It is known that Leonardo Pisano (Fibonacci) was challenged around 1220 by Johannes of Palermo to find a rational right triangle of area 5. He found the right triangle with sides of lenght 3
2 ,20 3 and 41 6 . Notice that the definition of a congruent number does not require the sides of the triangle to be integer, only rational. While n = 6 is the smallest possible area of a right triangle with integer sides of lenght 3,4,5 , n = 5 is the area of right triangle with rational sides of lenght 32 ,20 3 and 41 6 . So n = 5 is the smallest congruent number. In 1225,
Fibonacci wrote a general treatment about the congruent number problem, in which he stated out without proof that if n is a perfect square then n cannot be a congruent number. The proof of such a claim had to wait until Pierre de Fermat. He showed that n = 1 and so every square number is not a congruent number by using his method of infinite descent[6]. One can look at [4] and
[7] for Fermat’s descent method. In the present study we will show that if n is a congruent number then n can not be a perfect square by using the same method. Moreover, we proved Fermat’s last theorem for n = 4, which states that the equation x4 + y4 = z4 has no solutions in positive integers