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Difference between revisions of "2008 IMO Problems/Problem 3"
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=== Problem 3 ===
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== Problem ==
 
Prove that there are infinitely many positive integers <math>n</math> such that <math>n^{2} + 1</math> has a prime divisor greater than <math>2n + \sqrt {2n}</math>.
 
Prove that there are infinitely many positive integers <math>n</math> such that <math>n^{2} + 1</math> has a prime divisor greater than <math>2n + \sqrt {2n}</math>.
  
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Since <math>a</math> and <math>b</math> are (obviously) co-prime, there must exist integers <math>c</math> and <math>d</math> such that
 
Since <math>a</math> and <math>b</math> are (obviously) co-prime, there must exist integers <math>c</math> and <math>d</math> such that
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<cmath>ad+bc=1 \leqno{(1)}</cmath>
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<cmath>ad+bc=1. \quad(1)</cmath>
 
In fact, if <math>c</math> and <math>d</math> are such numbers, then <math>c\pm ma</math> and <math>d\mp mb</math> work as well for any integer <math>m</math>, so we can assume that <math>c \in \left[-\frac{a}{2}, \frac{a}{2}\right]</math>.
 
In fact, if <math>c</math> and <math>d</math> are such numbers, then <math>c\pm ma</math> and <math>d\mp mb</math> work as well for any integer <math>m</math>, so we can assume that <math>c \in \left[-\frac{a}{2}, \frac{a}{2}\right]</math>.
  
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<cmath>|d| \le \frac{b-1}{2}.</cmath>
 
<cmath>|d| \le \frac{b-1}{2}.</cmath>
  
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<cmath>n^2+1 = (a^2+b^2)(c^2+d^2) \le p\left( \frac{(a-1)^2}{4}+\frac{(b-1)^2}{4} \right).\label{(2)}</cmath>  
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<cmath>n^2+1 = (a^2+b^2)(c^2+d^2) \le p\left( \frac{(a-1)^2}{4}+\frac{(b-1)^2}{4} \right). \quad (2)</cmath>  
  
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Before we proceed, we would like to show that <math>a+b-1 > \sqrt{p}</math>. Observe that the function <math>x+\sqrt{p-x^2}</math> over <math>x\in(2,\sqrt{p-4})</math> reaches its minima on the ends, so <math>a+b</math> given <math>a^2+b^2=p</math> is minimized for <math>a = 2</math>, where it equals <math>2+\sqrt{p-2^2}</math>. So we want to show that <cmath>2+\sqrt{p-4} > \sqrt{p} + 1,</cmath>
+
Before we proceed, we would like to show first that <math>a+b-1 > \sqrt{p}</math>. Observe that the function <math>x+\sqrt{p-x^2}</math> over <math>x\in(2,\sqrt{p-4})</math> reaches its minima on the ends, so <math>a+b</math> given <math>a^2+b^2=p</math> is minimized for <math>a = 2</math>, where it equals <math>2+\sqrt{p-2^2}</math>. So we want to show that <cmath>2+\sqrt{p-4} > \sqrt{p} + 1,</cmath>
 
which obviously holds for large <math>p</math>.
 
which obviously holds for large <math>p</math>.
  

Revision as of 23:24, 3 September 2008

Problem

Prove that there are infinitely many positive integers $n$ such that $n^{2} + 1$ has a prime divisor greater than $2n + \sqrt {2n}$.

Solution

The main idea is to take a gaussian prime $a+bi$ and multiply it by a "twice as small" $c+di$ to get $n+i$. The rest is just making up the little details.

For each sufficiently large prime $p$ of the form $4k+1$, we shall find a corresponding $n$ satisfying the required condition with the prime number in question being $p$. Since there exist infinitely many such primes and, for each of them, $n \ge \sqrt{p-1}$, we will have found infinitely many distinct $n$ satisfying the problem.

Take a prime $p$ of the form $4k+1$ and consider its "sum-of-two squares" representation $p=a^2+b^2$, which we know to exist for all such primes. As $a\ne b$, assume without loss of generality that $b>a$. If $a=1$, then $n=b$ is what we are looking for, and $p=n^2+1 > 2n+\sqrt{2n}$ as long as $p$ (and hence $n$) is large enough. Assume from now on that $b>a>1$.

Since $a$ and $b$ are (obviously) co-prime, there must exist integers $c$ and $d$ such that \[ad+bc=1. \quad(1)\] In fact, if $c$ and $d$ are such numbers, then $c\pm ma$ and $d\mp mb$ work as well for any integer $m$, so we can assume that $c \in \left[-\frac{a}{2}, \frac{a}{2}\right]$.

Define $n=|ac-bd|$ and let's see why this was a good choice. For starters, notice that $(a^2+b^2)(c^2+d^2)=n^2+1$.


If $c=\pm\frac{a}{2}$, then from (1), we see that $a$ must divide $2$ and hence $a=2$. In turn, $d=-\frac{b-1}{2}$ and $n=\frac{b(b-1)}{2}-2$. Therefore, $b^2-b=2n+4>2n$ and so $\left(b-\frac{1}{2}\right)^2>2n$, from where $b > \sqrt{2n}+\frac{1}{2}$. Finally, $p=b^2+2^2 > 2n+\sqrt{2n}$ and the case $c=\pm\frac{a}{2}$ is cleared.

We can safely assume now that \[|c| \le \frac{a-1}{2}.\] As $b>a>1$ implies $b>2$, we have \[|d| = \left|\frac{1-bc}{a}\right| \le \frac{b(a-1)+2}{2a} < \frac{ba}{2a} = \frac{b}{2},\] so \[|d| \le \frac{b-1}{2}.\]

\[n^2+1 = (a^2+b^2)(c^2+d^2) \le p\left( \frac{(a-1)^2}{4}+\frac{(b-1)^2}{4} \right). \quad (2)\]

Before we proceed, we would like to show first that $a+b-1 > \sqrt{p}$. Observe that the function $x+\sqrt{p-x^2}$ over $x\in(2,\sqrt{p-4})$ reaches its minima on the ends, so $a+b$ given $a^2+b^2=p$ is minimized for $a = 2$, where it equals $2+\sqrt{p-2^2}$. So we want to show that \[2+\sqrt{p-4} > \sqrt{p} + 1,\] which obviously holds for large $p$.

Now armed with $a+b-1>\sqrt{p}$ and (2), we get \[4(n^2+1) \le p( a^2+b^2 - 2(a+b-1) ) \le p( p-2\sqrt{p} ) < u^2(u-1)^2,\] where $u=\sqrt{p}.$

Finally, \[u^2(u-1)^2 > 4n^2+4 > 4n^2\Rightarrow \\ u(u-1) > 2n \Rightarrow u > \sqrt{2n} + \frac{1}{2} \Rightarrow \\ p = u^2 > 2n + \sqrt{2n}.\]

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