author={James P. Jones and Daihachiro Sato and Hideo Wada and Douglas P. Wiens},
  journal={American Mathematical Monthly},
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Formalizing a Diophantine Representation of the Set of Prime Numbers

This work shows that the exponential function is diophantine, together with the same properties for the binomial coefficient and factorial, in the set of prime numbers and its explicit representation using 10 variables, the smallest representation known today.

Register machine proof of the theorem on exponential diophantine representation of enumerable sets

A new, simple proof of the theorem of M. Davis, Putnam and Robinson, which states that every recursively enumerable relation A is exponential diophantine, is given, 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.

Universal diophantine equation

Matijasevic's theorem implies the existence of a diophantine equation U such that for all x and v, x ∈ W v is also recursively enumerable, and the nonexistence of such an algorithm follows immediately from theexistence of r.e. nonrecursive sets.

Polynomials and Second Order Linear Recurrences

One of the most interesting results of the last century was the proof completed by Matijasevich that computably enumerable sets are precisely the diophantine sets [MRDP Theorem, 9], thus settling,

A note on diophantine representations

In 1900 David Hilbert asked for an algorithm to decide whether a given diophantine equation is solvable or not and put this problem tenth in his famous list of 23. In 1970 it was proved that such an

A Diophantine representation of Wolstenholme's pseudoprimality

It is shown that the upper bound for the number of variables in a prime-representing polynomial could be lowered to 8 if the converse of Wolstenholme's theorem (1862) holds, as conjectured by James P. Jones.

Prime Representing Polynomial

Summary The main purpose of formalization is to prove that the set of prime numbers is diophantine, i.e., is representable by a polynomial formula. We formalize this problem, using the Mizar system

Diophantine representations of recursive enumerable sets

In this thesis we will discuss various results found by other mathematicians about the connection between recursively enumerable sets and diophantine representation. As a starting point, we will use

Diophantine Representation of Fibonacci Numbers Over Natural Numbers

The sequence of Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, …, defined by F0 = 0, F1 = 1, F n+2 = F n + F n+1, played an important role in the solution of one of the Hilbert Problems. The


The Davis-Putnam-Robinson theorem showed that every partially computable m-ary function f (a1, . . . ,am) = c on the natural numbers can be specified by means of an exponential Diophantine formula



An unsolvable problem in number theory

  • H. Putnam
  • Mathematics
    Journal of Symbolic Logic
  • 1960
This paper presents several results whose common element is their connection with Hubert's tenth problem.1 The one that seems most striking (perhaps because it can be stated with a minimum of

Hilbert's Tenth Problem is Unsolvable

When a long outstanding problem is finally solved, every mathematician would like to share in the pleasure of discovery by following for himself what has been done. But too often he is stymied by the

An Introduction to the Theory of Numbers

This is the fifth edition of a work (first published in 1938) which has become the standard introduction to the subject. The book has grown out of lectures delivered by the authors at Oxford,

Formula for the Nth Prime Number

In this note we give a simple formula for the nth prime number. Let pn denote the nth prime number (p 1=2, p 2 = 3, etc.). We shall show that p n is given by the following formula.

Topics in complex function theory

Develops the higher parts of function theory in a unified presentation. Starts with elliptic integrals and functions and uniformization theory, continues with automorphic functions and the theory of

Prime-Representing Functions

The Decision Problem for Exponential Diophantine Equations