Contemporary Mathematics

Volume 270, 2000

Hilbert's Tenth Problem:

What Was Done and What Is to Be Done

Yuri Matiyasevich

ABSTRACT. The paper describes the history of Hilbert's tenth problem and

its "negative" solution based on Diophantine representations for effectively

enumerable sets. Two constructions of such representations are described in

terse but complete form.

In his famous address

[24]

delivered to the Second International Congress of

Mathematicians (which was held in Paris in 1900), David Hilbert pointed out 23

most important, in his opinion, unsolved mathematical problems which the pending

20th century was to inherit from the passing 19th century.

The formulation of the lOth problem was so short that it can be reproduced

here entirely.

10.

ENTSCHEIDUNG DER LOSBARKEIT EINER DIOPHANTISCHEN

GLEICHUNG.

Eine diophantische Gleichung mit irgendwelchen Unbekannten

und mit ganzen rationalen Zahlkoefficienten sei vorgelegt: man soll

ein Verfahren angeben, nach welchen sich mittels einer endlichen

Anzahl von Operationen entscheiden liijlt, ob die Gleichung in gan-

zen rationalen Zahlen los bar ist.

1

A Diophantine equation is an equation of the form

(1)

D(x1, ... , Xm)

=

0,

where D is a polynomial with integer coefficients. These equations were named

after greek mathematician Diophantus who lived, most likely, in the 3rd century

A.C.

During the period between Diophantus' life and Hilbert's address to the Con-

gress, number-theorists had found solutions for plenty of Diophantine equations

and also had proved the unsolvability of a large number of other equations. So why

did Hilbert still consider solving Diophantine equations as an open problem?

1991 Mathematics Subject Classification. Primary 03-02, 03D25, 03D35, 11D41, 11D61,

11U05.

1

10. DETERMINATION OF THE SOLVABILITY OF A DIOPHANTINE EQUATION. Given a dio-

phantine equation with any number of unknown quantities and with rational integral numerical

coefficients:

To devise a process according to which it can be determined by a finite number of

operations whether the equation is solvable in rational integers. (Translation was taken from [25].)

©

2000 American Mathematical Society

http://dx.doi.org/10.1090/conm/270/04368