Mathematics has fascinated the human race nearly as long as our existence. Some of the coincidences between numbers and their applications are incredibly neat, and some of the most deceptively simple ones continue to stump us and even our modern computers. Here are three famous math problems that people struggled with for a long time but were finally resolved, followed by two simple concepts that continue to boggle mankind's best minds.
1. Fermat's Last Theorem
In 1637, Pierre de Fermat scribbled a note in the margin of his copy of the book Arithmetica. He wrote (conjectured, in math terms) that for an integer n greater that two, the equation an + bn = cn had no whole number solutions. He wrote a proof for the special case n = 4, and claimed to have a simple, "marvellous" proof that would make this statement true for all integers. However, Fermat was fairly secretive about his mathematic endeavors, and no one discovered his conjecture until his death in 1665. No trace was found of the proof Fermat claimed to have for all numbers, and so the race to prove his conjecture was on. For the next 330 years, many great mathematicians, such as Euler, Legendre, and Hilbert, stood and fell at the foot of what came to be known as Fermat's Last Theorem. Some mathematicians were able to prove the theorem for more special cases, such as n = 3, 5, 10, and 14. Proving special cases gave a false sense of satisfaction; the theorem had to be proved for all numbers. Mathematicians began to doubt that there were sufficient techniques in existence to prove theorem. Eventually, in 1984, a mathematician named Gerhard Frey noted the similarity between the theorem and a geometrical identity, called an elliptical curve. Taking into account this new relationship, another mathematician, Andrew Wiles, set to work on the proof in secrecy in 1986. Nine years later, in 1995, with help from a former student Richard Taylor, Wiles successfully published a paper proving Fermat's Last Theorem, using a recent concept called the Taniyama-Shimura conjecture. 358 years later, Fermat's Last Theorem had finally been laid to rest.
2. The Enigma Machine
The Enigma machine was developed at the end of World War I by a German engineer, named Arthur Scherbius, and was most famously used to encode messages within the German military before and during World War II.
The Enigma relied on rotors to rotate each time a keyboard key was pressed, so that every time a letter was used, a different letter was substituted for it; for example, the first time B was pressed a P was substituted, the next time a G, and so on. Importantly, a letter would never appear as itself-- you would never find an unsubstituted letter. The use of the rotors created mathematically driven, extremely precise ciphers for messages, making them almost impossible to decode. The Enigma was originally developed with three substitution rotors, and a fourth was added for military use in 1942. The Allied Forces intercepted some messages, but the encoding was so complicated there seemed to be no hope of decoding.
Enter mathematician Alan Turing, who is now considered the father of modern computer science. Turing figured out that the Enigma sent its messages in a specific format: the message first listed settings for the rotors. Once the rotors were set, the message could be decoded on the receiving end. Turing developed a machine called the Bombe, which tried several different combinations of rotor settings, and could statistically eliminate a lot of legwork in decoding an Enigma message. Unlike the Enigma machines, which were roughly the size of a typewriter, the Bombe was about five feet high, six feet long, and two feet deep. It is often estimated that the development of the Bombe cut the war short by as much as two years.
3. The Four Color Theorem
The four color theorem was first proposed in 1852. A man named Francis Guthrie was coloring a map of the counties of England when he noticed that it seemed he would not need more than four ink colors in order to have no same-colored counties touching each other on the map. The conjecture was first credited in publication to a professor at University College, who taught Guthrie's brother. While the theorem worked for the map in question, it was deceptively difficult to prove. One mathematician, Alfred Kempe, wrote a proof for the conjecture in 1879 that was regarded as correct for 11 years, only to be disproven by another mathematician in 1890.
By the 1960s a German mathematician, Heinrich Heesch, was using computers to solve various math problems. Two other mathematicians, Kenneth Appel and Wolfgang Haken at the University of Illinois, decided to apply Heesch's methods to the problem. The four-color theorem became the first theorem to be proved with extensive computer involvement in 1976 by Appel and Haken.
...and 2 That Still Plague Us
1. Mersenne and Twin Primes
Prime numbers are a ticklish business to many mathematicians. An entire mathematic career these days can be spent playing with primes, numbers divisible only by themselves and 1, trying to divine their secrets. Prime numbers are classified based on the formula used to obtain them. One popular example is Mersenne primes, which are obtained by the formula 2n - 1 where n is a prime number; however, the formula does not always necessarily produce a prime, and there are only 47 known Mersenne primes, the most recently discovered one having 12,837,064 digits. It is well known and easily proved that there are infinitely many primes out there; however, what mathematicians struggle with is the infinity, or lack thereof, of certain types of primes, like the Mersenne prime. In 1849, a mathematician named de Polignac conjectures that there might be infinitely many primes where p is a prime, and p + 2 is also a prime. Prime numbers of this form are known as twin primes. Because of the generality if this statement, it should be provable; however, mathematicians continue to chase its certainty. Some derivative conjectures, such as the Hardy-Littlewood conjecture, have offered a bit of progress in the pursuit of a solution, but no definitive answers have arisen so far.
2. Odd Perfect Numbers
Perfect numbers, discovered by the Euclid of Greece and his brotherhood of mathematicians, have a certain satisfying unity. A perfect number is defined as a positive integer that is the sum of its positive divisors; that is to say, if you add up all the numbers that divide a number, you get that number back. One example would be the number28— it is divisible by 1, 2, 4, 7, and 14, and 1 + 2 + 4 + 7 + 14 = 28. In the 18th century, Euler proved that the formula 2(n-1)(2n-1) gives all even perfect numbers. The question remains, though, whether there exist any odd perfect numbers. A couple of conclusions have been drawn about odd perfect numbers, if they do exist; for example, an odd perfect number would not be divisible by 105, its number of divisors must be odd, it would have to be of the form 12m + 1 or 36m + 9, and so on. After over two thousand years, mathematicians still struggle to pin down the odd perfect number, but seem to still be quite far from doing so.