The G . C . D . in Lucas Sequences and Lehmer Number Sequences

Using basic identities, Lucas proved Theorem 0 in the first of his two 1878 articles in which he developed the general theory of second-order linear recurrences [5]; Lucas had previously proven parts (i) and (iii) in his 1875 article [4], Nearly four decades later, Carmichael [1] used the theory of cyclotomic polynomials to obtain both new results and results confirming and generalizing many of Lucas theorems; Theorem 0 was among the results obtained using cyclotomic polynomials. Curiously, the value of gcd(7TO, Vn) when m and n are not divisible by the same power of 2, and of gcd([/w, Vn) for m * n, do not appear in the literature, and have, apparently, never been established. It is interesting that the values of all three of these gcd's can be rather easily found, for all pairs of positive integers m and n, by the application of an approach similar to that used in establishingthe Euclidean algorithm to a single sequence of equations. We shall prove the following result.