Fermat's right triangle theorem

Fermat's right triangle theorem is a non-existence proof in number theory, published in 1670 among the works of Pierre de Fermat, soon after his death. It is the only complete proof given by Fermat.^[1] It has many equivalent formulations, one of which was stated (but not proved) in 1225 by Fibonacci. In its geometric forms, it states:

• A right triangle in the Euclidean plane for which all three side lengths are rational numbers cannot have an area that is the square of a rational number. The area of a rational-sided right triangle is called a congruent number, so no congruent number can be square.
• A right triangle and a square with equal areas cannot have all sides commensurate with each other.
• There do not exist two integer-sided right triangles in which the two legs of one triangle are the leg and hypotenuse of the other triangle.

More abstractly, as a result about Diophantine equations (integer or rational-number solutions to polynomial equations), it is equivalent to the statements that:

• If three square numbers form an arithmetic progression, then the gap between consecutive numbers in the progression (called a congruum) cannot itself be square.
• The only rational points on the elliptic curve ${\displaystyle y^{2}=x(x-1)(x+1)}$ are the three trivial points with ${\displaystyle x\in \{-1,0,1\}}$ and ${\displaystyle y=0}$.
• The quartic equation ${\displaystyle x^{4}-y^{4}=z^{2}}$ has no nonzero integer solution.

An immediate consequence of the last of these formulations is that Fermat's Last Theorem is true in the special case that its exponent is 4.