After nearly two centuries of work, mathematicians have finally gotten all the answers for the famous version of the Diophantine equation known as the "sum of three cubes problem." According to its conditions, for an equation of the form x3 + y3 + z3 = k, where k varies from 1 to 100, you need to find x, y and z. All numbers are integers, without fractions, and since then mathematicians have been able to find solutions for all variants of k, except 33 and 42. Andrew Booker from the University of Bristol managed the first - it took him a week of working with a supercomputer. But the indestructible 42 did not give in in any way.
Booker turned to MIT mathematics professor Andrew Sutherland for help. With his participation, they gained access to the Charity Engine, a distributed computing project that uses the resources of 500, 000 home computers for environmental calculations. And even with such computing power, the search for a solution took a total of millions of hours.
Here is the solution x = -80538738812075974, y = 80435758145817515 and z = 12602123297335631.
Booker himself describes his feelings after solving the problem as "a huge relief." Not that this decision was so important to science - rather, the search process itself did. Development of algorithms, organization of calculations, analysis of results - all attempts to solve "unsolvable" problems invariably benefit science, developing the mathematical apparatus and methods of working with it.