In other words, the 10th problem of Hilbert is undecided.
Mathematicians hoped to take the same approach to prove the extensive, rings-of-intellectors version of the problem-but they hit a limp.
Gumming on the works
The useful correspondence between Turing machines and theophantine comparisons falls apart when the comparisons are allowed to have non-integer solutions. For example, consider the equation again Y = x2. If you work in a ring of integers that include √2, you will have some new solutions, such as x = √2, Y = 2. The comparison no longer corresponds to a Turing machine that calculates perfect squares – and more commonly the diophantine comparisons can no longer encode the stop problem.
But in 1988, a graduate student at the University of New York named Sasha Shlapentokh began to play with ideas to get this problem. By 2000, she and others formulated a plan. Say that you would add a bunch of extra terms to a comparison Y = x2 which is magically forced x To be an integer again, even in a different number system. Then you can save the correspondence with a Turing machine. Can the same be done for all diophantine comparisons? If this is the case, it would mean that Hilbert’s problem could encode the stop problem in the new number system.
Illustration: Myriam real for Quanta Magazine
Over the years, Shlapentokh and other mathematicians have determined which terms they had to add to the diophantine comparisons for different types of rings, which enabled them to demonstrate that Hilbert’s problem in those institutions was still indisputable. They then dropped all remaining rings of integers in one case: rings involving the imaginary number I. Mathematicians realized that in this case the terms they had to add can be determined using a special comparison called an elliptical curve.
But the elliptical curve will have to satisfy two qualities. First, it will have to have infinitely many solutions. Second, if you switched to another ring of integers – if you removed the imaginary number from your number system – then all the solutions to the elliptical curve will have to maintain the same underlying structure.
It seems 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 elliptical curves who have worked closely together in the postgraduate school – have just set the right instrument to try.
Sleepless nights
Since his time as an undergraduate student, Kaymans has thought of Hilbert’s 10th problem. During the postgraduate school, and during his cooperation with Pagano, it arose. “I thought about it for a few days every year and got terribly stuck,” Kaymans said. “I would try three things and they would all blow up in my face.”
In 2022, while at a conference in Banff, Canada, he and Pagano finally talked about the problem. They hoped they could build the special elliptical curve needed to solve the problem. After completing a few other projects, they got to work.