Universal diophantine equation

1982 ◽  
Vol 47 (3) ◽  
pp. 549-571 ◽  
Author(s):  
James P. Jones
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Recursively Enumerable Set ◽  
Integer Coefficients ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Image Position ◽  
Positive Integers ◽  
Martin Davis ◽  
Exponential Relation

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 formHere P is a polynomial with integer coefficients and the variables range over positive integers.In 1970 Ju. V. Matijasevič used this result to establish the unsolvability of Hilbert's tenth problem. Matijasevič proved [11] that the exponential relation y = 2x is diophantine This together with [2] implies that every recursively enumerable set is diophantine, i.e. every r.e. set Wcan be represented in the formFrom 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 W1, W2, W3, … can be enumerated in such a way that the binary relation x ∈ Wv is also recursively enumerable. Thus Matijasevič's theorem implies the existence of a diophantine equation U such that for all x and v,

Some representations of Diophantine sets

1972 ◽  
Vol 37 (3) ◽  
pp. 572-578 ◽  
Author(s):  
Raphael M. Robinson
Keyword(s):  
Universal Quantifier ◽  
Natural Numbers ◽  
Recursively Enumerable Set ◽  
Minor Error ◽  
Integer Coefficients ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Image Position ◽  
A Minor

A set D of natural numbers is called Diophantine if it can be defined in the formwhere P is a polynomial with integer coefficients. Recently, Ju. V. Matijasevič [2], [3] has shown that all recursively enumerable sets are Diophantine. From this, it follows that a bound for n may be given.We use throughout the logical symbols ∧ (and), ∨ (or), → (if … then …), ↔ (if and only if), ⋀ (for every), and ⋁ (there exists); negation does not occur explicitly. The variables range over the natural numbers 0,1,2,3, …, except as otherwise noted.It is the purpose of this paper to show that if we do not insist on prenex form, then every Diophantine set can be defined existentially by a formula in which not more than five existential quantifiers are nested. Besides existential quantifiers, only conjunctions are needed. By Matijasevič [2], [3], the representation extends to all recursively enumerable sets. Using this, we can find a bound for the number of conjuncts needed.Davis [1] proved that every recursively enumerable set of natural numbers can be represented in the formwhere P is a polynomial with integer coefficients. I showed in [5] that we can take λ = 4. (A minor error is corrected in an Appendix to this paper.) By the methods of the present paper, we can again obtain this result, and indeed in a stronger form, with the universal quantifier replaced by a conjunction.

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

1984 ◽  
Vol 49 (3) ◽  
pp. 818-829 ◽  
Author(s):  
J. P. Jones ◽  
Y. V. Matijasevič
Keyword(s):  
Natural Number ◽  
Polynomial Time ◽  
Simple Proof ◽  
Diophantine Equations ◽  
Polynomial Equation ◽  
Exponential Diophantine Equations ◽  
Recursively Enumerable ◽  
Image Position ◽  
The Right ◽  
Exponential Relation

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(a1, …, an) is exponential diophantine, i.e. can be represented in the formwhere a1 …, an, x1, …, xm 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, AB. We refer to the variables a1,…,an as parameters and the variables x1 …, xm as unknowns.Historically, the Davis, Putnam and Robinson theorem was one of the important steps in the eventual solution of Hilbert's tenth problem by the second author [1970], who proved that the exponential relation, a = bc, 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. Matijasevič [1971a], M. Davis [1973], Y. Matijasevič and J. Robinson [1975] or C. Smoryński [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.

Reductions of Hilbert's tenth problem

1958 ◽  
Vol 23 (2) ◽  
pp. 183-187 ◽  
Author(s):  
Martin Davis ◽  
Hilary Putnam
Keyword(s):  
Diophantine Equation ◽  
Hilbert's Tenth Problem ◽  
Recursively Enumerable ◽  
Hilbert’S Tenth Problem ◽  
Essential Change ◽  
Recursive Predicate ◽  
Image Position ◽  
Positive Integers

Hilbert's tenth problem is to find an algorithm for determining whether or not a diophantine equation possesses solutions. A diophantine predicate (of positive integers) is defined to be one of the formwhere P is a polynomial with integral coefficients (positive, negative, or zero). Previous work has considered the variables as ranging over nonnegative integers; but we shall find it more useful here to restrict the range to positive integers, no essential change being thereby introduced. It is clear that the recursive unsolvability of Hilbert's tenth problem would follow if one could show that some non-recursive predicate were diophantine. In particular, it would suffice to show that every recursively enumerable predicate is diophantine. Actually, it would suffice to prove far less.

scholarly journals Formalization of the MRDP Theorem in the Mizar System

2019 ◽  
Vol 27 (2) ◽  
pp. 209-221
Author(s):  
Karol Pąk
Keyword(s):  
Normal Form ◽  
Linear Equations ◽  
Negative Solution ◽  
Recursively Enumerable Set ◽  
Pell’S Equation ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Finite Products ◽  
Special Case ◽  
Martin Davis

Summary This article is the final step of our attempts to formalize the negative solution of Hilbert’s tenth problem. In our approach, we work with the Pell’s Equation defined in [2]. We analyzed this equation in the general case to show its solvability as well as the cardinality and shape of all possible solutions. Then we focus on a special case of the equation, which has the form x2 − (a2 − 1)y2 = 1 [8] and its solutions considered as two sequences $\left\{ {{x_i}(a)} \right\}_{i = 0}^\infty ,\left\{ {{y_i}(a)} \right\}_{i = 0}^\infty$ . We showed in [1] that the n-th element of these sequences can be obtained from lists of several basic Diophantine relations as linear equations, finite products, congruences and inequalities, or more precisely that the equation x = yi(a) is Diophantine. Following the post-Matiyasevich results we show that the equality determined by the value of the power function y = xz is Diophantine, and analogously property in cases of the binomial coe cient, factorial and several product [9]. In this article, we combine analyzed so far Diophantine relation using conjunctions, alternatives as well as substitution to prove the bounded quantifier theorem. Based on this theorem we prove MDPR-theorem that every recursively enumerable set is Diophantine, where recursively enumerable sets have been defined by the Martin Davis normal form. The formalization by means of Mizar system [5], [7], [4] follows [10], Z. Adamowicz, P. Zbierski [3] as well as M. Davis [6].

Arithmetical representation of recursively enumerable sets

1956 ◽  
Vol 21 (2) ◽  
pp. 162-186 ◽  
Author(s):  
Raphael M. Robinson
Keyword(s):  
Recursive Function ◽  
Natural Numbers ◽  
General Recursive Function ◽  
Recursively Enumerable Set ◽  
Integer Coefficients ◽  
Primitive Recursive Function ◽  
Recursively Enumerable ◽  
Image Position ◽  
The Right ◽  
Primitive Recursive

A set S of natural numbers is called recursively enumerable if there is a general recursive function F(x, y) such thatIn other words, S is the projection of a two-dimensional general recursive set. Actually, it is no restriction on S to assume that F(x, y) is primitive recursive. If S is not empty, it is the range of the primitive recursive functionwhere a is a fixed element of S. Using pairing functions, we see that any non-empty recursively enumerable set is also the range of a primitive recursive function of one variable.We use throughout the logical symbols ⋀ (and), ⋁ (or), → (if…then…), ↔ (if and only if), ∧ (for every), and ∨(there exists); negation does not occur explicitly. The variables range over the natural numbers, except as otherwise noted.Martin Davis has shown that every recursively enumerable set S of natural numbers can be represented in the formwhere P(y, b, w, x1 …, xλ) is a polynomial with integer coefficients. (Notice that this would not be correct if we replaced ≤ by <, since the right side of the equivalence would always be satisfied by b = 0.) Conversely, every set S represented by a formula of the above form is recursively enumerable. A basic unsolved problem is whether S can be defined using only existential quantifiers.

A note on the diophantine equation (xm − l) / (x − 1) = yn + l

1994 ◽  
Vol 116 (3) ◽  
pp. 385-389 ◽  
Author(s):  
Le Maohua
Keyword(s):  
Diophantine Equation ◽  
Rational Numbers ◽  
Prime Factors ◽  
Image Position ◽  
Positive Integers

Let ℤ, ℕ, ℚ denote the sets of integers, positive integers and rational numbers, respectively. Solutions (x, y, m, n) of the equation (1) have been investigated in many papers:Let ω(m), ρ(m) denote the number of distinct prime factors and the greatest square free factor of m, respectively. In this note we prove the following results.

Degrees of formal systems

1958 ◽  
Vol 23 (4) ◽  
pp. 389-392 ◽  
Author(s):  
J. R. Shoenfield
Keyword(s):  
Sufficient Conditions ◽  
Formal System ◽  
Independent Interest ◽  
Recursively Enumerable Set ◽  
First Order ◽  
Formal Systems ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Axiomatizable Theory

In this paper we answer some of the questions left open in [2]. We use the terminology of [2]. In particular, a theory will be a formal system formulated within the first-order calculus with identity. A theory is identified with the set of Gödel numbers of the theorems of the theory. Thus Craig's theorem [1] asserts that a theory is axiomatizable if and only if it is recursively enumerable.In [2], Feferman showed that if A is any recursively enumerable set, then there is an axiomatizable theory T having the same degree of unsolvability as A. (This result was proved independently by D. B. Mumford.) We show in Theorem 2 that if A is not recursive, then T may be chosen essentially undecidable. This depends on Theorem 1, which is a result on recursively enumerable sets of some independent interest.Our second result, given in Theorem 3, gives sufficient conditions for a theory to be creative. These conditions are more general than those given by Feferman. In particular, they show that the system of Kreisel described in [2] is creative.

ON EXPONENTIAL DIOPHANTINE EQUATIONS CONTAINING THE EULER QUOTIENT

2014 ◽  
Vol 91 (1) ◽  
pp. 11-18
Author(s):  
NOBUHIRO TERAI
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Euler’S Totient Function ◽  
Exponential Diophantine Equations ◽  
Euler’S Theorem ◽  
Positive Integers

AbstractLet $a$ and $m$ be relatively prime positive integers with $a>1$ and $m>2$. Let ${\it\phi}(m)$ be Euler’s totient function. The quotient $E_{m}(a)=(a^{{\it\phi}(m)}-1)/m$ is called the Euler quotient of $m$ with base $a$. By Euler’s theorem, $E_{m}(a)$ is an integer. In this paper, we consider the Diophantine equation $E_{m}(a)=x^{l}$ in integers $x>1,l>1$. We conjecture that this equation has exactly five solutions $(a,m,x,l)$ except for $(l,m)=(2,3),(2,6)$, and show that if the equation has solutions, then $m=p^{s}$ or $m=2p^{s}$ with $p$ an odd prime and $s\geq 1$.

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