# Bynoalgorithmexistswemeanthereisnta

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Unformatted text preview: pervades all of computation Has much more broad consequences than it at first appears Hilbert’s 10th problem Hilbert’s 10th problem Finding integer solutions to a polynomial with integer coefficients: 15x4y2 ­ 3xy3z2 + 2x2z = 1 No algorithm exists for finding such solutions, or determining whether there are any. in some cases you can find solutions, or tell that there is no solution. By “No algorithm exists” we mean there isn’t a general method that works for all possible polynomials. Mathematical theorems… Mathematical theorems… Given a possible theorem T, we’d like to knowif T is true. We need to find a proof of T. Can write a computer program P to enumerate all possible proofs for T, and check to see if any is a correct one. If P ever halts, it has found a correct proof. Thus, halting problem for P is question of whether or not T is provably true. Towards prov...
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