Diophantine Equations

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  • Topic: Euclidean algorithm, Diophantine equation, Prime number
  • Pages : 11 (3293 words )
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  • Published : February 9, 2013
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1.INTRODUCTION:
The mathematician Diophantus of Alexandria around 250A.D. started some kind of research on some equations involving more than one variables which would take only integer values.These equations are famously known as “DIOPHANTINE EQUATION”,named due to Diophantus.The simplest type of Diophantine equations that we shall consider is the Linear Diophantine equations in two variables: ax+by=c, where a,b,c are integers and a,b are not both zero. We also have many kinds of Diophantine equations where our main goal is to find out its solutions in the set of integers.Interestingly we can see some good theoretical discussion in Euclid’s “ELEMENTS” but no remark had been cited by Diophantus in his research works regarding this type of equations.

2.Whole Numbers:
In number theory, we are usually concerned with the properties of the integers, or whole numbers: Z = {. . . ,−3,−2,−1, 0, 1, 2, 3, . . .}. Let us begin with a very simple problem that should be familiar to anyone who has studied elementary algebra. • Suppose that dolls sell for 7 dollars each, and toy train sets sell for 18 dollars. A storesells 25 total dolls and train sets, and the total amount received is 208 dollars. How many of each were sold?

The standard solution is straight-forward: Let x be the number of dolls and y be the number of train sets. Then we have two equations and two unknowns: x + y = 25
7x + 18y = 208
The equations above can be solved in many ways, but perhaps the easiest is to note that the first one can be converted to: x = 25−y and then that value of x is substituted into the other equation and solved:

7(25 − y) + 18y = 208,
i.e.175 − 7y + 18y = 208,
i.e.−7y + 18y = 208 − 175,
i.e.11y = 33,
i.e.y = 3,
Then if we substitute y = 3 into either of the original equations, we obtain x = 22, and it is easy to check that those values satisfy the conditions in the original problem. Now let’s look at a more interesting problem:

• Suppose that dolls sell for 7 dollars each, and toy train sets sell for 18 dollars. A store sells only dolls and train sets, and the total amount received is 208 dollars. How many of each were sold?

This time there is only one equation: 7x+18y = 208. We probably learned in algebra class that you need as many equations as unknowns to solve problems like this, so at first it seems hopeless, but there is one additional key piece of information: the number of dolls and the number of train sets must be non-negative whole numbers. With that in mind, let’s see what we can do, ignoring for the moment the fact that we already have a solution, namely: x = 22 and y = 3.

Again ,the all other solutions will be of the form given below: X=22+18c
Y=3-7c ,where c is any integer
If we want to have positive integral solutions then by an easy computation we observe that c= -1,0,1.
When an equation of this sort is solvable by this method, there is no limit to the number of steps that need to be taken to obtain the solution. In the example above, we needed to introduce integers a, b and c, but other equations might require more or fewer of these intermediate values.

3.Linear Diophantine Equations:
What we have just solved is known as a Diophantine equation – an equation whose roots are required to be integers. Probably the most famous Diophantine equation is the one representing Fermat’s last theorem, finally proved hundreds of years after it was proposed by AndrewWiles:

If n > 2, there are no non-trivial1 solutions in integers to the equation: x n + y n = z n
There are many, many forms of Diophantine equations, but equations of the sort that we just solved are called “linear Diophantine equations”: all the coefficients of the variables are integers.
Let’s look a little more closely at the equation we just solved: 7x + 18y = 208. If the only requirementwere that the roots be integers (not necessarily non-negative integers), then our solution: x = 22 + 18c and y = 3 − 7c represent an infinte set of solutions, where every...
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