Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2008 IMO Problems/Problem 3"
Line 3: Line 3:
 
For each sufficiently large prime <math>p</math> of the form <math>4k+1</math>, we shall find a corresponding <math>n</math> satisfying the required condition with the prime number in question being <math>p</math>. Since there exist infinitely many such primes and, for each of them, <math>n \ge \sqrt{p-1}</math>, we will have found infinitely many distinct <math>n</math> satisfying the problem.
 
For each sufficiently large prime <math>p</math> of the form <math>4k+1</math>, we shall find a corresponding <math>n</math> satisfying the required condition with the prime number in question being <math>p</math>. Since there exist infinitely many such primes and, for each of them, <math>n \ge \sqrt{p-1}</math>, we will have found infinitely many distinct <math>n</math> satisfying the problem.
  
−
Take a prime <math>p</math> of the form <math>4k+1</math> and consider its "sum-of-two squares" representation <math>p=a^2+b^2</math>, which we know to exist for all such primes. If <math>a=1</math> or <math>b=1</math>, then <math>n=b</math> or <math>n=a</math> is our guy, and <math>p=n^2+1 > 2n+\sqrt(2n)</math> as long as <math>p</math> (and hence <math>n</math>) is large enough.
+
Take a prime <math>p</math> of the form <math>4k+1</math> and consider its "sum-of-two squares" representation <math>p=a^2+b^2</math>, which we know to exist for all such primes. If <math>a=1</math> or <math>b=1</math>, then <math>n=b</math> or <math>n=a</math> is our guy, and <math>p=n^2+1 > 2n+\sqrt{2n}</math> as long as <math>p</math> (and hence <math>n</math>) is large enough. Let's see what happens when both <math>a>1</math> and <math>b>1</math>.
 +
 
 +
Since <math>a</math> and <math>b</math> are apparently co-prime, there must exist integers <math>c</math> and <math>d</math> such that
 +
<cmath>ad+bc=1.</cmath>
 +
In fact, if <math>c</math> and <math>d</math> are such numbers, then <math>c\pm a</math> and <math>d\mp b</math> work as well, so we can assume that <math>c \in \left(\frac{-a}{2}, \frac{a}{2}</math>.
 +
 
 +
Define <math>n=|ac-bd|</math> and let's see what happens.

Revision as of 20:44, 3 September 2008

(still editing...)

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. If $a=1$ or $b=1$, then $n=b$ or $n=a$ is our guy, and $p=n^2+1 > 2n+\sqrt{2n}$ as long as $p$ (and hence $n$) is large enough. Let's see what happens when both $a>1$ and $b>1$.

Since $a$ and $b$ are apparently co-prime, there must exist integers $c$ and $d$ such that \[ad+bc=1.\] In fact, if $c$ and $d$ are such numbers, then $c\pm a$ and $d\mp b$ work as well, so we can assume that $c \in \left(\frac{-a}{2}, \frac{a}{2}$ (Error compiling LaTeX. Unknown error_msg).

Define $n=|ac-bd|$ and let's see what happens.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2008_IMO_Problems/Problem_3&oldid=27729"