The Jacobi-Perron Algorithm Its Theory and Application by L. Bernstein

By L. Bernstein

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Accumulator, and the numbers Al, ••• ,A _ -the elements of the n l accumulator. 18) t-l n a. + ~ a . i=t i i=O ~ - c ~ ai The main result of this paragraph is stated in THEOREM 5. 20 ) = [a(k)], ~ 2(n-l)d. of the vector ~(n-l-s+i)(Dl)iS-i . w , ••• ) a (0)_( ••• , Li=O (s=l, ••• ,n-l) ~ 2 is purely periodic and its primitive length equals n , if not both m and d 1, and n if d = m = 1. 1) of n fugues. 21). This rather complicated Theorem was proved by the author in [2,cJ. We shall illustrate Theorem 5 for the cases n = 2 and n 1 (i) = (D 2 - d- l m)2; diD; D n = 2; a a (0) = ~ 5.

22) n-1 > 1 e + ••• + en 1 + K. 23) l. 23) holds for any v > vo' we also have A~v+n+l) l. Al v+n-l) > K K ( A{v+n-1) [ o Atv+n- A(v+n-l) K2 and generally ( o ) 1) - m. l. 24) A~v+n+h) .. A6v+n+h) i [o A~V') max ( _ m > K +1 ~ A~v+n-l) _ m ~ A6v+n-l) ) i . h -£(l+K+···+K). A~v' +1) ~ ~ A (v') '-A':;(-v-''''''')- , ••• , 0 Among the n numbers in the brackets, one must equal Mi v '), say A~v'+j) (v' ) ~ M. ~ h €(l+K+"'+K) > Kh+1 ( Mi v ') - m ) i ~ h+1 h K (M i -mil - £(l+K+' "+K ); o > - K(h+l) (M .

4 6 Z Z 5 3 Z A6 ) = 1; A6 )= 4D t; A6 6 )= l6D 4 t +6Dt; A6 7)= 64D t 3 +48D t +4t; A68)=256D9t3+288D6tZ+68D3t+l; A69}=1024Dllt4+l536D8t3+6Z4D5tZ+56DZt. Most of the classical investigations of periodicity of the JPA with f(a(k» = [a{k)] concentrated on a vector a(O) E:O(a) of the form 55 a (0) = 2 (a, a , ••• , a n-l) ~ E _ , n l 2 n-l which may seem a natural approach with the numbers 1, a, a , ••• ,a forming a basis of Q(cr). As we shall see, the JPA of such a(O) also becomes periodic, but has a pre-period of length n-l.

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