The great mathematician, Herman Minkowski, once told his students that the 4-Color Conjecture had not been settled because only third-rate mathematicians had concerned themselves with it. "I believe I can prove it," he declared. After a long period, he admitted, "Heaven is angered by my arrogance; my proof is also defective."
(The Four Color Problem: Assaults and Conquest. Saaty & Kainen, 1986, p.8)
Ever notice that most maps aren't really that colorful? Your average, run-of-the-mill map usually only has four colors. If you just start coming up with fake maps, you can always find a way to color the whole thing with four colors, and two adjacent countries will never have the same color.

Way the heck back in 1856, a few mathematically-inclined brothers noticed this. They couldn't prove it, so they brought it to a teacher of theirs, the storied professor Augustus De Morgan. Morgan could not prove this conjecture. Word of this odd problem slowly spread, until it was presented to the London Mathematical Society in 1878, and the problem was then published. One by one, mathematicians attacked the problem, only to fall short. Not that their effort was wasted; Kempe chains were first formulated to try and solve this problem, and a man named John Tait solved a similar problem based on three colors.

In 1976, two men named Kenneth Appel and Wolfgang Haken came up with a solution. This solution involved breaking up the problem into about 1500 smaller problems, and then programming a computer to directly prove all of these cases. In other words, Appel and Haken made a computer check each and every single case they could think of. It took 1200 hours of processing to finish, and they triumphantly published their work - solving a problem that has lasted over 100 years.

This, however, brought up another problem. Mathematicians just won't trust you when you say that you've solved a problem. No, they gotta check the work. By hand. But how do you check the work of a computer? If a computer had to do the work in the first place, how can a human go through and make sure that everything the computer did was right? After a few tries, most mathematicians just accepted that they were probably right. This is the only proof in existence that people believe is solved without anyone actually checking all the work by hand.