In other words, Hilbert's 10th problem is undecidable.
The mathematicians hoped to follow the same approach to prove the extensive version of the rings of the titles of the problem, but they hit a hitch.
Eraser the works
The useful correspondence between the Turing machines and the diophantine equations collapses when the equations are authorized to have non -whole solutions. For example, consider the equation again Y = X2. If you work in a ring of whole that includes √2, you will end up with new solutions, such as X = √2, Y = 2. The equation no longer corresponds to a Turing machine which calculates perfect squares – and, more generally, the diophantine equations can no longer code the stop problem.
But in 1988, a student graduated from New York University appointed Sasha Shlapentokh started playing with ideas to get around this problem. In 2000, she and others had formulated a plan. Say that you had to add a lot of additional terms to an equation like Y = X2 that magically forced X To be an integer again, even in a system of different numbers. Then you can recover the correspondence from a Turing machine. Could we do the same for all Diophantine equations? If this is the case, this would mean that Hilbert's problem could code the stop problem in the new number system.
Illustration: Myriam Wares for How many magazine
Over the years, Shlapentokh and other mathematicians have understood what terms they had to add to the Diophantine equations for various types of rings, which allowed them to demonstrate that Hilbert's problem was always undecidable in these contexts. They then resolved all the remaining rings of the integers in one case: rings that involve the imaginary number I. The mathematicians have realized that in this case, the terms they should add could be determined using a special equation called the elliptical curve.
But the elliptical curve should satisfy two properties. First, it would be necessary to have an infinity of solutions. Second, if you go to another ring of whole – if you have deleted the imaginary number of your number system – then all the solutions to the elliptical curve should maintain the same underlying structure.
It turned out that the construction of such an elliptical curve that worked for each remaining ring was an extremely subtle and difficult task. But Koymans and Pagano – Experts on the elliptical curves who had worked in close collaboration since they were in higher education – had just the right tool to try.
Sleepless nights
Since his first cycle time, Koymans thought of the 10th Hilbert problem. Throughout higher education, and throughout his collaboration with Pagano, he was bearing. “I spent a few days each year thinking about it and staying horribly stuck,” said Koymans. “I would try three things and they would all explode on my face.”
In 2022, at a conference in Banff, Canada, he and Pagano ended up discussing the problem. They hoped that together, they could build the special elliptical curve necessary to solve the problem. After finishing other projects, they got to work.
