One of the most exciting features of mathematics is its ability to generate challenging problems. Many have to do with important real-life applications of mathematics. Others are the result of pure curiosity, the human thirst for knowledge. The funnest ones are those formulated with few requirements, so everyone understands the issue. Which, incidentally, does not prevent the answer from being difficult, requiring sophisticated mathematical tools.
Fermat’s Last Theorem is a fine example. It was formulated in 1637 by the French amateur lawyer and mathematician Pierre de Fermat (1607 – 1665), in a famous marginal notation in Diophantus’ “Arithmetic”: if n is an integer greater than 2 then there are no positive integers A, B, and C such that An+Bn=Cn. But proof of this fact was only found in 1993/94, by the English mathematician Andrew Wiles, and it uses several advanced mathematical ideas developed in these more than 350 years.
Among the many people who were trying to solve the problem around 1993 and “lost” to Wiles was another amateur mathematician, American banker Andrew Beal. He then posed an even more difficult question: if p, q and r are integers greater than 2 and A, B and C are positive integers such that Ap+Bq=Cr so A, B, and C have some prime factor in common.
To encourage the study of this issue, Beal offered a cash prize: initially it was US$ 5,000, but given the difficulty encountered, it has increased and is currently at US$ 1 million — it won’t be needed, after all he has US$ 10.2 billion , according to Forbes magazine. Even so, we still do not know whether Beal’s conjecture (statement) is true or false.
Proving that it is true cannot be an easy task, because it would entail a new proof of Fermat’s theorem. To prove it false, it would be enough to find specific solutions in which A, B and C do not have common prime factors. But to this day, no one has succeeded, even with the use of supercomputers.
The main breakthrough was achieved by Henri Darmon and Andrew Granville in 1995: they showed that if we fix the values p, q and r, then the set of solutions A, B, and C without common prime factors is finite (Beal’s conjecture states that this set is empty). Their argument just doesn’t hold when p=q=r=3, but in this case Euler and probably Fermat already knew that there are no solutions.
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