In the 17th century, while studying the work "Arithmetic" by the ancient Greek mathematician Diophantus, French mathematician Fermat noticed a problem about Pythagorean triples: "Given a square number, how can it be written as the sum of two other square numbers?" He then wrote in the margin of the book: "... Generally speaking, any integer power greater than 2 cannot be written as the sum of two integers of the same power. I found a truly wonderful proof, but the margin is too small to write it down." Using modern mathematical symbols, Fermat was saying: x^n + y^n = z^n (n > 2) (1) has no solution for positive integers. This is the famous "Fermat's Last Theorem", also known as "Fermat's Great Theorem".
Now it is believed that Fermat did not prove his "Great Theorem", because it was really too difficult. In the 18th century, mathematician Euler only gave a proof for n=3. In the 19th century, mathematician Gauss also studied it but gave up because he couldn't get a result. When the great mathematician Hilbert of the 20th century was persuaded to solve Fermat's Last Theorem, he said he didn't want to "kill the goose that lays the golden eggs". The "golden eggs" Hilbert referred to were the wonderful mathematical concepts and theories developed to prove Fermat's Last Theorem. Firstly, the concept of "integers" was generalized. Mathematicians found that if n-th unit root ω (that is, ω^n = 1) was regarded as an "integer", then (1) could be decomposed into z^n = x^n + y^n = (x + y)(x + yω)(x + yω^2)... (x + yω^(n-1)), and further analysis could be carried out. We know that any integer greater than 1 can be uniquely decomposed into the product of some prime numbers, such as 6 = 2 × 3, 20 = 2^2 × 5, etc. If the extended "integers" still maintain this unique decomposition, then Fermat's Last Theorem would have been solved long ago. Unfortunately, this is not the case. For example, for 6 in the extended "integers", there are two decompositions: 6 = 2 × 3 and Thus, in the 19th century, the German mathematician Kummer invented a new concept of "ideal numbers". On this basis, he proved Fermat's Last Theorem for n ≤ 100 (except for a few cases). Kummer thus opened up an important branch of modern mathematics - algebraic number theory.
In 1994, after eight years of closed-door research, British mathematician Wiles finally completely proved Fermat's Last Theorem, solving a problem that had puzzled mathematicians for more than 350 years. The key to Wiles's proof was that he successfully applied "Galois group representation" to establish a correspondence between "elliptic curves" and "modular forms", thus revealing the profound connections between different fields of modern mathematics. This was the "last golden egg" born from Fermat's Last Theorem.