Birch and Swinnerton-Dyer conjecture

"The conjecture discovered jointly by Peter Swinnerton-Dyer and Bryan Birch in the early 1960s both surprised the mathematical world, and also forcefully reminded mathematicians that computations remained as important as ever in uncovering new mysteries in the ancient discipline of number theory. Although there has been some progress on their conjecture, it remains today largely unproven, and is unquestionably one of the central open problems of number theory. It also has a different flavour from most other number-theoretic conjectures in that it involves exact formulae, rather than inequalities or asymptotic questions."

"Just as Weil's conjectures were about counting solutions to equations in a situation where the number of solutions is known to be finite, the BSD conjecture concerns the simplest class of polynomial equations—elliptic curves—for which there is no simple way to decide whether the number of solutions is finite or infinite."

"Thanks to their conjecture, Birch and Swinnerton-Dyer are two names that (to mathematicians) are as inextricably linked as the names of Laurel and Hardy, although many have been tricked into believing that there are in fact three mathematicians behind the conjecture - Birch, Swinnerton and Dyer. Birch, with his rather bumbling manner, plays Stan Laurel to Swinnerton-Dyer's rather dour Oliver Hardy."

"This remarkable conjecture relates the behaviour of a function L at a point where it is not at present known to be defined to the order of a group Ш which is not known to be finite."