Vol. 16, No. 3, 1966

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ISSN: 0030-8730
New infinite classes of periodic Jacobi-Perron algorithms

Leon Bernstein

Vol. 16 (1966), No. 3, 439–469
Abstract

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, dD, 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) = 0s(n − s − 1+ i)
iwsi(D 1)i,
Fs(w,D) = 0s(           )
n − s − 1+ i
iwsiDi, (s = 1,,n 1)
of which each is an inverse function of the other.

Mathematical Subject Classification
Primary: 10.32
Secondary: 10.66
Milestones
Received: 18 June 1964
Published: 1 March 1966
Authors
Leon Bernstein