The Diophantine Equation

Maxwell’s equations, Einstein’s field equations and Schrödinger’s equations are some of the most difficult equations for anyone to attempt to solve, and if you don’t have a degree in mathematics, then forget about it.  Maxwell’s equations require an understanding of some calculus, vector calculus, light differential equations, and basic electromagnetism.  Working with Einstein’s field equations which relate spacetime curvature to matter and energy would require you to know about tensor calculus (or differential geometry).  To understand Schrödinger’s equations, you would have to know Linear Algebra, Partial Differential Equations, Hilbert space, the Dirac- delta function, and Fourier transforms.  To work on a Diophantine equation, all you need to know is what constitutes an integer and the basic arithmetic of Euclidean division, where you divide the dividend by the divisor, however this equation can be very difficult to solve.

Around 270, Diophantus of Alexandria wrote a collection of books called Arithmetica, that contains approximately 260 assorted algebra problems that deal with finding integer solutions to equations in several unknowns.  Diophantine equations are polynomial equations that involve only sums, products, and powers in which all the constants have integer coefficients and where only integer solutions are accepted.  An example would be, ax + by = c or x² − y² = z³, where a, b, c, x, y, and z are integers.  Diophantus was the first to introduce algebraic symbolism.  A Diophantine equation is an equation relating integer (or sometimes natural number or whole number) quantities.  Finding the solution or solutions to a Diophantine equation is closely tied to modular arithmetic, number theory, Galois theory, Fermat’s Last Theorem, theta-functions and Abelian functions.  Often, when a Diophantine equation has infinitely many solutions, parametric form is used to express the relation between the variables of the equation.  The equation ax + by = c, has many solutions and if we choose a to be 3 and x to be 4 and b to be 6 and y to be 1, then c would equal 18 as shown here: 3·4+6·1=18, but I am not going to get into how to solve Diophantine equations in this post.

The German mathematician David Hilbert (1862–1943) was a speaker at the International Congress of Mathematicians in 1900, where he proposed a list of 23 as yet unsolved problems that he saw as being the greatest challenges for twentieth-century mathematics and this proved to be very influential in the development of mathematics in the last century.  The tenth problem in his list relates to a Diophantine equation, where Hilbert asked for a computing algorithm which will tell if a given polynomial Diophantine equation with integer coefficients has a solution in integers or not, or they would have to prove that a solution was impossible.  Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.  This tenth problem was solved in 1970 by Yuri Matiyasevich and his ingenious solution reduced the proof to polynomials, proving that no such algorithm exists.  Matiyasevich built on the work of others including Julia Robinson, Martin Davis, and Hilary Putnam, eventually showing that one could model Turing machines using Diophantine equations where a solution to a Diophantine equation existing was equivalent to the corresponding Turing machine halting.  The halting problem tried to determine from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever.  A general procedure to solve Diophantine equations is equivalent to solving the Halting problem.

Written for FOWC with Fandango – Equation.