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A Lower Bound for the Scholz-Brauer Problem

Published online by Cambridge University Press:  20 November 2018

Kenneth B. Stolarsky
Kenneth B. Stolarsky*
Affiliation:
The Institute for Advanced Study, Princeton, New Jersey
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In (6)Scholz asked if the inequality

1.1

held for all positive integers q, where l(n)is the number of multiplications required to raise xto the nth power (a precise definition of l(n)in terms of addition chains is given in § 2). Soon afterwards, Brauer (2) showed, among other things, that l(n) ∼(log n)/(log2). This suggests the problem of Calculating

1.2

It can be deduced from (2) that θ≦ 1. If θ <1, (1.1) follows immediately for infinitely many q.My main result,Theorem 5 of § 4, merely shows that θ is slightly larger than ⅓.Actually, I know of no case where (1.1) is not in fact an equality; a tedious calculation verifies this for 1 ≦ q≦ 8.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969, pp. 675 - 683
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Bellman, R., Advanced problem 5125, Amer. Math. Monthly 70 (1963), 765.Google Scholar
2. Brauer, A. T., On addition chains, Bull. Amer. Math. Soc. 45 (1939), 736–739.Google Scholar
3. Gioia, A. A., Subbarao, M. V., and Sugunamma, M., The Scholz-Brauer problem in addition chains, Duke Math. J. 29 (1962), 481–487.Google Scholar
4. Hansen, W., Zum Scholz-Brauerschen problem, J. Reine Angew. Math. 202 (1959), 129–136.Google Scholar
5. Il'in, A. M., On additive number chains, Problemy Kibernet. 13 (1965), 245–248. (Russian)Google Scholar
6. Scholz, A., Jahresbericht, Deutsche Math.-Verein. 47 (1937), 41.Google Scholar
7. Straus, E. G., Addition chains of vectors, Amer. Math. Monthly 71 (1964), 806–808.Google Scholar
8. Utz, W. R., A note on the Scholz-Brauer problem in addition chains, Proc. Amer. Math. Soc. 4 (1953), 462–463.Google Scholar
9. Whyburn, C. T., A note on addition chains, Proc. Amer. Math. Soc. 16 (1965), 1134.Google Scholar