It is by logic that we prove, but by intuition that we discover. To know how to criticize is good, to know how to create is better — H.Poincaré, Science and Method (1908, Part II. Ch. 2, p. 129)
A conjecture is an answer proposed for an important mathematical question. It is more than a guess: something supported by intuition, experience, perhaps examples and computations.
Fermat's last theorem was a famous conjecture. The question was whether there exist positive whole number solutions to the equation
xd + yd = zd
for d larger than two. For the case d = 2, the equation reads
x2 + y2 = z2,
and there are many solutions, e.g, x, y, z = 3, 4, 5 which comes from the lengths of the sides of a right triangle. For d = 3, the equation reads
x3 + y3 = z3
there are no positive whole number solutions. In 1994, Andrew Wiles showed that there are no positive whole number solutions for any d greater than two.
There are many unsolved problems in mathematics, some of which rise to the status of conjectures. One of these is the famous Riemann hypothesis. The Riemann hypothesis is concerned with the question of how prime numbers (2, 3, 5, 7, 11, 13, 17, 19, ...) are distributed along the number line. The Riemann Hypothesis was stated in 1859. It remains unsolved to this day. It is one of Hilbert's famous problems listed in 1900 and also one of the seven Millenniumm Prize problems announced in 2000. Both announcements were made in Paris.