Software 251 Can humans solve ‘unsolvable’ problems? The question of whether Fermat’s Last Theorem could be solved mechanically remained unanswered until 1970 when Yuri Matiyasevich filled in the missing piece in Julia Robinson's proof. Matiyasevich used an ingenious reduction method to match up sequences in Robinson's theorem with a set of Turing machines. This showed that if Robinson's theorem was false you could solve the halting problem and since you can't solve the halting problem, then Robinson's theorem must be true. All this effort proved Diophantine equations have no general algorithmic solution. This was a hugely important result but, as we noted earlier, Fermat’s Last Theorem is not, strictly speaking, a Diophantine. It is an exponential Diophantine equation. We still had no definitive answer to Fermat. In 1972 Keijo Ruohonen and again in 1993, Christoph Baxa demonstrated that Diophantine equations with exponential terms could be rewritten as regular Diophantine equations with one additional complication — the necessity of adding an infinite set of terms to the end of the equation. In 1993, J.P. Jones of the University of Calgary showed the logic limit for regular Diophantine equations lies at thirteen unknowns. Matiyasevich had already pointed this out but never completed his proof. Since infinity is greater than thirteen, all exponential Diophantine equations are above the logic limit and, therefore, undecidable. Finally, we have a proof that Fermat’s Last Theorem is unsolvable by a computer — or at least by a general purpose algorithm running on a computer. Matiyasevich went on to show many mathematical problems can be rewritten as exponential Diophantine equations and that much of mathematics is undecidable. For example, the Four Color Conjecture: “Given an arbitrary map on a Euclidean plane, show the map can be colored in a maximum of four colors such that no adjacent area shares the same color.” Meanwhile, Andrew Wiles, an English mathematics Pro