Fermat's Last Theorem


July 5, 2022

Fermat's Last Theorem (English: Fermat's Last Theorem) is one of the most famous theorems in mathematics, coined by Pierre de Fermat in the 17th century. This theorem says: In 1637, Fermat wrote the theorem in the margin of one of the pages of his book. He claims to have found evidence of the theory, only that he can't write it down because the margins of his book don't fit anymore. However, for the next 357 years, the world's mathematicians were unable to prove it, and this theorem became one of the greatest puzzles in mathematics. Finally, in 1994, British mathematician Andrew Wiles succeeded in proving this theorem true.


Trigger by Fermat

Around 1637, Fermat wrote the theorem in the margin of one of the pages of his Arithmetica (by Diophantus), which reads: However, it is unknown whether Fermat actually found evidence for all the ranks n {\displaystyle n} . Fermat's only surviving evidence of that is evidence for n 4 {\displaystyle n4} .

Proof for a certain rank

For numbers to the power of 4

The case for exponents 4 {\displaystyle 4} proved by Fermat himself. He uses the infinite descent technique to prove that the equation x 4 ️ y 4 z 2 {\displaystyle x^{4}-y^{4}z^{2}} does not have a primitive solution (the solution with x , y , z {\displaystyle x,y,z} each pair is relatively prime). This results in Fermat's Last Theorem valid for n 4 {\displaystyle n4} , because of the equation a 4 + b 4 c 4 {\displaystyle a^{4}+b^{4}c^{4}} can be written c 4 ️ b 4 ( a 2 ) 2 {\displaystyle c^{4}-b^{4}(a^{2})^{2}} .

Other exponents

After Fermat proved the case n 4 {\displaystyle n4} , remaining to prove the case that n {\displaystyle n} odd prime. In other words, it remains to prove that the equation a n + b n c n {\displaystyle a^{n}+b^{n}c^{n}} has no unanimous solution ( a , b , c ) {\displaystyle(a,b,c)} if n {\displaystyle n} odd prime number. This is because if there is a solution ( a , b , c ) {\displaystyle(a,b,c)} for rank n {\displaystyle n} , then there is a solution to the power of all positive factors n