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.References
- P. Erdős, Some problems and results on the irrationality of the sum of infinite series, J. Math. Sci. 10 (1975), 1–7 (1976). MR 539489 —, Sur l’irrationalité d’une certaine série, C. R. Acad. Sci. Paris Ser. I Math. 292 (1981), 65-768. P. Erdös and R. L. Graham, Old and new results in combinatorial number theory, Monograph No. 28 de d’Enseignement Mathématique, Genève, 1980.
- Donald E. Knuth, The art of computer programming. Vol. 2, 2nd ed., Addison-Wesley Series in Computer Science and Information Processing, Addison-Wesley Publishing Co., Reading, Mass., 1981. Seminumerical algorithms. MR 633878
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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