Solving the Diophantine Equation ax by 1 for Positive Integers x and y

Introduction to Diophantine Equations

Diophantine equations are equations where the solutions are sought in integers. The specific equation we will focus on today is ax by 1, where a and b are integers and x and y are also integers. For this equation to have a solution, particularly where x and y are positive integers, a mathematical condition must be satisfied: the greatest common divisor (gcd) of a and b must be 1. That is, gcd(a, b) 1.

Understanding the GCD Condition

The condition gcd(a, b) 1 is crucial because it ensures that a and b are relatively prime (i.e., they have no common factors other than 1). This property is essential in finding a solution to the equation ax by 1. If a and b are not relatively prime, the equation does not have a solution in integers. For instance, if a 6 and b 12, then gcd(6, 12) 6, and the equation 6x 12y 1 cannot have positive integer solutions.

Using the Euclidean Division Algorithm to Solve ax by 1

The Euclidean division algorithm, one of the oldest and most fundamental algorithms in mathematics, is a powerful tool for finding the greatest common divisor of two integers. It can also be used to find a solution to the equation ax by 1. Let's walk through the steps of the Euclidean division algorithm and then use it to solve a specific example.

Step 1: Apply the Euclidean Algorithm

To find the gcd of two numbers using the Euclidean algorithm, we repeatedly divide the larger number by the smaller number and replace the larger number with the remainder. We continue this process until the remainder is 0. The last non-zero remainder is the gcd of the original two numbers. For the example where ( a 77 ) and ( b 15 ), we can apply the Euclidean algorithm as follows: 1. ( 77 5 times 15 2 ) Here, 77 is divided by 15, and the remainder is 2. 2. ( 15 7 times 2 1 ) Now, 15 is divided by 2, and the remainder is 1. Since the remainder is now 1, we can stop. The gcd of 77 and 15 is 1, which means a solution to the equation ( 77x 15y 1 ) exists.

Step 2: Using Backward Substitution to Find a Solution

The Euclidean algorithm not only finds the gcd but also provides a way to express the gcd as a linear combination of the original numbers. By working backwards from the last non-zero remainder, we can find the coefficients ( x ) and ( y ) that satisfy the equation. From the second step: 1. ( 1 15 - 7 times 2 ) Substituting ( 2 ) from the first step: 2. ( 1 15 - 7 times (77 - 5 times 15) ) Simplifying, we get: 3. ( 1 15 - 7 times 77 35 times 15 ) Combining like terms: 4. ( 1 36 times 15 - 7 times 77 ) Thus, we have found a solution: ( x -7 ) and ( y 36 ).

Generating Infinitely Many Solutions

Once a solution ( (x, y) ) is found, we can generate infinitely many solutions to the equation ( ax by 1 ). This is because if ( (x_0, y_0) ) is a solution, then for any integer ( k ), the pair ( (x_0 frac{b}{text{gcd}(a, b)}k, y_0 - frac{a}{text{gcd}(a, b)}k) ) is also a solution. Given that gcd(a, b) 1, this simplifies to ( (x bk, y - ak) ). For our example where ( a 77 ), ( b 15 ), and the initial solution is ( (x, y) (-7, 36) ), any integer ( k ) can generate a new solution: 1. ( x -7 15k ) 2. ( y 36 - 77k ) This means that for any integer ( k ), the pair ( (-7 15k, 36 - 77k) ) is a solution to the equation ( 77x 15y 1 ).

Conclusion and Practical Application

The process of solving the Diophantine equation ax by 1, particularly for positive integers, is not only a fundamental concept in number theory but also has practical applications in various fields. For instance, it plays a role in cryptography, particularly in the RSA algorithm, where finding such integer solutions is crucial for generating public and private keys. Understanding and applying the Euclidean division algorithm is therefore essential for practitioners and students in mathematics, computer science, and related fields.

Keywords

Diophantine equation, Euclidean division algorithm, positive integers, gcd