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Some questions of Erdős and Graham on numbers of the form $ \sum g\sb n/2\sp {g\sb n}$

Authors: Peter Borwein and Terry A. Loring
Journal: Math. Comp. 54 (1990), 377-394
MSC: Primary 11A63; Secondary 11-04, 11D68, 11K16
MathSciNet review: 990598
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Abstract: Erdös in 1975 and Erdös and Graham in 1980 raised several questions concerning representing numbers as series of the form $ \Sigma _{n = 1}^\infty \;{g_n}/{2^{{g_n}}}$. For example, does the equation

$\displaystyle \frac{n}{{{2^n}}} = \sum\limits_{k = 1}^T {\frac{{{g_k}}}{{{2^{{g_k}}}}},\quad T > 1} ,$

have a solution for infinitely many n ? The answer to this question is affirmative; in fact, we conjecture that the above equation is solvable for every n. This conjecture is based on a more general conjecture, namely that the algorithm

$\displaystyle {a_{n + 1}} = 2({a_n}\bmod n)$

with initial condition $ {a_m} \in {\mathbf{Z}}$ always eventually terminates at zero. This, in turn, is based on an examination of how the "greedy algorithm" can be used to represent numbers in the form $ \sum {{g_n}/{2^{{g_n}}}} $. The analysis of this, reformulated as a "base change" algorithm, proves surprising. Some numbers have a unique representation, as above, others have uncountably many. Also, from this analysis we observe that $ \sum {{g_n}/{2^{{g_n}}}} $ is irrational if $ \lim {\sup _n}(({g_{n + 1}} - {g_n})/\log ({g_{n + 1}})) = \infty $ and conjecture that this is best possible.

References [Enhancements On Off] (What's this?)

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Article copyright: © Copyright 1990 American Mathematical Society