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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2024 MCQ for Mathematics of Computation is 1.78.

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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
  • Jeffrey C. Lagarias, The $3x+1$ problem and its generalizations, Amer. Math. Monthly 92 (1985), no. 1, 3–23. MR 777565, DOI 10.2307/2322189
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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