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An application of recursive function theory to Hilbert's tenth problem. It is proved that if a particular exhibited diophantine equation has no nontrivial solutions, then all recursively enumerable sets are diophantine. Hence, if the exhibited diophantine equation has no nontrivial solutions, then Hilbert's tenth problem is recursively unsolvable. The methods used can be readily adapted to obtain various other hypotheses about which demonstrations can be made similar to the one given in this study. It has not yet been proved that the "one equation to rule them all" has no nontrivial solution, but so far the search for counterexamples has been fruitless.

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