Volume 270, 2000
Hilbert's Tenth Problem:
What Was Done and What Is to Be Done
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
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
ENTSCHEIDUNG DER LOSBARKEIT EINER DIOPHANTISCHEN
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.
A Diophantine equation is an equation of the form
D(x1, ... , Xm)
where D is a polynomial with integer coefficients. These equations were named
after greek mathematician Diophantus who lived, most likely, in the 3rd century
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,
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
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 .)
2000 American Mathematical Society