Generalization of a Theorem of Drobot

It is well known that the Fibonacci number Fn can be a prime only If n 4 or n p, where p is an odd prime. Throughout this paper, p will denote a prime. In a very interesting paper, Drobot [2] proved that Fp is composite for certain primes p. In particular, he proved that if p > 7, p = 2 or 4 (mod 5), and 2p -1 is also a prime, then 2p -11 Fp and Fp > 2p . For example, 371 Fl9 = 4181-37-113. A similar result was proved by Euler and, independently, by Lagrange about the Mersenne numbers. It is easy to see that the Mersenne number Mn = 2" 1 can be a prime only if n is a prime. Euler and Lagrange proved that, if p = 3 (mod 4) and 2p +1 is also a prime, then 2p +11 Mp = 2p + l. A proof of this result is given in [5, pp. 90-91]. The primality of Mersenne numbers is of interest because of the following relationship to even perfect numbers. A positive integer is perfect if it is equal to the sum of its proper divisors. Euclid and Euler proved that the even integer n is perfect if and only if n is of the form 2"(2~ 1), where 2~ 1 is a Mersenne prime. Euclid proved that this condition is sufficient for n to be a perfect number and Euler proved the necessity of this condition. At the present time only thirty-eight Mersenne primes are known, with the largest known Mersenne prime being 2 9 7 2 5 9 -1, which has over two million digits. A list of all known Mersenne primes is given in the web site