Are there math problems that have proven unsolvable

Success after 300 years

The auditorium of the University of Göttingen was occupied up to the gallery, although neither a star among the rock singers nor any other famous artist or even politician was on the program - but a mathematician. But a very special one: he had solved a centuries-old riddle. So there was probably no one among the audience who had not heard of the “Great Fermat's Theorem” at some point in their life - a mathematical theorem that is so easy to formulate that you can explain it to children yourself, and therefore always about it guessed that it was also very easy to prove.

The French lawyer and hobby mathematician Pierre de Fermat (1601 to 1665) wrote it in the margin of the page of a book and remarked that he had found a wonderful proof of it, which he could not add due to lack of space.

Over the centuries mathematicians and laypeople have tried to prove the theorem - but that only succeeded more than 300 years later. The mathematician who proved the “Great Fermat's Theorem” in 1995 could not manage with the margin of a book page. It needed 108 printed pages. On June 27, 1997, he received the “Wolfskehl Prize” for this in the auditorium of the University of Göttingen.

In 1905, Paul Friedrich Wolfskehl (1856 to 1906), son of a Jewish banker in Darmstadt, offered 100,000 gold marks in his will as prize money for those who provided proof of Fermat's Great Theorem. The foundation came into force three years later.

Wolfskehl had studied medicine, but fell ill with multiple sclerosis shortly after completing his doctorate. Thereupon he began to study mathematics because he told himself that he could still work successfully in this science later in a wheelchair.

The Göttingen Academy of Sciences was responsible for administering the award. In the first year there were 621 submissions - but each of the evidence was flawed. To date, more than 5,000 unsuccessful demonstrations have arrived.

Much of the amount exposed for evidence had meanwhile fallen victim to contemporary history. The purchasing power of the original sum would be around three million marks today. But during the First World War all foundation assets had to be invested in war bonds. The academy managed to save a small part of the money. In the following 20 years, this remainder grew again thanks to the interest, until the state once again reached into the coffers of the foundations during the Second World War - and again it was possible to save a small part that bore interest in the peaceful half-century that followed. At last there were about 75,000 marks in the till.

This sum has now been paid out to 45-year-old mathematician Andrew John Wiles. Born in England, he is a professor at Princeton University in New Jersey. He had first heard of Fermat's unproven Theorem at the age of ten. The problem captivated him and did not let go of him when he later made mathematics his profession.

Building on the work of the mathematicians Gerd Faltings, Gerhard Frey and Kenneth Ribet, Wiles succeeded in nine years of work in finding an - albeit very long - argument. But it soon turned out that the evidence still had several loopholes. When these could also be closed, Fermat's Great Theorem was proven. The Wolfskehl Prize was awarded 89 years to the day after the foundation came into force.

Like any other science, mathematics thrives on unsolved problems. Otherwise she would be dead - because there would be no more questions whose answers weren't already in a mathematical lexicon. Even today, problems that have yet to be solved are formulated every day in original mathematical papers. It's always been like that. “Is the number of prime numbers finite or is there a largest one?” That was an unsolved problem in ancient times, for example, until the Greek mathematician Euclid was able to prove around 300 BC that there are an infinite number of prime numbers.

The outsider hardly learns anything about most of the unsolved problems in mathematics, partly because they are solved after a short time, partly because the question can hardly be made understandable to a non-mathematician.

This is different with the so-called four-color set. It too can be explained to every child, but it took a great deal of effort to prove it. In 1852 the English student Francis Guthrie tried to draw a map of the English island - showing the numerous counties into which the country is divided. For the sake of clarity, he wanted to display the individual counties in color, with fields that have a common border being given different colors. How many different colored pencils do you need to color the well over 50 counties so that each border separates two different colored fields? Is five enough or is twenty necessary?

Guthrie realized very quickly that he could get by with only four colors and found that this was not a special characteristic of the counties of England, but that the fields of any map can be colored with only four different colors so that two fields are never the same Color have a piece of common border.

In 1897, Arthur Kempe, a lawyer and respected amateur mathematician, published a proof of the theorem. That seemed to be the end of the problem.

But shortly afterwards it turned out that Kempe's "proof" contained an error. Thus the supposedly proven four-color theorem was again downgraded to the four-color conjecture. It stayed that way - until 1970, when the mathematician Wolfgang Haken, who worked at the University of Illinois, took up the problem again. In the meantime, the Mannheim mathematician Heinrich Heesch had not solved the problem, but had broken it down into numerous sub-problems. If someone were to solve all of these minor problems, the four-color theorem would be proven. Wolfgang Haken and the mathematician Kenneth Appel succeeded in solving the sub-problems with the help of a computer and thus proving that four colors are sufficient for every conceivable case of a map. The four-color conjecture was proven.

Not all mathematicians are happy with it. To prove a mathematical proposition means to make the proof in such a way that everyone can understand it and convince themselves of the correctness of the assertion. When Appel and Haken proved it at the end of the 1970s, computers needed around 300 hours of computing time to run through all the cases. The proof of the four-color theorem that has been carried out so far differs qualitatively from the mathematical proofs that anyone can check with paper and pencil.

Are all these useless gimmicks with numbers, no more than solutions to crossword puzzles? Mathematics is a whole. Anyone who wants to investigate their connections must try to solve the problems in all their areas, regardless of whether they are of practical use or not. When physicists discovered quantum mechanics, which dominates our lives today, in the first half of our century, they were able to rely on mathematical methods developed a century earlier to understand vibrating strings. Similar to the encryption methods used in money transactions and electronic information exchange: They are based on "number games" developed centuries earlier.

Solved problems The four-color problem: If you want to mark the countries on a map with colors and if countries with a piece of common border are to be given different colors, then four different colors are sufficient, regardless of how many countries are on the map and how they are spread over the map. This empirically found fact was proven in 1989 by the American mathematicians Kenneth Appel and Wolfgang Haken.

Fermat's theorem: The squares of the numbers 3, 4 and 5 have the property that 32 + 42 = 52 (you can easily do the math: 9 + 16 = 25). There are an infinite number of integers A, B and C, for which the following applies accordingly: A2 + B2 = C2, for example A = 32, B = 126 and C = 130. In 1670, the French lawyer and mathematician Pierre de Fermat wrote to the Edge of the edition of a book by the mathematician Diophant, who lived around AD 250, that he had proven the following assumption: There are no integers A, B and C for which An + Bn = Cn if n is greater than 2. Since then mathematicians have tried to find the proof of Fermat's theorem. Andrew Wiles finally delivered it in 1995.

Unsolvable problems Dividing the angle into three parts: In school we learned how to cut an angle in half with a compass and ruler. The construction is very easy. But can you also find a construction with compass and ruler that divides any angle into three equal angles?

Unsolved problems Goldbach's conjecture: In 1742, the conference secretary of the St. Petersburg Academy, Christian von Goldbach, wrote to the mathematician Leonhard Euler in a letter that every even number greater than 2 can be represented as the sum of two prime numbers. Examples: 20 = 13 + 7, 24 = 5 + 19, 872 = 199 + 673. Whichever even number you choose - to date none has been found that is not the sum of two prime numbers. Even so, this assumption has not been proven. Are there infinitely many prime twins? Among the infinite number of prime numbers, there are always some that differ by only 2 - so-called prime twins: 2/3, 3/5, 5/7, 11/13, 17/19, 29/31 and so on. One might think that this is a phenomenon that only occurs with low numbers, because there the prime numbers follow one another more closely than with the large numbers. But in the range between 800 and 900 there are the twin pairs 821/823, 827/829, 857/859 and 881/883. Even with even larger numbers, one can find prime twins: 9890641/9890643. Even in the area of ​​more than 11,000-digit numbers, there are prime twins. Up until now, no one has been able to mathematically decide whether the sequence of twin pairs ends somewhere.

Rudolf Kippenhahn

November 1, 1997

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