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Difference between revisions of "2017 USAJMO Problems/Problem 1"

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Let <math>a = 2^n - 1</math> and <math>b = 2^n + 1</math>. We see that <math>a</math> and <math>b</math> are relatively prime (they are consecutive positive odd integers).  
 
Let <math>a = 2^n - 1</math> and <math>b = 2^n + 1</math>. We see that <math>a</math> and <math>b</math> are relatively prime (they are consecutive positive odd integers).  
  
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Lemma: <math>(2^k + 1)^{-1} \equiv 2^k + 1 \pmod{2^{k+1}}</math>.  
 
Lemma: <math>(2^k + 1)^{-1} \equiv 2^k + 1 \pmod{2^{k+1}}</math>.  
  
 
Since every number has a unique modular inverse, the lemma is equivalent to proving that <math>(2^k+1)^2 \equiv 1 \pmod{2^{k+1}}</math>. Expanding, we have the result.  
 
Since every number has a unique modular inverse, the lemma is equivalent to proving that <math>(2^k+1)^2 \equiv 1 \pmod{2^{k+1}}</math>. Expanding, we have the result.  
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Substituting for <math>a</math> and <math>b</math>, we have  
 
Substituting for <math>a</math> and <math>b</math>, we have  

Revision as of 18:12, 19 April 2017

Problem

Prove that there are infinitely many distinct pairs $(a,b)$ of relatively prime integers $a>1$ and $b>1$ such that $a^b+b^a$ is divisible by $a+b$.

Solution

Let $a = 2^n - 1$ and $b = 2^n + 1$. We see that $a$ and $b$ are relatively prime (they are consecutive positive odd integers).

Lemma: $(2^k + 1)^{-1} \equiv 2^k + 1 \pmod{2^{k+1}}$.

Since every number has a unique modular inverse, the lemma is equivalent to proving that $(2^k+1)^2 \equiv 1 \pmod{2^{k+1}}$. Expanding, we have the result.

Substituting for $a$ and $b$, we have \[(2^k+1)^{2^k-1} + (2^k-1)^{2^k+1} \equiv 2^k - 1 + 2^k + 1 \equiv 0 \pmod{2^{k+1}},\] where we use our lemma and the Euler totient theorem: $a^\phi{n} \equiv 1 \pmod{n}$ when $a$ and $n$ are relatively prime.

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

See also

2017 USAJMO (Problems • Resources)
First Problem Followed by
Problem 2
1 • 2 • 3 • 4 • 5 • 6
All USAJMO Problems and Solutions
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