Riemann Hypothesis

Depending on who you ask, for example, present-day mathematicians have nearly as much chance of solving the Riemann hypothesis - the most famous unsolved problem in math - as da Vinci had of building a machine that could actually fly.

"As of yet there's not been a proposed strategy for handling the Riemann hypothesis that's even semi-plausible," said Jacob Tsimerman of the University of Toronto.

Abstractions: Exploring the world of scientific ideas

But while it may have been obvious in da Vinci's time that a functional version of the aerial screw would have to wait, often in math it's not clear what's possible and what's not.

Sometimes a problem can seem hopeless, only for a mathematician to realize that the ingredients of a solution have been hiding in plain sight. This is what happened with Vesselin Dimitrov's recent proof of a problem called the Schinzel-Zassenhaus conjecture, which Quanta covered in our article "Mathematician Measures the Repulsive Force Within Polynomials."

Mathematicians had long failed to prove the conjecture, and many believed that it would take a new mathematical invention to get there. But Dimitrov cracked the problem by finding a novel way of combining techniques that have been around for more than 40 years.

"Mathematicians are sometimes too quick to dismiss the possibility that we can solve something" Tsimerman said. "Math is really hard, and people sometimes overlook things."

So how do mathematicians know if a problem is currently impossible or just really hard? Obviously, there's no clear way to tell, so they have to rely on clues. And the biggest hint that a problem is out of reach is simply that lots of people have failed to solve it.