Main Index
Number Theory
Diophantine equations
Subject Index
comment on the page
Main Index
Number Theory
Diophantine equations
Subject Index
comment on the page
An indeterminate or an unknown is a (usually) independent variable that is not known.
An indeterminate equation is a mathematical statement stating that two expression (in modern era written in symbols) are exactly the same (or equivalent) and which expressions contain one or more unknowns or indeterminate variables.
To solve an (indeterminate) equation is the problem of finding what values (usually numbers or functions) fulfill a condition stated as an equation. The solution of an equation is the set of all values which, when substituted for unknowns, make an equation (or all equations of a system of equation) true. The idea of a solution of equations is very old, and it has existed since the time of the Egyptians and Babylonians.
A Diophantine equation is an indeterminate (usually algebraic with integer coefficients) equation that allows the variables to be integers only. The word Diophantine refers to Diophantus of Alexandria, a Hellenistic mathematician (born between 200 and 214, died between 284 and 298 CE) 
. His major work Arithmetica is a collection of 130 problems giving numerical solutions of determinate equations. Contrary to contemporary situation, Diophantus solved these equations not in integers but mostly in positive rational numbers.
Hilbert's 10th problem asked if an algorithm existed for determining whether an arbitrary Diophantine equation has a solution or not. The impossibility of existence of such algorithm was proven (based on earlier work by M.Davis, Hilary Putnam and Julia Robinson) .by Yuri Matiyasevich in 1970 [1] , [2] (for some improvements cf. [3] , [4] ). Later he and Robinson [5] proved that even for equations in thirteen variables, no algorithm can exist to determine whether there is a solution and with Jones he then improved this result to equations in only nine variables [6] (for more details about the solution of 10th Hilbert’s problem consult [7] ).
That Hilbert's problem, maybe, has negative solutions became more probable after the notion of algorithm was formalized in 1930s. Hilbert did use the word algorithm, he said ... man soll ein Verfahren angeben, nach welchen sich mittels einer endlichen Anzahl von Operationen entscheiden lässt, ob die Gleichung in ganzen rationalen Zahlen lösbar ist (devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers).
| [1] | Matiyasevich, Y. V. (1970). Enumerable sets are Diophantine (in Russian). Doklady Akad. Nauk SSSR, 191, 279-282. |
| [2] | Matiyasevich, Y. V. (1970). Enumerable sets are Diophantine. Soviet Math. Doklady, 11(2), 279-282. |
| [3] | Matiyasevich, Y. V. (1972). Diophantine sets, (Russian). Uspekhi Math. Nauk, 27, 185-222. |
| [4] | Matiyasevich, Y. V. (1972). Diophantine sets. Russian Mathematical Surveys, 27(5), 124-164. |
| [5] | Matiyasevich, Y. V., & Robinson, J. (1975). Reduction of an arbitrary Diophantine equation to one in 13 unknowns. Acta Arithmetica, 27, 521-553. |
| [6] | Jones, J. P., & Matiyasevich, Y. V. (1981). Exponential Diophantine Representation of Recursively Enumerable Sets. Proceedings of the Herbrand Symposium, Marseilles, North-Holland, Amsterdam, pp. 159-177. |
| [7] | Matiyasevich, Y. V. (1993). Hilbert's Tenth Problem. Cambridge: MIT Press. |
Cite this web-page as:
Štefan Porubský: Diophantine equations.