Göbel's sequence
Appearance
This article needs additional citations for verification. (January 2017) |
In mathematics, a Göbel sequence is a sequence of rational numbers defined by the recurrence relation
with starting value
Göbel's sequence starts with
The first non-integral value is x43.[1]
History
[edit]This sequence was developed by the German mathematician Fritz Göbel in the 1970s.[2] In 1975, the Dutch mathematician Hendrik Lenstra showed that the 43rd term is not an integer.[2]
Generalization
[edit]Göbel's sequence can be generalized to kth powers by
The least indices at which the k-Göbel sequences assume a non-integral value are
Regardless of the value chosen for k, the initial 19 terms are always integers.[3][2]
See also
[edit]References
[edit]- ^ Guy, Richard K. (1981). Unsolved Problems in Number Theory. Springer New York. p. 120. ISBN 978-1-4757-1740-2.
- ^ a b c Stone, Alex (2023). "The Astonishing Behavior of Recursive Sequences". Quanta Magazine. Retrieved 2023-11-17.
- ^ Matsuhira, Rinnosuke; Matsusaka, Toshiki; Tsuchida, Koki (19 July 2023). "How long can k-Göbel sequences remain integers?". arXiv:2307.09741 [math.NT].