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Mathematics of Computation

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A conjecture in addition chains related to Scholz's conjecture

Authors: Walter Aiello and M. V. Subbarao
Journal: Math. Comp. 61 (1993), 17-23, S1
MSC: Primary 11B83; Secondary 11Y55
MathSciNet review: 1189515
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Abstract: Let $ l(n)$ denote, as usual, the length of a shortest addition chain for a given positive integer n. The most famous unsolved problem in addition chains is Scholz's 1937 conjecture that for all natural numbers $ n,l({2^n} - 1) \leq l(n) + n - 1$. While this conjecture has been proved for certain classes of values of n, its validity for all n is yet an open problem. In this paper, we put forth a new conjecture, namely, that for each integer $ n \geq 1$ there exists an addition chain for $ {2^n} - 1$ whose length equals $ l(n) + n - 1$. Obviously, our conjecture implies (and is stronger than) Scholtz's conjecture. However, it is not as bold as conjecturing that $ l({2^n} - 1) = l(n) + n - 1$, which is known to hold, so far, for only the twenty-one values of n which were obtained by Knuth and Thurber after extensive computations. By utilizing a series of algorithms we establish our conjecture for all $ n \leq 128$ by actually computing the desired addition chains. We also show that our conjecture holds for infinitely many n, for example, for all n which are powers of 2.

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Keywords: Addition chain, Scholz conjecture, short chain
Article copyright: © Copyright 1993 American Mathematical Society