sequence $a_1, a_2, \ldots$ is eventually constant. Since every term of the sequence is $0 \mathrm{\ mod\ } 1$, the claim is true when $n = 1$. Now assume it’s true for $1, \ldots, n$. We want to show it’s true for $n + 1$ as well.

Suppose first that $n + 1$ is odd. Since $\varphi(n + 1) < n + 1$, by our inductive hypothesis there exists an $m$ such that

\[a_m = a_{m + 1} = a_{m + 2} = \cdots \pmod{\varphi(n + 1)}.\]

Since $n + 1$ is coprime to every term of the sequence, it follows from Euler’s theorem that

\[2^{a_m} = 2^{a_{m + 1}} = 2^{a_{m + 2}} = \cdots \pmod{n + 1},\]

or equivalently

\[a_{m + 1} = a_{m + 2} = a_{m + 3} = \cdots \pmod{n + 1},\]

which is what we wanted to show.

Now suppose that $n + 1$ is even. Write $n + 1 = 2^{k} \cdot s$, where $1 \leq s < n + 1$ is odd. The series must eventually be constant $\textrm{mod\ } 2^k$, since $a_m = 0 \textrm{\ mod\ } {2^k}$ for large enough $m$. And by our inductive hypothesis, the series must also eventually be constant $\textrm{mod\ } s$. So for large enough $m$,

\[a_m = a_{m + 1} = a_{m + 2} = \cdots \pmod{2^k},\] \[a_m = a_{m + 1} = a_{m + 2} = \cdots \pmod{s}.\]

Since $2^k$ and $s$ are coprime, these equations are also true modulo $2^k \cdot s = n + 1$. So

\[a_m = a_{m + 1} = a_{m + 2} = \cdots \pmod{n + 1},\]

which completes the proof.

See Also

1991 USAMO (Problems • Resources)
Preceded by
Problem 2 		Followed by
Problem 4
1 • 2 • 3 • 4 • 5
All USAMO Problems and Solutions

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