Diophantine equation — Safekipedia Adventurer

In mathematics, a Diophantine equation is a special type of polynomial equation. It uses whole numbers, called integers, both in the equation itself and as the answers we look for. These equations can be simple, like adding numbers, or very complex, with numbers used in exponents.

Diophantine problems often have fewer equations than unknowns. The goal is to find whole numbers that work for all equations at the same time. Because these problems can describe shapes and patterns, they are part of a bigger area of math called algebraic geometry, especially a part known as Diophantine geometry.

The word Diophantine comes from the work of Diophantus, a Hellenistic mathematician from Alexandria. He lived in the 3rd century and was one of the first to use symbols in algebra. He studied these kinds of equations a lot. Today, working on these problems is called Diophantine analysis.

Solving Diophantine equations has been a fun challenge for many years. Making general theories to understand all of them, especially the hard ones, was a major accomplishment in the twentieth century.

Examples

In Diophantine equations, we look for whole number solutions. For example, in the equation ax + by = c, where a, b, and c are known numbers, we want to find whole numbers x and y that make the equation true. These are called linear Diophantine equations.

Sometimes, unknowns can appear in exponents, like in x2 + y2 = z2, which is the famous Pythagorean theorem. These are called exponential Diophantine equations. Both types of equations challenge us to find whole numbers that fit perfectly, making them fun puzzles for mathematicians!

a x + b y = c {\displaystyle ax+by=c}	This is a linear Diophantine equation, related to Bézout's identity.
w 3 + x 3 = y 3 + z 3 {\displaystyle w^{3}+x^{3}=y^{3}+z^{3}}	The smallest nontrivial solution in positive integers is 12³ + 1³ = 9³ + 10³ = 1729. It was famously given as an evident property of 1729, a taxicab number (also named Hardy–Ramanujan number) by Ramanujan to Hardy while meeting in 1917. There are infinitely many nontrivial solutions.
x n + y n = z n {\displaystyle x^{n}+y^{n}=z^{n}}	For n = 2 there are infinitely many solutions (x, y, z): the Pythagorean triples. For larger integer values of n, Fermat's Last Theorem (initially claimed in 1637 by Fermat and proved by Andrew Wiles in 1995) states there are no positive integer solutions (x, y, z).
x 2 − n y 2 = ± 1 {\displaystyle x^{2}-ny^{2}=\pm 1}	This is Pell's equation, which is named after the English mathematician John Pell. It was studied by Brahmagupta in the 7th century, as well as by Fermat in the 17th century.
4 n = 1 x + 1 y + 1 z {\displaystyle {\frac {4}{n}}={\frac {1}{x}}+{\frac {1}{y}}+{\frac {1}{z}}}	The Erdős–Straus conjecture states that, for every positive integer n ≥ 2, there exists a solution in x, y, and z, all as positive integers. Although not usually stated in polynomial form, this example is equivalent to the polynomial equation 4 x y z = n ( y z + x z + x y ) . {\displaystyle 4xyz=n(yz+xz+xy).}
x 4 + y 4 + z 4 = w 4 {\displaystyle x^{4}+y^{4}+z^{4}=w^{4}}	Conjectured incorrectly by Euler to have no nontrivial solutions. Proved by Elkies to have infinitely many nontrivial solutions, with a computer search by Frye determining the smallest nontrivial solution, 95800⁴ + 217519⁴ + 414560⁴ = 422481⁴.

Linear Diophantine equations

The simplest linear Diophantine equation looks like ax + by = c, where a, b, and c are whole numbers. To find whole number solutions for x and y, c must be a multiple of the greatest common divisor of a and b. If one solution is found, others can be found by adding or subtracting multiples of special numbers related to a and b.

The Chinese remainder theorem helps solve sets of these equations. It tells us that if we have several numbers that don’t share factors, there is exactly one whole number that satisfies all the equations at once. This theorem is very useful in many areas of mathematics and computer science.

Main article: Chinese remainder theorem

Further information: Hermite normal form, Integer linear programming

Homogeneous equations

A homogeneous Diophantine equation uses a special kind of math expression called a homogeneous polynomial. One famous example is Fermat's Last Theorem. It asks if there are whole numbers that solve the equation x^d + y^d − z^d = 0 when d is bigger than 2. This problem was very hard and took more than three hundred years for mathematicians to solve.

For equations of degree two, there are better ways to find answers. These ways help check if a solution exists and then find all the possible answers. An example is the equation x² + y² = 3z². The only answer is when x, y, and z are all zero.

Another well-known example is the equation for Pythagorean triples, x² + y² − z² = 0. This describes triangles with sides that are whole numbers.

Diophantine analysis

Main article: Hilbert's tenth problem

Diophantine analysis is a part of mathematics that looks at equations where we want only whole number answers. It asks questions such as: Are there any solutions? Are there more than the easy ones we can find? Are there only a few solutions or many?

One famous example is from Pierre de Fermat. He said that you cannot split a cube into two smaller cubes, or a fourth power into two smaller fourth powers. This is called Fermat's Last Theorem and was proven true in 1995 by Andrew Wiles. Another fun problem involves ages, where a father's age is linked to his son's age by flipping the digits. These puzzles show how hard it can be to find whole number answers to some equations.

Over time, mathematicians have created ways to try and solve these puzzles, but many Diophantine equations are still very hard to solve. Today, researchers use ideas from geometry to learn more about them.

Exponential Diophantine equations

If a Diophantine equation has variables that are used as exponents, it is called an exponential Diophantine equation. Famous examples include the Ramanujan–Nagell equation, which is 2ⁿ − 7 = x², and equations linked to the Fermat–Catalan conjecture and Beal's conjecture.

There is no single method to solve all these equations. Some special cases, like Catalan's conjecture and Fermat's Last Theorem, have been solved. Others need special ways to find answers or sometimes just careful checking.