In other words, Hilbert's 10th problem is not solved.
Mathematicians hope to follow the same approach to prove the extended version, the ring of the problem but they have hit a push.
Gumming on works
The useful correspondence between turing machines and diophantine equations collapses when the equations are allowed to have non -intact solutions. For example, please review the equation y = x2. If you are working in a round of integers including √2, then you will end with some new solutions, such as x = 2, y = 2. The equation no longer corresponds to the Turing machine calculating the perfect square and in general, the diophantine equations can no longer encrypt the suspension problem.
But in 1988, a graduated at New York University was named Sasha shlapentokh Start playing with the idea of solving this problem. By 2000, she and others built a plan. Say that you have added a series of additional terms to an equation like y = x2 That miraculously x Become an integer again, even in another digital system. You can then save the corresponding to a turing machine. Can the same thing can be done for all diophantine equations? If so, that means Hilbert's problem can encrypt the suspension problem in a new digital system.
Illustration: Myriam Wares How many magazines
Over the past years, Shlapentokh and other mathematicians have found the terms they have to add to the diophantine equations for different types of rings, allowing them to prove that Hilbert's problem still cannot be solved in those settings. After that, they boiled all the remaining integers in one case: the rings related to imagination numbers I. Mathematicians realize that in this case, the terms they must be added can be determined by using a special equation called ellipse curve.
But the elliptical curve will have to meet the two attributes. First, it will need countless solutions. Secondly, if you switch to another integer round, if you remove the imagination from your digital system, all solutions for elliptical curves will have to maintain the same basic structure.
When it pops out, building such an elliptical curve works for every remaining ring is an extremely delicate and difficult task. But Koymans and Pagano, who have been on the elliptical curves, who have worked closely since they were in graduate school, were just a suitable tool to try.
Night without sleep
Since college college, Koymans have thought about Hilbert's 10th issue. During the university, and during his cooperation with Pagano, it waved. I spent a few days a year thinking about it and trapped, Mr. Koy Koyman said. “I tried three things and they would all explode on my face.”
In 2022, when at a conference in BanFF, Canada, England and Pagano ended a conversation about this issue. They hope that together, they can build an especially elliptical curve needed to solve the problem. After completing a number of other projects, they have to work.