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Diophantine set - Wikipedia

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In mathematics, a Diophantine equation is an equation of the form P(x1, ..., xj, y1, ..., yk) = 0 (usually abbreviated P(x, y) = 0) where P(x, y) is a polynomial with integer coefficients, where x1, ..., xj indicate parameters and y1, ..., yk indicate unknowns. A Diophantine set is a subset S of 𝑁 𝑗 , the set of all j-tuples of natural numbers, so that for some Diophantine equation P(x, y) = 0, That is, a parameter value is in the Diophantine set S if and only if the associated Diophantine equation is satisfiable under that parameter value. The use of natural numbers both in S and the existential quantification merely reflects the usual applications in computability theory and model theory. It does not matter whether natural numbers refer to the set of nonnegative integers or positive integers since the two definitions for Diophantine sets are equivalent. We can also equally well speak of Diophantine sets of integers and freely replace quantification over natural numbers with quanti

Diophantine set - Wikipedia Jump to content From Wikipedia, the free encyclopedia Solution of some Diophantine equation In mathematics , a Diophantine equation is an equation of the form P ( x 1 , ..., x j , y 1 , ..., y k ) = 0 (usually abbreviated P ( x , y ) = 0) where P ( x , y ) is a polynomial with integer coefficients , where x 1 , ..., x j indicate parameters and y 1 , ..., y k indicate unknowns. A Diophantine set is a subset S of N j {\displaystyle \mathbb {N} ^{j}} , the set of all j -tuples of natural numbers, so that for some Diophantine equation P ( x , y ) = 0, x ¯ ∈ S ⟺ ( ∃ y ¯

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