Binary quadratic Diophantine equations are of interest from the viewpoint of computational complexity theory. This class of equations includes as special cases many of the known examples of natural problems apparently occupying intermediate stages in the P − NP hierarchy, i.e., problems not known to be solvable in polynomial time nor to be NP-complete, for example the problem of factoring integers. Let L(F) denote the length of the binary encoding of the binary quadratic Diophantine equation F given by ax 2 +bx1x2+cx 2 +dx1+ex2+f = 0. Suppose F is such an equation having a nonnegative integer solution. This paper shows that there is a proof (i.e., “certificate”) that F has such a solution which can be checked in O(L(F) 5 logL(F)log logL(F)) bit operations. A corollary of this result is that the set � = {F : F has a nonnegative integer solution} is in

Additional excerpts

The Decision Problem for Exponential Diophantine Equations

Succinct certificates for the solvability of binary quadratic Diophantine equations

We shall show that there is no general method of telling whether an arbitrary polynomial P(xl, , Xk) with integer coefficients is ever a power of 2 for xi, , xk natural numbers. At present there is no general method known even in the special case with k = 1. Actually we shall show directly that the relation given by r =2 is diophantine in the set S of powers of 2. Hence every recursively enumerable set is diophantine in i by Corollary 5 of Davis, Putnam, and Robinson [4]. We call a relation p(x1, , xn) among natural numbers diophantine if there is a polynomial P with integer coefficients such that p(xi, , xn) if and only if there are natural numbers yi, , Yk with P(xi, , xn, yi, , yk) = 0. A set 8 (function F) is diophantine if xCS (the graph of F) is diophantine. Also, p is diophantine in a set 8 if there is a polynomial P with integer coefficients such that p(xi, , xn) if and only if there are natural numbers yi, , Yk, Z, ... z1 with both P(xi, , xn, Yl, , Yk, zl, , z,)=0 and Zi, , z,CS. In [6], the term "existentially definable" was used instead of "diophantine". The definitions given there are easily seen to be equivalent to these. In both [1] and [4], relations over the positive integers were considered but the definitions and theorems hold for natural numbers with only the obvious modifications. The logical symbols V (there exists), A (and), 1 and an, a ' be defined by

background or methods

/pdf/unsolvable-diophantine-problems-jl8axt6fw3.pdf

Unsolvable diophantine problems

We formalise the undecidability of solvability of Diophantine equations, i.e. polynomial equations over natural numbers, in Coq's constructive type theory. To do so, we give the first full mechanisation of the Davis-Putnam-Robinson-Matiyasevich theorem, stating that every recursively enumerable problem - in our case by a Minsky machine - is Diophantine. We obtain an elegant and comprehensible proof by using a synthetic approach to computability and by introducing Conway's FRACTRAN language as intermediate layer.

background

https://drops.dagstuhl.de/opus/volltexte/2019/10534/pdf/LIPIcs-FSCD-2019-27.pdf/

Hilbert's Tenth Problem in Coq.

Suppose we wish to build up the class of recursively enumerable sets by starting with the set DI of natural numbers and constructing new sets from those already obtained using as little auxiliary machinery as possible. One way would be to start with a finite number of functions F1, * * *, Fk (of one variable, from and to 9t) such that every recursively enumerable set can be obtained from t by constructing new sets Fj [3 I where 5 is a previously obtained set. We can think of F1, * * *, Fk as unary operations on sets of natural numbers. Any set 8 obtained in this way is the range of a function F obtained by composition from F1, **, Fk. If we consider the values of F1, * * *, Fk as given, then the number of steps needed to compute Fn does not depend on n. Hence for all xe8, there exists a proof that xCS of bounded length in terms of F1, * * *, Fk (just as there is a one-step proof that a composite number is composite in terms of multiplication). We say a set of natural numbers is generated by F1, * * *, Fk if it is the range of a function obtained by composition from F1, * * * , Fk. Also a class e of sets is generated by F1, * * *, Fk if every nonempty set of e is generated by F1, l * , Fk and every set generated by Fl, @ , Fk is in e. EXAMPLE. Let Go, G1, * * * be the primitive recursive functions listed systematically so that the function G given by

Finite generation of recursively enumerable sets

As is well known, there is an algorithm for deciding if a system of linear equations with coefficients from the set Z of integers has a solution in integers. The purpose of this paper is to answer the following question: does this remain true if ‘linear’ is replaced by ‘quadratic’?

The Decision Problem for Exponential Diophantine Equations

Citations

Succinct certificates for the solvability of binary quadratic Diophantine equations

Unsolvable diophantine problems

Hilbert's Tenth Problem in Coq.

Finite generation of recursively enumerable sets

Integer solutions of systems of quadratic equations

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