The research on prime numbers is an interesting topic in Analytic Number Theory. The Fibonacci sequence is one of the most popular sequences in mathematics. In this paper, we will discuss the prime numbers of the Fibonacci sequence, which we call Fibonacci primes. The goal of this paper is to prove that starting with F5 = 5, all the Fibonacci primes p are satisfied with p≡1 (mod4), on the basis of the Pythagorean theorem.