The word Diophantine refers to Diophantus, the third century mathematician from Alexandria. Numbers are the fundamental building blocks of mathematics, so he started thinking about problems for which we need integer solutions. For example, how many horses do we need to achieve a particular task? Even if the exact answer is 15.5, we cannot really cut the last horse into half. So he started studying these problems in order to develop a sound framework around it. This went on to become one of the most enigmatic and endearing fields in the history of mathematics. Diophantine equations have become so fundamental that we cannot imagine our world without them. Let’s go ahead and see what they are all about, shall we?
What is a Diophantine equation?
Diophantine equations are polynomial equations that seek only integer solutions. For example, consider the following equation:
3x + 4y = 10
It has many solutions. Some of them are integers and some of them are not. For example, x = 2.1 and y = 0.925 is one possible non-integer solution.
But if we characterize this as a Diophantine equation, then we only seek integer values for x and y. In this case, x = 2 and y = 1 is one possible integer solution. Also, x = -2 and y = 4 is another possible integer solution. This is an example of linear Diophantine equation.
A non-linear Diophantine equation would be something like:
x2 + y2 = z2
Bear in mind that x, y, and z have to be integers here. For example, x = 3, y = 4, and z = 5 is one possible solution here. In fact, all Pythagorean triplets satisfy the above non-linear Diophantine equation.
How to visualize it?
Diophantine problems define an algebraic variety of dimension ‘n’ and probe into the lattice points on it. We’ll talk about both the concepts here — algebraic variety and lattice points. Algebraic variety is a mathematical object that’s used in algebraic geometry. It is simply a set of solutions to a given equation. An algebraic variety of dimension 1 is called an algebraic curve. This is a simple 2D curve on a plane. We have all seen it!
An algebraic variety of dimension two is called an algebraic surface. It’s just a simple surface that can be defined with an equation. We have seen these as well. Just think of any curved or planar surface in 3D space. Now algebraic varieties of dimension three or higher are tricky to imagine because we live in a 3-dimensional world. We would need to imagine a 4D space to understand algebraic varieties of dimension three.
Lattice points are the integer points on those algebraic varieties. It’s like drawing a smooth curve and highlighting the integer points on it. The actualy theory of lattice in the mathematical sense involves understanding the concept of isomorphism, which in turn would involve discussing the concept of morphism. But we can think of lattice as a regular tiling on a space, thereby discretizing the space. Diophantine equations look for lattice points that satify a given set of properties.
Where do we use it?
It’s a fundamental tool that finds applications in fields like physics, material science, economics, finance, construction, and many more. For example, the field of cryptography utilizes Diophantine equations to build security systems. Many great conjectures and theorems are based on Diophantine equations. The entire field of Number Theory extensively uses Diophantine Equations to design and develop mathematical frameworks. These frameworks find applications in many real world applications.