Note on the Euclidean Algorithm

1. The E.A. (Euclidean Algorithm) holds for a quadratic field R(2/.m)? when, for any two integers a, ,G in R(~/m.) with /3 + 0, a third integer y in B(~/w) can be so det,ermined that (1) 1 N(a-/3y) / < j N(P) (or \ N(a/fi-y) 1 < 1. The existence of the E .A. is undecided in the following cases; p, q denoting primes : y I. m=p= 13+24n (,n> 1); II. m=p= 1+&L (n> 7); III. m =pq wit#h 23 = q-3 or p = qz 7 (mod S), and pq > 57. In this paper, we show that the E.,4. does not exist for large p in thefirst two mm, i.e. it can exist only in a finite number of quadratic fields 3(1/Ip). The integers in R(dp), where pr 1 (mod 4), are given by +(x+ydp), where x: y a,re rational integers and x-y {mod 2). Instead of (l), we can write 1 h'(a+b dp-&(z+y2/P))1= i (a-$~)~-p(b-&#j < 1, xvhere a, b. denote rational numbers, i.e. (2) j(x-22a)2-p(y-22b)2] < 4. ..-A ___