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2011 USAMO Problems/Problem 4

Revision as of 18:35, 12 April 2014 by Suli (talk | contribs) (→‎Solution 2)

This problem is from both the 2011 USAJMO and the 2011 USAMO, so both problems redirect here.

Problem

Consider the assertion that for each positive integer $n \ge 2$, the remainder upon dividing $2^{2^n}$ by $2^n-1$ is a power of 4. Either prove the assertion or find (with proof) a counterexample.

Solution

We will show that $n = 25$ is a counter-example.

Since $\textstyle 2^n \equiv 1 \pmod{2^n - 1}$, we see that for any integer $k$, $\textstyle 2^{2^n} \equiv 2^{(2^n - kn)} \pmod{2^n-1}$. Let $0 \le m < n$ be the residue of $2^n \pmod n$. Note that since $\textstyle m < n$ and $\textstyle n \ge 2$, necessarily $\textstyle 2^m < 2^n -1$, and thus the remainder in question is $\textstyle 2^m$. We want to show that $\textstyle 2^m \pmod {2^n-1}$ is an odd power of 2 for some $\textstyle n$, and thus not a power of 4.

Let $\textstyle n=p^2$ for some odd prime $\textstyle p$. Then $\textstyle \varphi(p^2) = p^2 - p$. Since 2 is co-prime to $\textstyle p^2$, we have \[{2^{\varphi(p^2)} \equiv 1 \pmod{p^2}}\] and thus \[\textstyle 2^{p^2} \equiv 2^{(p^2 - p) + p} \equiv 2^p \pmod{p^2}.\]

Therefore, for a counter-example, it suffices that $\textstyle 2^p \pmod{p^2}$ be odd. Choosing $\textstyle p=5$, we have $\textstyle 2^5 = 32 \equiv 7 \pmod{25}$. Therefore, $\textstyle 2^{25} \equiv 7 \pmod{25}$ and thus \[\textstyle 2^{2^{25}} \equiv 2^7 \pmod {2^{25} - 1}.\] Since $\textstyle 2^7$ is not a power of 4, we are done.

Solution 2

Lemma (useful for all situations): If x and y are positive integers such that $2^x - 1$ divides $2^y - 1$, then x divides y. Proof: $2^y \equiv 1 \pmod{2^x - 1}$. Replacing the 1 with a $2^x$ and dividing out the powers of two should create an easy induction proof which will be left to the reader as an Exercise.

Consider n = 25. We will prove that this case is a counterexample via contradiction.

Because 4 = $2^2$, we will assume there exists a positive integer k such that $2^{2^n} - 2^{2k}$ divides $2^n - 1$ and $2^{2k} < 2^n - 1$. Dividing the powers of 2 from LHS gives $2^{2^n - 2k} - 1$ divides $2^n - 1$. Hence, $2^n - 2k$ divides n. Because n = 25 is odd, $2^{24} - k$ divides 25. Modular arithmetic (in particular, Euler's totient function for 25 equals 20) gives $2^{24} \equiv 2^4 \equiv 16 \pmod{25}$ and so k ≥ 16. However, $2^{2k} >= 2^{32} > 2^{25} - 1$, a contradiction. Thus, n = 25 is a valid counterexample.

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

See also

2011 USAMO (Problems • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions
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