tailieunhanh - ON THE APPEARANCE OF PRIMES IN LINEAR RECURSIVE SEQUENCES JOHN H. JAROMA Received 16 August 2004 and

ON THE APPEARANCE OF PRIMES IN LINEAR RECURSIVE SEQUENCES JOHN H. JAROMA Received 16 August 2004 and in revised form 5 December 2004 We present an application of difference equations to number theory by considering the set √ √ of linear second-order recursive relations, Un+2 ( √ R,Q) = RUn+1 − QUn , U0 = 0, U1 = 1, √ √ and Vn+2 ( R,Q) = RVn+1 − QVn , V0 = 2,V1 = R, where R and Q are relatively prime integers and n ∈ {0,1,.}. These equations describe the set of extended Lucas sequences, or rather, the Lehmer sequences. We. | ON THE APPEARANCE OF PRIMES IN LINEAR RECURSIVE SEQUENCES JOHN H. JAROMA Received 16 August 2004 and in revised form 5 December 2004 We present an application of difference equations to number theory by considering the set oflinear second-order recursive relations Un 2 y R Q fRU 1 - QUn U0 0 U1 1 and Vn 2 y R Q ỵfRVn 1 - QVn V0 2 V1 ỵfR where R and Q are relatively prime integers and n e 0 1 . . These equations describe the set of extended Lucas sequences or rather the Lehmer sequences. We add that the rank of apparition of an odd prime p in a specific Lehmer sequence is the index of the first term that contains p as a divisor. In this paper we obtain results that pertain to the rank of apparition of primes of the form 2np 1. Upon doing so we will also establish rank of apparition results under more explicit hypotheses for some notable special cases of the Lehmer sequences. Presently there does not exist a closed formula that will produce the rank of apparition of an arbitrary prime in any of the aforementioned sequences. 1. Introduction Linear recursive equations such as the family of second-order extended Lucas sequences described above have attracted considerable theoretic attention for more than a century. Among other things they have played an important role in primality testing. For example the prime character of a number is often a consequence of having maximal rank of apparition that is rank of apparition equal to N 1. The first objective of this paper is to provide a general rank-of-apparition result for primes of the form N 2np 1 where p is a prime. Then using more explicit criteria we will determine when such primes have maximal rank of apparition in the specific Lehmer sequences ffnl Un 1 -1 1 1 2 3 . and Ln Vn 1 -1 1 3 4 7 . . Respectively ffn and Ln represent the Fibonacci and the Lucas numbers. 2. The Lucas and Lehmer sequences In 4 Lucas published the first set of papers that provided an in-depth analysis of the numerical factors of the set of .