| Popis: |
A proof of the Lucas-Lehmer test can be di- cult to find, for most textbooks that state the result do not prove it. Over the past two decades, there have been some eorts to produce elementary versions of this famous result. However, the two that we acknowledge in this note did so by using either algebraic numbers or group theory. It also appears that in the process of trying to develop an elemen- tary proof of this theorem, the original version provided by D. H. Lehmer's in 1930 has been overlooked. Furthermore, it is quite succinct and elementary, although by its style, it ap- pears to have been written for an audience that has expertise in the theory of the extended Lucas sequences. Therefore, it is the primary objective of this paper to provide a brief intro- duction into the theory that underlies the said sequences, as well as to present an annotated version of Lehmer's original proof of the Lucas-Lehmer test. In conclusion, we show how the test may be utilized in order to identify certain compos- ite terms of the Lucas numbers, Ln = 1;3;4;7;11;18;29;:::, with index equal to 2n. A generalization to the companion Lehmer sequences is oered, as well. |
