Questions of decidability and undecidability in Number Theory

1994 ◽  
Vol 59 (2) ◽  
pp. 353-371 ◽  
Author(s):  
B. Mazur
Keyword(s):  
Number Theory ◽  
Upper Bound ◽  
Diophantine Equations ◽  
Computable Function ◽  
Integral Solution ◽  
Survey Article ◽  
Image Position ◽  
Parameter Values ◽  
Computable Functions ◽  
Integral Solutions

Davis, Matijasevic, and Robinson, in their admirable survey article [D-M-R], interpret the negative solution of Hilbert's Tenth Problem as a resounding positive statement about the versatility of Diophantine equations (that any listable set can be coded as the set of parameter values for which a suitable polynomial possesses integral solutions).One can also view the Matijasevic result as implying that there are families of Diophantine equations parametrized by a variable t, which have integral solutions for some integral values t = a > 0, and yet there is no computable function of t which provides an upper bound for the smallest integral solution for these values a. The smallest integral solutions of the Diophantine equation for these values are, at least sporadically, too large to be bounded by any computable function. This is somewhat difficult to visualize, since there is quite an array of computable functions. But let us take an explicit example. Consider the functionMatijasevic's result guarantees the existence of parametrized families of Diophantine equations such that even this function fails to yield an upper bound for its smallest integral solutions (for all values of the parameter t for which there are integral solutions).Families of Diophantine equations in a parameter t, whose integral solutions for t = 1, 2, 3,… exhibit a certain arythmia in terms of their size, have fascinated mathematicians for centuries, and this phenomenon (the size of smallest integral solution varying wildly with the parameter-value) is surprising, even when the equations are perfectly “decidable”.

Bounds for the least solutions of homogeneous quadratic equations

1955 ◽  
Vol 51 (2) ◽  
pp. 262-264 ◽  
Author(s):  
J. W. S. Cassels
Keyword(s):  
Integral Solution ◽  
Quadratic Equations ◽  
Diagonal Forms ◽  
Image Position ◽  
Integral Solutions

In this note I obtain bounds for the least integral solutions of the equationin terms ofFor ternary diagonal forms, such bounds have been given by Axel Thue (4) and, more recently, by Holzer (1), Mordell (2) and Skolem (3), but these lead only to bad estimates for general ternaries. So far as I know there have not been given estimates for n≽4. Here I generalize Thue's method to prove:Theorem. Suppose that n ≥ 2 and that f(ɛ) represents zero. Then there is an integral solution of f(a) = 0 with

An application of the Thue–Siegel–Roth theorem to elliptic functions

1971 ◽  
Vol 69 (1) ◽  
pp. 157-161 ◽  
Author(s):  
J. Coates
Keyword(s):  
Number Theory ◽  
Diophantine Equations ◽  
Elliptic Functions ◽  
Rational Numbers ◽  
Algebraic Numbers ◽  
Number Of Solutions ◽  
Transcendental Numbers ◽  
Linearly Independent ◽  
Image Position ◽  
Effective Proof

Let α1, …, αn be n ≥ 2 algebraic numbers such that log α1,…, log αn and 2πi are linearly independent over the field of rational numbers Q. It is well known (see (6), Ch. 1) that the Thue–Siegel–Roth theorem implies that, for each positive number δ, there are only finitely many integers b1,…, bn satisfyingwhere H denotes the maximum of the absolute values of b1, …, bn. However, such an argument cannot provide an explicit upper bound for the solutions of (1), because of the non-effective nature of the theorem of Thue–Siegel–Roth. An effective proof that (1) has only a finite number of solutions was given by Gelfond (6) in the case n = 2, and by Baker(1) for arbitrary n. The work of both these authors is based on arguments from the theory of transcendental numbers. Baker's effective proof of (1) has important applications to other problems in number theory; in particular, it provides an algorithm for solving a wide class of diophantine equations in two variables (2).

scholarly journals Linear diophantine equations with cyclic coefficient matrices and its applications to Riemann surfaces

1985 ◽  
Vol 28 (3) ◽  
pp. 369-380
Author(s):  
Nobumasa Takigawa
Keyword(s):  
Riemann Surfaces ◽  
Diophantine Equations ◽  
Linear Equations ◽  
Integral Solution ◽  
Complex Numbers ◽  
Necessary And Sufficient Condition ◽  
Linear Diophantine Equations ◽  
Fixed Complex ◽  
Image Position ◽  
Necessary And Sufficient

Let co, c1, …, cn-1 be the nonzero complex numbers and let C = (cu+1,v+1) = (cn+u-v), O≦u,v≦n — 1, be a cyclic matrix, where n + u — v is taken modulo n. In this paper we shall give the solution of the linear equationswhere Lu (0≦u≦n —1) is a fixed complex number. In Theorem 1 weshall give a necessary and sufficient condition for (1) to have an integral solution.

The S-unit equation over function fields

1984 ◽  
Vol 95 (1) ◽  
pp. 3-4 ◽  
Author(s):  
Joseph H. Silverman
Keyword(s):  
Algebraic Geometry ◽  
Diophantine Equations ◽  
Function Fields ◽  
Number Fields ◽  
Upper Bounds ◽  
Linear Forms In Logarithms ◽  
Linear Forms ◽  
Image Position ◽  
The Given ◽  
Integral Solutions

In the study of integral solutions to Diophantine equations, many problems can be reduced to that of solving the equationin S-units of the given ring. To accomplish this over number fields, the only known effective method is to use Baker's deep results on linear forms in logarithms, which yield relatively weak upper bounds. For function fields, R. C. Mason [2] has recently given a remarkably strong effective upper bound. In this note we give an independent proof of Mason's bound, relying only on elementary algebraic geometry, principally the Riemann-Hurwitz formula.

scholarly journals On the System of Diophantine Equationsx2-6y2=-5andx=az2-b

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Silan Zhang ◽  
Jianhua Chen ◽  
Hao Hu
Keyword(s):  
Number Theory ◽  
Algebraic Number ◽  
Class Number ◽  
Diophantine Equations ◽  
Algebraic Number Theory ◽  
The Family ◽  
Integral Solutions

Mignotte and Pethö used the Siegel-Baker method to find all the integral solutions(x,y,z)of the system of Diophantine equationsx2-6y2=-5andx=2z2-1. In this paper, we extend this result and put forward a generalized method which can completely solve the family of systems of Diophantine equationsx2-6y2=-5andx=az2-bfor each pair of integral parametersa,b. The proof utilizes algebraic number theory andp-adic analysis which successfully avoid discussing the class number and factoring the ideals.

Addendum to “The truth is never simple”

1988 ◽  
Vol 53 (2) ◽  
pp. 390-392
Author(s):  
John P. Burgess
Keyword(s):  
Upper Bound ◽  
Present Note ◽  
Recent Survey ◽  
Survey Article ◽  
Inductive Definitions ◽  
Complete Set ◽  
Image Position

The present note outlines an answer to a question listed as open in our recent survey article [1], familiarity with which is assumed.Let Γ be the class of all X ⊆ ω such that X is reducible to for some some arithmetical and some positive with Y ⊆ p(Y) for all Y.A.1. Theorem. is a complete set of class Γ.A.2. Corollary, (a) is-in-a--parameter.(b) (i) Every set-in-a--parameter is reducible to.(ii) Every set-in-a--parameter is reducible to.Remarks. A.1 answers 7.3 of [1]. A.2 says as much as can be said about in terms of the coarse classifications of the analytical hierarchy. A.2 follows from A.1 by general methods and results in the theory of inductive definitions (having nothing specifically to do with truth), and its proof will be omitted.Proof of A.1. We omit subscripts vF.Upper Bound. Define

Isolated minima of the product of n linear forms

1953 ◽  
Vol 49 (1) ◽  
pp. 59-62 ◽  
Author(s):  
E. S. Barnes
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Linear Forms ◽  
Image Position

Letbe n linear forms with real coefficients and determinant Δ = ∥ aij∥ ≠ 0; and denote by M(X) the lower bound of | X1X2 … Xn| over all integer sets (u) ≠ (0). It is well known that γn, the upper bound of M(X)/|Δ| over all sets of forms Xi, is finite, and the value of γn has been determined when n = 2 and n = 3.

Post completeness in modal logic

1972 ◽  
Vol 37 (4) ◽  
pp. 711-715 ◽  
Author(s):  
Krister Segerberg
Keyword(s):  
Modal Logic ◽  
Upper Bound ◽  
Modus Ponens ◽  
Proper Subset ◽  
Proper Extension ◽  
Image Position

Let ⊥, →, and □ be primitive, and let us have a countable supply of propositional letters. By a (modal) logic we understand a proper subset of the set of all formulas containing every tautology and being closed under modus ponens and substitution. A logic is regular if it contains every instance of □A ∧ □B ↔ □(A ∧ B) and is closed under the ruleA regular logic is normal if it contains □⊤. The smallest regular logic we denote by C (the same as Lemmon's C2), the smallest normal one by K. If L and L' are logics and L ⊆ L′, then L is a sublogic of L', and L' is an extension of L; properly so if L ≠ L'. A logic is quasi-regular (respectively, quasi-normal) if it is an extension of C (respectively, K).A logic is Post complete if it has no proper extension. The Post number, denoted by p(L), is the number of Post complete extensions of L. Thanks to Lindenbaum, we know thatThere is an obvious upper bound, too:Furthermore,.

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.

Sublattices and Initial Segments of the Degrees of Unsolvability

1970 ◽  
Vol 22 (3) ◽  
pp. 569-581 ◽  
Author(s):  
S. K. Thomason
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Lattice Theory ◽  
Initial Segment ◽  
Finite Lattice ◽  
Equivalence Relations ◽  
Finite Lattices ◽  
Image Position ◽  
Degrees Of Unsolvability

In this paper we shall prove that every finite lattice is isomorphic to a sublattice of the degrees of unsolvability, and that every one of a certain class of finite lattices is isomorphic to an initial segment of degrees.Acknowledgment. I am grateful to Ralph McKenzie for his assistance in matters of lattice theory.1. Representation of lattices. The equivalence lattice of the set S consists of all equivalence relations on S, ordered by setting θ ≦ θ’ if for all a and b in S, a θ b ⇒ a θ’ b. The least upper bound and greatest lower bound in are given by the ⋃ and ⋂ operations:

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