Göbel's sequence

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In mathematics, a Göbel sequence is a sequence of rational numbers defined by the recurrence relation

[math]\displaystyle{ x_n = \frac{ x_0^2+x_1^2+\cdots+x_{n-1}^2}{n-1},\!\, }[/math]

with starting value

[math]\displaystyle{ x_0 = x_1 = 1. }[/math]

Göbel's sequence starts with

1, 1, 2, 3, 5, 10, 28, 154, 3520, 1551880, ... (sequence A003504 in the OEIS)

The first non-integral value is x43.[1]

History

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

Göbel's sequence can be generalized to kth powers by

[math]\displaystyle{ x_n = \frac{x_0^k+x_1^k+\cdots+x_{n-1}^k}{n}. }[/math]

The least indices at which the k-Göbel sequences assume a non-integral value are

43, 89, 97, 214, 19, 239, 37, 79, 83, 239, ... (sequence A108394 in the OEIS)

Regardless of the value chosen for k, the initial 19 terms are always integers.[3][2]

See also

References

  1. ↑ Guy, Richard K. (1981). Unsolved Problems in Number Theory. Springer New York. p. 120. ISBN 978-1-4757-1740-2. 
  2. ↑ 2.0 2.1 2.2 Stone, Alex (2023). "The Astonishing Behavior of Recursive Sequences" (in en). https://www.quantamagazine.org/the-astonishing-behavior-of-recursive-sequences-20231116/. 
  3. ↑ Matsuhira, Rinnosuke; Matsusaka, Toshiki; Tsuchida, Koki (19 July 2023). "How long can k-Göbel sequences remain integers?". arXiv:2307.09741 [math.NT].

External links



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