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

On some connections between Legendre symbols and continued fractions 21 Since (P

m_i,Q

m_i) = 1 then by (9) it follows that P

m-i and Q

m-i are odd and consequently we obtain P^-i = QM-I = 1 (mod 4). Since pq = 3 (mod 4) then by (9) it follows that 1 = = pqQ

2

m_

l + (-l)

m4 = 3 (mod 4) and we get a contradiction. Therefore, we have c

m = p,q,2. Let c

m—p then from (6) we obtain (10) pX

2 - qQ

2

m= (-1)-, where P

m

= pX. Prom (10) and the well-known properties of Legendre's symbol we obtain (?) - m - (T) In similar way, for the case c

m = q we get (12) By (12) and the reciprocity law of Gauss we obtain f P\ . . P-1 s + q-l (.9/ = ff c

m = 2 then by (6) it follows that ) = (

2

(~

t

j

i

r ) = 1. Hence, in virtue of pq = 3 (mod 4) we obtain f 2= ( — l)

m and the proof is complete. References [1] P. CHOWLA AND S. CHOWLA, Problems on periodic simple continued fractions, Proc. Nat. Acad. Sei. USA 69 (1972), 37-45. [2] C. FRIESEN, Legendre symbols and continued fractions, Acta Arith. 59 4. (1991), 365-379. [3] A. SCHINZEL

, On two conjectures of P. Chowla and S. Chowla concerning continued fractions, Ann! Math. Pure Appl. 98 (1974), 111-117. INTITUTE OF MATHEMATICS DEPARTMENT OF ALGEBRA AND NUMBER THEORY T. KOTARBINSKI PEDAGOGICAL UNIVERSITY PL. SLOWIANSKI 9, 65-069 ZIELONA GORA POLAND