Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 2001. Sectio Mathematicae (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 28)

DRESS, A. & LUCA, L., Real numbers that have good diophantine approximations

18 A. Dress fe F. Luca Lucas sequence of the first kind for a given by formula (6), mostly because the constant c appearing in inequality (8) is arbitrary. Of course, if one imposes that the constant c appearing in inequality (8) is small enough (for example, c. = 1/2), then the rational numbers r

n+i/r

n are exactly the convergents of a, therefore r

n is indeed given by formula (6) for all n (up to some linear shift in the index n). Proof of Theorem 2. If one replaces the sequence (r

n)

n>o by the sequence {Rn)n>o = {i'nt

x)n>o» then the first inequality (9) together with part (i) of Theorem 1 show that c*

S

l is an algebraic integer, different than 0 or ±1, of degree at most 2. Similarly, if one replaces the sequence (r

n)

n>o by the sequence (Rn)n>o = (?'ní

2)n>0? then the second part of inequality (9) together with part (ii) of Theorem 1 show that a

5

2 is an algebraic, integer, different that 0 or ±1, of degree at most 2. Prom here on, all we need to establish is that Q is itself algebraic of degree at most 2. Assume that this is not so and let K :— Q[a] and Ki := Q[a

Ä

I] for i — 1, 2. Since Si and «2 are coprime, we get that K = Q[a

S

l, a

i

3]. Moreover, we must have [Ki : Q] = 2 for both i — 1 and 2, i.e. K is a biquadratic real extension of Q and Gal(/\/Q) = Z2 0Z2. Hence, there exist two non-trivial elements <ti and er

2 in Gal(/v

/Q) with cr^cv

5') = a

s', i.e. for i = 1, 2. Since K is a real field and is non-trivial, formula (17) implies that (Ji(a) = —<y for i = 1, 2. Hence, <ri(a) = a2(a), which implies a 1 = cr

2. But this is a contradiction. The remaining of the assertions of Theorem 2 follow from Theorem 1. Theorem 2 is therefore established. Acknowledgements Work by the second author was done while he visited Bielefeld. He would like to thank the Graduate College Strukturbildungsprozesse and the Forschungsschwerpunkt Mathematisierug there for their hospitality and the Alexander von Humboldt Foundation for support. [1] DRESS, A., LUCIA, F., Unbounded Integer Sequences (Ai)n>O with A

N

+IA

n_i —A^ Bounded are of Fibonacci Type, to appear in the Proceedings of ALCOMA99. [2] DRESS, A., LUCA, F., A Characterization of Certain Binary Recurrence Sequences, to appear in the Proceedings of ALCOMA99. (17) References