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hilbert-tenth
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No such algorithm exists: every computably enumerable set is Diophantine, so a decision procedure would solve the halting problem. The version over the rationals is still open.
Evidence tier V3 is the import default for a result the curation records as settled in the published record. No formal proof or certificate is attached to this page yet.
Posed 1900, David Hilbert. Closed 1970, Yuri Matiyasevich, completing Martin Davis, Hilary Putnam and Julia Robinson.