Fermat's Last Theorem
Like most mathematicians, Pierre de Fermat studied many problems for his own amusement. Indeed, many of his most important contributions were initially scribbled in the margins of books or in notes to friends.
One day in 1637, he made a curious note in his copy of Diophantus's Arithmetic: "The equation x^n + y^n = z^n, where x, y, and z are positive integers, has no solution if n is greater than 2... I have discovered a most remarkable proof, but this margin is too narrow to contain it."
Between bouts of insanity and frequent hospitalizations, Georg Cantor laid the foundations of set theory and the study of infinity. In 1878, the young mathematician discovered that there are in fact as many points on the minutest line segment as exist in all of space. Cantor, too, was incredulous. "I see it," he wrote to a colleague, "but I don't believe it!"
Such was his admiration of Karl Friedrich Gauss that the German mathematician Peter Dirichlet is said to have slept with Gauss's Disquisitiones Arithmeticae under his pillow.
Srinivasa Ramanujan: 1729
Srinivasa Ramanujan was a mathematical prodigy. "I remember once going to see him when he was lying ill at Putney," the mathematician G. H. Hardy once remarked. "I had ridden in taxicab number 1729, and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen.
"'No,' he replied, 'it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.'"
Paul Erdos : 2½ Billion year old
Paul Erdos :"In 1970, I preached in Los Angeles on `my first two and a half billion years in mathematics.' When I was a child, the Earth was said to be two billion years old. Now scientists say it's four and a half billion. So that makes me two and a half billion. The students at the lecture drew a timeline that showed me riding a dinosaur. I was asked, `How were the dinosaurs?' Later, the right answer occurred to me: `You know, I don't remember, because an old man only remembers the very early years, and the dinosaurs were born yesterday, only a hundred million years ago.'"