Fermat's right triangle theorem [SEP] named after
Fermat's right triangle theorem
Fermat's right triangle theorem is a non-existence proof in number theory, the only complete proof left by Pierre de Fermat. It has several equivalent formulations:
- If three square numbers form an arithmetic progression, then the gap between consecutive numbers in the progression (called a congruum) cannot itself be square.
- There do not exist two Pythagorean triangles in which the two legs of one triangle are the leg and hypotenuse of the other triangle.
- A right triangle
descent to show that the area of a right triangle with integer sides can never equal the square of an integer. This result is known as Fermat's right triangle theorem. As shown below, his proof is equivalent to demonstrating that the equation
has no primitive solutions in integers (no pairwise coprime solutions). In turn, this is sufficient to prove Fermat's Last Theorem for the case "n" = 4, since the equation "a" + "b" = "c" can be written as "