Difference between revisions of "2017 USAJMO Problems/Problem 2"

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==Problem:==
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Prove that there are infinitely many distinct pairs <math>(a,b)</math> of relatively prime positive integers <math>a > 1</math> and <math>b > 1</math> such that <math>a^b + b^a</math> is divisible by <math>a + b</math>.
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==Solution==
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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).
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Lemma: <math>(2^k + 1)^{-1} \equiv 2^k + 1 \pmod{2^{k+1}}</math>.
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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
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<cmath>(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}},</cmath>
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where we use our lemma and the Euler totient theorem: <math>a^\phi{n} \equiv 1 \pmod{n}</math> when <math>a</math> and <math>n</math> are relatively prime.
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==See also==
 
==See also==
 
{{USAJMO newbox|year=2017|num-b=1|num-a=3}}
 
{{USAJMO newbox|year=2017|num-b=1|num-a=3}}

Revision as of 19:10, 19 April 2017

Problem:

Prove that there are infinitely many distinct pairs $(a,b)$ of relatively prime positive 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.

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See also

2017 USAJMO (ProblemsResources)
Preceded by
Problem 1
Followed by
Problem 3
1 2 3 4 5 6
All USAJMO Problems and Solutions
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