The Subtle Art of the Mathematical Conjecture

In mathematics, the role of these highest peaks is played by the great conjectures — sharply formulated statements that are most likely true but for which no conclusive proof has yet been found. It included all-time favorites like the Riemann hypothesis — often considered the greatest of great conjectures, one that has remained the Everest of mathematics for over a century. In a perverse way, the refutation of a great conjecture can be even better news than its success, since the failure makes clear that our imagined map of the mathematical world is seriously wrong.

Source: www.quantamagazine.org