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Universal Diophantine Equation
TLDR
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.
DIOPHANTINE REPRESENTATION OF THE SET OF PRIME NUMBERS
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Proof or recursive unsolvability of Hilbert's tenth problem
(1991). Proof of Recursive Unsolvability of Hilbert's Tenth Problem. The American Mathematical Monthly: Vol. 98, No. 8, pp. 689-709.
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 Hilbert's
Register Machine Proof of the Theorem on Exponential Diophantine Representation of Enumerable Sets
TLDR
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.
Variants of Robinson's essentially undecidable theoryR
TLDR
An essentially undecidable theory based on three axiom schemes involving only multiplication and less than or equals is given.
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.
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