Diophantine Equations
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
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