Difference between revisions of "Diophantine equation"

Revision as of 16:52, 16 December 2007

This is an AoPSWiki Word of the Week for Dec 13-19

A Diophantine equation is an multi-variable equation for which integer solutions are to be found.

Finding the solution or solutions to a Diophantine equation is closely tied to modular arithmetic and number theory. Often, when a Diophantine equation has infinitely many solutions, parametric form is used to express the relation between the variables of the equation.

Types of Diophantine Equations

Linear Combination

A Diophantine equation in the form $ax+by=c$ is known as a linear combination. If two relatively prime integers $a$ and $b$ are written in this form with $c=1$ , the equation will have an infinite number of solutions. More generally, there will always be an infinite number of solutions when $\gcd(a,b)=c$ . If $\gcd(a,b)=c$ , then there are no solutions to the equation. To see why, consider the equation $3x-9y=3(x-3y)=17$ . 3 is a divisor of the LHS (also notice that $x-3y$ must always be an integer). However, 17 will never be a multiple of 3, hence, no solutions exist.

Now consider the case where $c=0$ . Thus, $ax=-by$ . If $a$ and $b$ are relatively prime, then all solutions are obviously in the form $(bk,-ak)$ for all integers $k$ . If they are not, we simply divide them by their greatest common divisor.

Pythagorean Triples

Main article: Pythagorean triple

A Pythagorean triple is a set of three integers that satisfy the Pythagorean Theorem, $a^2+b^2=c^2$ . There are three main methods of finding Pythagorean triples:

Method of Pythagoras

If $m>1$ is an odd number, then $m, \frac {m^2 + 1}{2}, \frac {m^2 - 1}{2}$ is a Pythagorean triple.

Method of Plato

If $n>1$ , $2n, n^2 + 1, n^2 - 1$ is a Pythagorean triple.

Babylonian Method

For any $m,n$ , $m^2 - n^2, 2mn, m^2 + n^2$ is a Pythagorean triple.

$x^4+y^4=z^2$

A equation of form $x^4+y^4=z^2$ has no integer solutions, as follows: We assume that the equation does have integer solutions, and consider the solution which minimizes $z$ . Let this solution be $(x_0,y_0,z_0)$ . If $\gcd (x_0,y_0)\nequiv 1$ (Error compiling LaTeX. Unknown error_msg) then their GCD $d$ must satsify $d^2|z$ . The solution $\left(\frac{x_0}{d},\frac{y_0}{d},\frac{z_0}{d}\right)$ would then be a solution less than $z_0$ , which contradicts our assumption. THus, this equation has no integer solutions.

Pell Equations

Main article: Pell equation

Methods of Solving

Coordinate Plane

Note that any linear combination can be transformed into the linear equation $y=\frac{-b}{a}x+\frac{c}{a}$ , which is just the slope-intercept equation for a line. The solutions to the diophantine equation correspond to lattice points that lie on the line. For example, consider the equation $-3x+4y=4$ or $y=\frac{3}{4}x+1$ . One solution is (0,1). If you graph the line, it's easy to see that the line intersects a lattice point as x and y increase or decrease by the same multiple of 4 and 3, respectively (wording?). Hence, the solutions to the equation may be written parametrically $x=4t, y=3t+1$ (if we think of $(0,1)$ as a "starting point").

Modular Arithmetic

Sometimes, modular arithmetic can be used to prove that no solutions to a given Diophantine equation exist. Specifically, if we show that the equation in question is never true mod $m$ , for some integer $m$ , then we have shown that the equation is false. However, this technique cannot be used to show that solutions to a Diophantine equation do exist.

Induction

Sometimes, when a few solutions have been found by brute force, induction can be used to find a family of solutions. Techniques such as infinite descent can also show that no solutions to a particular equation exist, or that no solutions outside of a particular family exist.

It is natural to ask whether there is a general solution for Diophantine equations. This is known as Hilbert's Tenth Problem. The answer, however, is no.

Fermat's Last Theorem

Main article: Fermat's Last Theorem

$x^n+y^n=z^n$ is known as Fermat's Last Theorem for the condition $n\geq3$ . In the 1600s, Fermat, as he was working through a book on Diophantine Equations, wrote a comment in the margins to the effect of "I have a truly marvelous proof of this proposition which this margin is too narrow to contain." Fermat actually made many conjectures and proposed plenty of "theorems," but wasn't one to write down the proofs or much other than scribbled comments. After he died, all his conjectures were re-proven (either false or true) except for Fermat's Last Theorem. After over 350 years of failing to be proven, the theorem was finally proven by Andrew Wiles after he spent over 7 years working on the 200-page proof, and another year fixing an error in the original proof.

Problems

Introductory

Two farmers agree that pigs are worth $300$ dollars and that goats are worth $210$ dollars. When one farmer owes the other money, he pays the debt in pigs or goats, with "change" received in the form of goats or pigs as necessary. (For example, a $390$ dollar debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way?

$\mathrm{(A) \ } 5\qquad \mathrm{(B) \ } 10\qquad \mathrm{(C) \ } 30\qquad \mathrm{(D) \ } 90\qquad \mathrm{(E) \ } 210$

(Source)

Intermediate

Let $P(x)$ be a polynomial with integer coefficients that satisfies $P(17)=10$ and $P(24)=17.$ Given that $P(n)=n+3$ has two distinct integer solutions $n_1$ and $n_2,$ find the product $n_1\cdot n_2$ . (Source)

Olympiad

Determine the maximum value of $m^2 + n^2$ , where $m$ and $n$ are integers satisfying $m, n \in \{ 1,2, \ldots , 1981 \}$ and $( n^2 - mn - m^2 )^2 = 1$ . (Source)

References

Proof of Fermat's Last Theorem

@@ Line 1: / Line 1: @@
 {{WotWAnnounce|week=Dec 13-19}}
-A '''Diophantine equation''' is an [[equation]] which must be solved using only [[integer]]s.
+A '''Diophantine equation''' is an multi-variable [[equation]] for which [[integer]] solutions are to be found.
 Finding the solution or solutions to a Diophantine equation is closely tied to [[modular arithmetic]] and [[number theory]]. Often, when a Diophantine equation has infinitely many solutions, [[parametric form]] is used to express the relation between the variables of the equation.
-== Linear combination ==
+== Types of Diophantine Equations ==
-A Diophantine equation in the form <math>ax+by=c</math> is known as a linear combination.  If two [[relatively prime]] integers a and b are written in this form with c=1, the equation will have an infinite number of solutions.  More generally, there will always be an infinite number of solutions when gcd(a,b)=c.  If gcd(a,b)=c, then there are no solutions to the equation.  To see why, consider the equation <math>3x-9y=3(x-3y)=17</math>.  3 is a divisor of the LHS (also notice that <math>x-3y</math> must always be an integer).  However, 17 will never be a multiple of 3, hence, no solutions exist.
+== Linear Combination ==
-=== Methods of Solving ===
+A Diophantine equation in the form <math>ax+by=c</math> is known as a linear combination.  If two [[relatively prime]] integers <math>a</math> and <math>b</math> are written in this form with <math>c=1</math>, the equation will have an infinite number of solutions.  More generally, there will always be an infinite number of solutions when <math>\gcd(a,b)=c</math>.  If <math>\gcd(a,b)=c</math>, then there are no solutions to the equation.  To see why, consider the equation <math>3x-9y=3(x-3y)=17</math>.  3 is a divisor of the [[LHS]] (also notice that <math>x-3y</math> must always be an integer).  However, 17 will never be a multiple of 3, hence, no solutions exist.
-==== Coordinate Plane ====
-Note that any linear congruence can be transformed into the linear equation <math>y=\frac{-b}{a}x+\frac{c}{a}</math>, which is just the slope-intercept equation for a line.  The solutions to the diophantine equation correspond to [[lattice point]]s that lie on the line.  For example, consider the equation <math>-3x+4y=4</math> or <math>y=\frac{3}{4}x+1</math>. One solution is (0,1).  If you graph the line, it's easy to see that the line intersects a [[lattice point]] as x and y increase or decrease by the same multiple of 4 and 3, respectively (wording?).  Hence, the solutions to the equation may be written [[parametrically]] <math>x=4t, y=3t+1</math> (if we think of <math>(0,1)</math> as a "starting point").
-==== Modular Arithmetic ====
-{{incomplete|section}}
-== Examples ==
+Now consider the case where <math>c=0</math>. Thus, <math>ax=-by</math>. If <math>a</math> and <math>b</math> are relatively prime, then all solutions are obviously in the form <math>(bk,-ak)</math> for all integers <math>k</math>. If they are not, we simply divide them by their greatest common divisor.
-<math>a^2</math><math>+b^2=c^2</math> is the general form of any Pythagorean triple <math>(a,b,c)</math>.
-<math>x^n+y^n=z^n</math> is known as [[Fermat's Last Theorem]] for the condition <math>n\geq3</math>.  In the 1600s, Fermat, as he was working through a book on Diophantine Equations, wrote a comment in the margins to the effect of "I have a truly marvelous proof of this proposition which this margin is too narrow to contain."  Fermat actually made many conjectures and proposed plenty of "theorems," but wasn't one to write down the proofs or much other than scribbled comments.  After he died, all his conjectures were re-proven (either false or true) except for Fermat's "''Last''" Theorem.  After over 350 years of failing to be proven, FLT was finally solved by [[Andrew Wiles]] after he spent over 7 years working on the 200-page proof, and another year fixing an error in the original proof. There are several good books on the history of this problem.
+== Pythagorean Triples ==
+{{main|Pythagorean triple}}
+A Pythagorean triple is a set of three [[integer]]s that satisfy the [[Pythagorean Theorem]], <math>a^2+b^2=c^2</math>. There are three main methods of finding Pythagorean triples:
+=== Method of Pythagoras ===
+If <math>m>1</math> is an [[odd]] number, then <math>m, \frac {m^2 + 1}{2}, \frac {m^2 - 1}{2}</math> is a Pythagorean triple.
-== Solving Diophantine Equations ==
+=== Method of Plato ===
+If <math>n>1</math>, <math>2n, n^2 + 1, n^2 - 1</math> is a Pythagorean triple.
+=== Babylonian Method ===
+For any <math>m,n</math>, <math>m^2 - n^2, 2mn, m^2 + n^2</math> is a Pythagorean triple.
+== <math>x^4+y^4=z^2</math> ==
+A equation of form <math>x^4+y^4=z^2</math> has no [[integer]] solutions, as follows:
+<!-- taken from AoPS Vol. 2 -->
+We assume that the equation does have integer solutions, and consider the solution which minimizes <math>z</math>. Let this solution be <math>(x_0,y_0,z_0)</math>. If <math>\gcd (x_0,y_0)\nequiv 1</math> then their [[GCD]] <math>d</math> must satsify <math>d^2|z</math>. The solution <math>\left(\frac{x_0}{d},\frac{y_0}{d},\frac{z_0}{d}\right)</math> would then be a solution less than <math>z_0</math>, which contradicts our assumption. THus, this equation has no integer solutions.
+== Pell Equations ==
+{{main|Pell equation}}
+== Methods of Solving ==
+=== Coordinate Plane ===
+Note that any linear combination can be transformed into the linear equation <math>y=\frac{-b}{a}x+\frac{c}{a}</math>, which is just the slope-intercept equation for a line.  The solutions to the diophantine equation correspond to [[lattice point]]s that lie on the line.  For example, consider the equation <math>-3x+4y=4</math> or <math>y=\frac{3}{4}x+1</math>. One solution is (0,1).  If you graph the line, it's easy to see that the line intersects a [[lattice point]] as x and y increase or decrease by the same multiple of 4 and 3, respectively (wording?).  Hence, the solutions to the equation may be written [[parametrically]] <math>x=4t, y=3t+1</math> (if we think of <math>(0,1)</math> as a "starting point").
+=== Modular Arithmetic ===
 Sometimes, [[modular arithmetic]] can be used to prove that no solutions to a given Diophantine equation exist.  Specifically, if we show that the equation in question is never true mod <math>m</math>, for some integer <math>m</math>, then we have shown that the equation is false.  However, this technique cannot be used to show that solutions to a Diophantine equation do exist.
-Sometimes, when a few solutions have been found, [[induction]] can be used to find a family of solutions.  Techniques such as [[infinite descent]] can also show that no solutions to a particular equation exist, or that no solutions outside of a particular family exist.
+===Induction===
+Sometimes, when a few solutions have been found by [[brute force]], [[induction]] can be used to find a family of solutions.  Techniques such as [[infinite descent]] can also show that no solutions to a particular equation exist, or that no solutions outside of a particular family exist.
+It is natural to ask whether there is a general solution for Diophantine equations.  This is known as [[Hilbert's Tenth Problem]]. The answer, however, is no.
+== Fermat's Last Theorem ==
+{{main|Fermat's Last Theorem}}
+<math>x^n+y^n=z^n</math> is known as [[Fermat's Last Theorem]] for the condition <math>n\geq3</math>.  In the 1600s, Fermat, as he was working through a book on Diophantine Equations, wrote a comment in the margins to the effect of "I have a truly marvelous proof of this proposition which this margin is too narrow to contain."  Fermat actually made many conjectures and proposed plenty of "theorems," but wasn't one to write down the proofs or much other than scribbled comments.  After he died, all his conjectures were re-proven (either false or true) except for Fermat's Last Theorem.  After over 350 years of failing to be proven, the theorem was finally proven by [[Andrew Wiles]] after he spent over 7 years working on the 200-page proof, and another year fixing an error in the original proof.
-It is natural to ask whether there is a general solution for Diophantine equations.  This was [[Hilbert's Tenth Problem]].  Unfortunately, the answer to this question is "no."
 == Problems ==
 === Introductory ===
-* [[2006_AMC_10A_Problems/Problem_22 | 2006 AMC 10A Problem 22]]
+* Two farmers agree that pigs are worth <math>300</math> dollars and that goats are worth <math>210</math> dollars. When one farmer owes the other money, he pays the debt in pigs or goats, with "change" received in the form of goats or pigs as necessary. (For example, a <math>390</math> dollar debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way?
+<math> \mathrm{(A) \ } 5\qquad \mathrm{(B) \ } 10\qquad \mathrm{(C) \ } 30\qquad \mathrm{(D) \ } 90\qquad \mathrm{(E) \ }  210</math>
+([[2006_AMC_10A_Problems/Problem_22|Source]])
 === Intermediate ===
+*Let <math> P(x) </math> be a polynomial with integer coefficients that satisfies <math> P(17)=10 </math> and <math> P(24)=17. </math> Given that <math> P(n)=n+3 </math> has two distinct integer solutions <math> n_1 </math> and <math> n_2, </math> find the product <math> n_1\cdot n_2</math>. ([[2005 AIME II Problems/Problem 13|Source]])
 === Olympiad ===
+*Determine the maximum value of <math>m^2 + n^2 </math>, where <math>m </math> and <math>n </math> are integers satisfying <math> m, n \in \{ 1,2, \ldots , 1981 \} </math> and <math>( n^2 - mn - m^2 )^2 = 1 </math>. ([[1981 IMO Problems/Problem 3|Source]])
 ==References==
 *[http://www.ams.org/notices/199507/faltings.pdf Proof of Fermat's Last Theorem]
@@ Line 40: / Line 65: @@
 * [[Pell equation]]
-{{stub}}
 [[Category:Number Theory]]

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