Do there exist any Composite Numbers such that ? No such numbers are known. In 1932, Lehmer showed that such an must be Odd and Squarefree, and that the number of distinct Prime factors . This was subsequently extended to . The best current results are and (Cohen and Hagis 1980), if , then (Wall 1980), and if then and (Lieuwens 1970).
References
Cohen, G. L. and Hagis, P. Jr. ``On the Number of Prime Factors of is .''
Nieuw Arch. Wisk. 28, 177-185, 1980.
Lieuwens, E. ``Do There Exist Composite Numbers for which Holds?'' Nieuw. Arch. Wisk.
18, 165-169, 1970.
Ribenboim, P. The Book of Prime Number Records, 2nd ed. New York: Springer-Verlag, pp. 27-28, 1989.
Wall, D. W. ``Conditions for to Properly Divide .'' In A Collection of Manuscripts Related to the Fibonacci Sequence
(Ed. V. E. Hoggatt and M. V. E. Bicknell-Johnson). San Jose, CA: Fibonacci Assoc., pp. 205-208, 1980.