Introduction An integer n ? 1 is said to be a prime number (or simply prime) if the only divisors of n are \Sigma1 and \Sigman. There are infinitely many prime numbers, the first four being 2, 3, 5, and 7. If n ? 1 and n is not prime, then n is said to be composite. The integer 1 is neither prime nor composite. The Fundamental Theorem of Arithmetic states that every positive integer can be expressed as a finite (perhaps empty) product of prime numbers, and that this factorization is unique except for the ordering of the factors. Table 1.1 has some sample factorizations. 1990 = 2 \Delta 5 \Delta 199 1995 = 3 \Delta 5 \Delta 7 \Delta 19 2000 = 2 4 \Delta 5 3 2005 = 5 \Delta 401

Introduction We follow the notations of both [2] and [3]. Let F n (resp. L n ) be the n-th Fibonacci number (resp. Lucas number). The aim of this informal note is to describe a short proof of primality for both F 4723 and F 5387 . See the paper [5] for more on this topic. 2 F 4723 From [3, (4.1)], one has F 4k+3 \Gamma 1 = F k+1 L k+1 L 2k+1 : (1) Here k = 1180, k + 1 = 1181 and 2k + 1 = 2361 = 3 \Theta 787. From [3] and with the help of factors found by Montgomery and Silverman [7, 8, 9], we get F 1181 = 5453857 \Theta C 240 ; L 1181 = 59051 \Theta<F27.43