Lehmer's Problem

Do there exist any Composite Numbers such that ? No such numbers areknown. In 1932, Lehmer showed that such an must be Odd and Squarefree, and that the number of distinctPrime 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.

• Monotonic Function
• Monotonic Sequence
• Monotonic Voting
• Monster Group
• Monte Carlo Integration
• Monte Carlo Method
• Montel's Theorem
• Monty Hall Dilemma
• Monty Hall Problem
• Moore Graph
• Moore-Penrose Generalized Matrix Inverse
• Mordell Conjecture
• Mordell Integral
• Mordell-Weil Theorem
• Morera's Theorem
• Morgado Identity
• Morgan-Voyce Polynomial
• Morley Centers
• Morley's Formula
• Morley's Theorem
• Morley's Triangle
• Morphism
• Morrie's Law
• Morse Inequalities
• Morse Theory