Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1997. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 24)

GRYTCZUK, A., On, some connections between Legendre symbols and continued fractions

20 Aleksander Grytczuk 2. Proof of the Theorem In the proof of the Theorem we use the following lemmas: Lemma 1. Let \fd = [<?o5 <Zi > • • • ? Qs] be the representation of \fd as a simple continued fraqtion. Then (1) q

n ^^ , b

n + b

n

+1 = c

nq

n, d = b

2

n+

1 + c

nc

n+1, for any integer n> 0 (2) if s = 2r +

1 then minimal number k, for which ck = c^+i is k = (3) if s = 2r then minimal number k, for which b^ = is k = ^ (4) 1 < c

n < 2

\fd, for 1 < n < s — 1 (5) P

2-\ - dQn-1 = (—l)

n

cn) where Pn/Q

n is n-th convergent of \[d. This Lemma is a collection of the well-known results of the theory of continued fractions. Lemma 2. Let \fd — [goi <7i

, • • •, q

s]- The equation x

2 — dy

2 = —1 is solvable if and only if the period s is odd. Moreover, if p = 3 (mod 4) and p is a divisor of d then this equation is unsolvable. This Lemma is well-known result given by Legendre in 1785. For the proof of the Theorem we remark that by the condition d = pq = 3 (mod 4) it follows that p = 3 (mod 4) or q = 3 (mod 4) and consequently from Lemma 2 we obtain that the period s — 2m. From (5) of Lemma 1 we get (6) P

2

m-x -pqQ

2

m-i =(-l)

mc

m. On the other hand by (1) and (3) of Lemma 1 it follows that (7) 2b

m

+i - q

mc

m

: d-pq- b

2

m+

l + c

mc

m+

1

. From (7) we obtain (8) 4

pq = c

m(q^c

m + 4c

m+

1). By (8) it follows that c

m = l,2,4,p, q,pq,2pq,4pq. Using (4) of Lemma 1 we get that c

m = 1,2,4,p,q. If c

m = 1 then it is easy to see that (6) is impossible. If c

m = 4 then from (6) we obtain (9) e

2

m-i - pqQl-i = (-I)

M4.