Corrigenda to "Calculation of the regulator of Q(√D) by use of the nearest integer continued fraction algorithm"

There are some minor errors in one of the algorithms and two of the tables in a paper by Williams and Buhr. These errors do not affect the major conclusions of the paper. We present corrections to one of the algorithms and two of the tables in [1]. These corrections do not affect the major conclusions of the paper. In the algorithm for computing the NICF of √ D on the bottom half of page 373, when Qk < 0, Tk should be defined as • If Qk + F + 1 is even, then Tk = d+ (|Qk|+ F + 1)/2 . • If Qk + F + 1 is odd, then Tk = 1 + d+ (|Qk|+ F + 1)/2 . R′ k+1 should be defined as • If Qk+1 < 0 and Qk+1 divides P ′ k+1 + Tk+1 evenly, then R′ k+1 = |Qk+1|. • Otherwise, R′ k+1 is, as in [1], the remainder on dividing P ′ k+1 + Tk+1 by Qk+1. Note that the formula for R′ k+1 has to be used with k = −1 in order to set the value of R′ 0. In the other formulas in this algorithm k ≥ 0. Also, P ′ k+1 = Tk −R′ k. The description of Table 1 in [1] should read, “In Table 1 we give the frequency of occurrence of each of these criteria for the NICF expansion of √ D for each nonsquare 10 ≤ D < M .” Corrected values for the Table 1 in [1] are given in “Table 1 (with corrections)”. Table 1 (with corrections) Condition M = 10,000 M = 100,000 M = 1,000,000 M = 10,000,000 1 7,370 76,155 776,894 7,882,803 2 880 9,698 101,347 1,032,817 3 324 2,340 18,093 146,161 4 785 6,819 60,702 552,135 5 153 1,302 11,734 106,995 6 382 3,363 30,224 275,920 Received by the editor January 15, 2008. 2000 Mathematics Subject Classification. Primary 11A55. c ©2008 American Mathematical Society 615 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 616KEITH R. MATTHEWS, JOHN P. ROBERTSON, AND JIM WHITE The corrected Table 1 agrees with that in [1] for M = 10, 000, but most of thevalues to 100,000 and to 1,000,000 in the corrected table are slightly different fromthose in [1]. We have added counts to 10 million.In Table 2 of [1] each Θ should be 2Θ. For Case 6, the log(√D +|Qρ−1/2|) in[1] should be log(√D −|Qρ−1/2|).References[1] H. C. Williams and P. A. Buhr, Calculation of the regulator of Q(√D) by use of the nearestinteger continued fraction algorithm, Math. Comp. 33 (145) (1979), 369–381. MR514833(80e:12003) Department of Mathematics, University of Queensland, St. Lucia, Brisbane, QLD4072, Australia, and Centre for Mathematics and its Applications, Australian NationalUniversity, Canberra, ACT 0200, AustraliaE-mail address: [email protected] Actuarial and Economic Services Division, National Council on Compensation Insur-ance, Boca Raton, Florida 33487E-mail address: [email protected] Mathematical Sciences Institute, Australian National University, Canberra, ACT0200, AustraliaE-mail address: [email protected] License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use