James P. Jones

Universal Diophantine Equation

In 1961 Martin Davis, Hilary Putnam and Julia Robinson [2] proved that every recursively enumerable set W is exponential diophantine, i.e. can be represented in the form Here P is a polynomial with integer coefficients and the variables range over positive integers. In 1970 Ju. V. Matijasevic used this result to establish the unsolvability of Hilberts tenth problem. Matijasevic proved [11] that the exponential relation y = 2 x is diophantine This together with [2] implies that every recursively enumerable set is diophantine, i.e. every r.e. set W can be represented in the form From this it follows that there does not exist an algorithm to decide solvability of diophantine equations. The nonexistence of such an algorithm follows immediately from the existence of r.e. nonrecursive sets. Now it is well known that the recursively enumerable sets W 1 , W 2 , W 3 , … can be enumerated in such a way that the binary relation x ∈ W v is also recursively enumerable. Thus Matijasevics theorem implies the existence of a diophantine equation U such that for all x and v ,

DIOPHANTINE REPRESENTATION OF THE SET OF PRIME NUMBERS

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. [4] [81 have proven that every recursively enumerable set is Diophantine, and hence that the set of prime numbers is Diophantine. From this, and work of Putnam [12], it follows that the set of prime numbers is representable by a polynomial formula. In this article such a prime representing polynomial will be exhibited in explicit form. We prove (in Section 2)

Register Machine Proof of the Theorem on Exponential Diophantine Representation of Enumerable Sets

The purpose of the present paper is to give a new, simple proof of the theorem of M. Davis, H. Putnam and J. Robinson [1961], which states that every recursively enumerable relation A ( a 1 , …, a n ) is exponential diophantine , i.e. can be represented in the form where a 1 …, a n , x 1 , …, x m range over natural numbers and R and S are functions built up from these variables and natural number constants by the operations of addition, A + B , multiplication, AB , and exponentiation, A B . We refer to the variables a 1 ,…, a n as parameters and the variables x 1 …, x m as unknowns . Historically, the Davis, Putnam and Robinson theorem was one of the important steps in the eventual solution of Hilberts tenth problem by the second author [1970], who proved that the exponential relation, a = b c , is diophantine, and hence that the right side of (1) can be replaced by a polynomial equation. But this part will not be reproved here. Readers wishing to read about the proof of that are directed to the papers of Y. Matijasevic [1971a], M. Davis [1973], Y. Matijasevic and J. Robinson [1975] or C. Smorynski [1972]. We concern ourselves here for the most part only with exponential diophantine equations until §5 where we mention a few consequences for the class NP of sets computable in nondeterministic polynomial time.

Proof or recursive unsolvability of Hilbert's tenth problem

JAMES P. JONES received his Ph.D. in 1968 at the University of Washington, Seattle. His supervisor was R. W. Ritchie. He is presently in the Department of Mathematics of the University of Calgary, Alberta, Canada. He has spent leaves in the Steklov Mathematical Institute of the Academy of Sciences of the U.S.S.R., Leningrad, the Tata Institute of Fundamental Research, the University of California, Berkeley and Academia Sinica, Beijing. He has given lectures in Moscow State University, the Institute of Cybernetics in Tallinn, U.S.S.R., the Steklov Mathematical Institute in Leningrad, U.S.S.R., and in the Main Computing Center of the Academy of Sciences of the U.S.S.R. in Riga. He has also lectured at the Stefan Banach Center in Warsaw and at universities in England, France, Italy, Germany, China and Nepal. His current research area is logic and number theory.

Some undecidable determined games

Computing machines using algorithms play games and even learn to play games. However, the inherent finiteness properties of algorithms impose limitations on the game playing abilities of machines. M. Rabin illustrated this limitation in 1957 by constructing a two-person win-lose game with decidable rules but no computable winning strategies. Rabins game was of the type where two players take turns choosing integers to satisfy some decidable but very complicated winning condition. In the present paper we obtain similar theorems of this type but the winning conditions are extremely simple relations (polynomial equations). Specific examples are given.

Exponential Diophantine Representation of Recursively Enumerable Sets

Publisher Summary This chapter discusses the exponential diophantine representation of recursively enumerable sets. The chapter presents several theorems and states that in the case of certain particular recursive sets one may be able to delete a quantifier. The chapter shows that this is the case for primes, Mersenne primes, perfect numbers, and certain other recursive sets occurring in classical number theory. These sets are all particular examples of Kalmar Elementary Relations. The results are essentially the same as those of Jones–Matijasevic.

Undecidable diophantine equations

In 1900 Hubert asked for an algorithm to decide the solvability of all diophantine equations, P(x1, . . . , xv) = 0, where P is a polynomial with integer coefficients. In special cases of Hilberts tenth problem, such algorithms are known. Siegel [7] gives an algorithm for all polynomials P(xx, . . . , xv) of degree < 2. From the work of A. Baker [1] we know that there is also a decision procedure for the case of homogeneous polynomials in two variables, P(x, y) = c. The first steps toward the eventual negative solution of the entire (unrestricted) form of Hilberts tenth problem, were taken in 1961 by Julia Robinson, Martin Davis and Hilary Putnam [2]. They proved that every recursively enumerable set, W can be represented in exponential diophantine form

Variants of Robinson's essentially undecidable theoryR

AbstractCobham has observed that Raphael Robinsons well known essentially undecidable theoryR remains essentially undecidable if the fifth axiom scheme

Complexities of winning strategies in diophantine games

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