On the Infinitude of Lucas Pseudoprimes

It is well-known that properties (1) and (2) are satisfied if n is prime. If (1) is satisfied for some composite n, then n is called a Fibonacci pseudoprime (or FPP). If (2) is satisfied for some composite n, then n is called a Lucas pseudoprime (or LPP). Let U and V denote the sets of FPP's and LPP's, respectively. It must be remarked that, the above terminology is different from that used by many other authors; frequently, the term "Fibonacci pseudoprime" is used to describe numbers that satisfy (2), and/or "Lucas pseudoprime" sometimes is used to describe numbers that satisfy (1). There are, no doubt, some very good reasons for describing such numbers by one term versus another. In most papers that this author has seen, the subject matter is only one of the types of numbers here described, which tends to minimize confusion. When both types of numbers are being discussed, as is the case in this paper, it seems preferable to adopt the terminology defined above. Readers of this journal may tend to be more sympathetic to this usage, for obvious reasons. Apologies are made here and now to those readers who may take exception to the nomenclature adopted here. In a 1964 paper [2], E. Lehmer showed that f/is an infinite set, specifically by proving that n=F2p satisfies (1) for any prime p>5. In a 1970 paper [3], E. A..Parberry proved some interesting results related to those of Lehmer, indirectly commenting on the infinitude of U, by a different approach. It is informative to paraphrase that portion of Parberry's results that touches on the subject of this paper; we state this as a theorem.