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.

The Perfect Bayesian equilibrium is not a refinement of the Nash equilibrium. The Perfect Bayesian equilibrium requires players to have beliefs that are consistent with the equilibrium strategies of other players. The Nash equilibrium does not explicitly specify the beliefs of the players. In any given Nash equilibrium, the default beliefs held by players are that the other players are playing the equilibrium strategies (or a particular subset of the equilibrium strategies in the case of multiple equilibriums). The default beliefs and equilibrium strategies in Nash equilibrium could be more narrowly defined than those allowed under Perfect Bayesian equilibrium. Consequently, the set of equilibriums under Perfect Bayesian equilibrium is not a subset of the set of equilibriums under Nash equilibrium.

This paper analyzes a generalized Stackelberg model of market competition with noisy observation. The current prevailing Stackelberg model and Cournot market are two extreme cases of this generalized Stackelberg model. In this generalized model, as the follower relies less on the inaccurate noisy observation and more on prior information and conjecture for his statistical inference and decision, the value of moving first and strategic commitment of the Stackelberg leader decreases and his profit lowers. On the other hand, the profit of the follower increases and the equilibrium moves towards the Cournot model and away from the Stackelberg model.