The question whether a
system of n − 1 real algebraic numbers (n = 2,3,⋯) chosen from an algebraic
field of degree not higher than n, yields periodicity by Jacobi’s Algorithm is
still as open and challenging as hundred years ago. The present paper gives
an affirmative answer to this problem in the following case: let K(w) be
an algebraic number field generated by w = (Dn− d : m)1∕n, where m, n,
d, D are natural numbers satisfying the conditions m ≧ 1, n ≧ 3, d∣D,
1 ≦ d ≦ D∕2(n − 1). Then n − 1 numbers can be chosen from K(w), so that
their Jacobi Algorithm becomes purely periodic. The length of the period
equals n2 (or n, if d = m = 1). This is the longest period of a periodic Jacobi
Algorithm ever known. In three corollaries the following special cases are
investigated
w
= (Dn− dr)1∕n,
(r = 0,1,⋯,n)
w
= (Dn− drD)1∕n,
(r = 0,1,⋯,n − 2)
w
= (Dn− pd∕m)1∕n.
(n = pu,p a prime, u = 1,2,⋯,m as before)
In all these three cases the Algorithm of Jacobi remains purely periodic with length
equal to n2.
The main tools in proving these results are the polynomials
fs(w,D − 1)
=∑0sws−i(D − 1)i,
Fs(w,D)
=∑0sws−iDi, (s = 1,⋯,n − 1)
of which each is an inverse function of the other.
