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Can someone provide a proof, or a link to a proof, of why does the Matiyasevich polynomial always generate primes for the nonnegative results? Any help will be appreciated.

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Have you read Matiyasevich's original paper? Or his book? –  Chris Eagle Oct 9 '12 at 20:25
    
The book Hilbert's Tenth Problem goes through all that rather well. –  Ross Millikan Oct 9 '12 at 20:29
    
Thank you for pointing me to the right side, I´m going to investigate. –  chubakueno Oct 11 '12 at 1:29

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