Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1993. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 21)

Bui Minh Phong: Recurrence sequences and pseudoprimes

1.3. Some Diophantine equations concerning recurrence sequences A linear recurrence W = {W

n}™

0 of order k(> 1) is defined by integers A

Q,A

l,..., A

k_

, and by recursion K = A

0W^

+A,W

n^...

+A

k_

lW

n_

k Cn>k

), where the initial values W

Q,W

u...,W

k_

x are fixed not all zero integers and 0. Denote the distinct roots of characteristic polynomial by a

0,a

]

y...,a

t, where a. has multiplity w.. It is known that for « > 0 K=/, (») <+/

2 («) +• • •+/, oo where / (n) is a polynomial of degree at most w -1, furthermore the coefficients of f

t(n) are algebraic numbers from the field Q(a

x,...,a

t). We say that the sequence IT is non-degenerate if t > 1 and a, / a

} is not a root of unity for t> j >i> 1. Let p

l,p

2,...,p

r be primes and we denote by S the set of integers which have only these primes as prime factors. K. Győry, P. Kiss and A. Schinzel (Colloq. Math. 45, 1981, 75-—80) showed that if W is a non-degenerate Lucas sequence R, then (1.5) W

xeS 118