β Scribed by Neela S Sukthankar (REGE)
No coin nor oath required. For personal study only.
Let 2 be the ring of rational integers. For n E Z, w(n) denotes the number of distinct prime factors of n. In [4] Ramachandra, Shorey and Tijdeman proved that if 1 k, where c > 0 is an effectively computable constant. In this paper, we will generalise this problem to the case of imaginary quadratic fields. The more general case of algebraic number fields will be published separately. For simplicity we will discuss this for the field of gaussian numbers but the same proof can be carried over for arbitrary imaginary quadratic fields. Let k= Q(i) be the field of gaussian numbers and o={a+6i[a,b E Z} be the ring of gaussian integers. For z E o, w(z) will denote the number of distinct prime factors of Z. Let c(z,R) denote the disc {C E 01 /z-cl exp (c(log R)6+8), where c> 0 is a constant, then it follows that none of the 5'9's is a unit nor any two cf's are associates.
In this note we will prove:
π SIMILAR VOLUMES
Let k be an algebraic number field of degree n over &, n =rl+ 2rs and let WI, . . . . Wn be a fixed basis of the ring of integers o over 2. For z E k, z= zB1 qwt, at E (1, c(z, R) will denote the disc {c E k, c= 2 biwi, bt E Q, ~i1~-bt12=zfi} and ~(0, R) will be the disc (CE k, 5= zbtiwr, xb; (log ]