## Some questions of Erdős and Graham on numbers of the form $\sum g_ n/2^ {g_ n}$

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- by Peter Borwein and Terry A. Loring PDF
- Math. Comp.
**54**(1990), 377-394 Request permission

## 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 \[ \frac {n}{{{2^n}}} = \sum \limits _{k = 1}^T {\frac {{{g_k}}}{{{2^{{g_k}}}}},\quad T > 1} ,\] 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 \[ {a_{n + 1}} = 2({a_n}\bmod n)\] 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.

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## Additional Information

- © Copyright 1990 American Mathematical Society
- Journal: Math. Comp.
**54**(1990), 377-394 - MSC: Primary 11A63; Secondary 11-04, 11D68, 11K16
- DOI: https://doi.org/10.1090/S0025-5718-1990-0990598-9
- MathSciNet review: 990598