Diophantine equation — Safekipedia Discoverer

In mathematics, a Diophantine equation is a special kind of polynomial equation that uses whole numbers, called integers, as its coefficients. We are only interested in solutions where the answers are also whole numbers. These equations can be simple, like adding a few numbers together, or much more complex, where numbers appear in exponents.

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

The term Diophantine honors the work of the Hellenistic mathematician Diophantus from Alexandria, who lived in the 3rd century. He was one of the first to use symbols in algebra and studied these kinds of equations. Today, the study of these problems is called Diophantine analysis.

While solving individual Diophantine equations has been a favorite puzzle for centuries, creating general theories to understand them all — especially the more complicated ones — was a big achievement 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 because they involve adding and multiplying unknowns in a straight-line way.

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 mathematical expression called a homogeneous polynomial. One famous example is Fermat's Last Theorem, which asks whether there are integers that satisfy the equation x^d + y^d − z^d = 0 for d greater than 2. This problem was very hard and took over three hundred years for mathematicians to solve.

For equations of degree two, there are better methods to find solutions. These methods involve checking if a solution exists and then finding all possible solutions. An example is the equation x² + y² = 3z², which only has the trivial solution where x, y, and z are all zero.

Another well-known example is the equation for Pythagorean triples, x² + y² − z² = 0, which 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 are only interested in whole number solutions. It asks questions like: 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, who claimed in 1637 that you cannot split a cube into two smaller cubes, or a fourth power into two smaller fourth powers. This became known as Fermat's Last Theorem and was finally proven true in 1995 by Andrew Wiles. Another interesting problem involves ages, where a father's age is connected to his son's age by reversing the digits. These puzzles show how tricky it can be to find whole number answers to certain equations.

Over time, mathematicians have developed methods to try and solve these puzzles, but many Diophantine equations remain very hard to solve. Even today, researchers use advanced ideas from geometry to understand them better.

Exponential Diophantine equations

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

There isn't a general way to solve all these equations. Some special cases, like Catalan's conjecture and Fermat's Last Theorem, have been solved, but many are tackled using specific methods or even careful testing.