Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2004. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 31)

LUCA, F., Primitive divisors of Lucas sequences and prime factors of ... and ...

20 F. Luca is X = 24208144. (ii) The largest positive integer solution x of the inequality P(x

4 -f 1) < 233 (2) is x = 10. Proof. We start with the first question. Assume that £ is a positive integer such that P(x

2 + 1) < 101. The only prime numbers p that can divide a number of the form x

2 + 1 are either p = 2, or p = 1 (mod 4). There are only 12 such primes p less than 101 and they are pev = {2, 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97}. In particular, the number x has the property that x

2 + l = dy

2, (3) where d > 1 and y > 1 are integers whose factors belong to and d is squarefree. If we rewrite equation (3) as x

2-dy

2 = -1, (4) it follows that the pair (x, y) is a positive integer solution of a Fell equation of the form (4) for some squarefree d > 1 whose prime factors are in the set V. Let A be the set of all the squarefree positive integers d > 1 whose prime factors are in the set V. Clearly, A contains precisely 2^1 — 1 = 2

1

2 — 1 = 4095 elements. For each d £ -4 let (Ai (d)

, Yi (d)

) be the first, positive integer solution of the Pell equation X

2-dY

2 = ± 1. (5) It is wellknown that if we denote by rrid the length of the continued fraction of s/d, then (A'i(t/), V'i(c/)) = (P

m

d_i, Q

m

d-1), where for a nonnegative integer k we have denoted by Pk/Qk the A;th convergent to Vd. Moreover, if md is even, then equation (5) has no integer solution (A", Y) with the sign —1 appearing on the right hand side. Of the totality of 4095 elements d of A, only 2672 of them have the property that the period nid is odd. Let us denote by B the subset of A consisting of only these elements. We used Mathematica to compute (A'i(t/), Fi(c/)) for all d £ B. These computations took about 7 hours. Assume now that (x

, y) is a solution of equation (4) for some d £ B. It then follows that (x, y) = (A

n(ei), Y

n(d)) for some odd value of n > 1, where X

n(d) and Y

n(d) can be computed using the formulae X

n(d) = and y„(d) = - W*»" 2 2 Vd