mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a Göbel sequence is a sequence of

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...

s defined by the

recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...

x_n = \frac,\!\,

with starting value :

x_0 = x_1 = 1.

Göbel's sequence starts with : 1, 1, 2, 3, 5, 10, 28, 154, 3520, 1551880, ... The first non-integral value is ''x''₄₃.

History

This sequence was developed by the German mathematician Fritz Göbel in the 1970s. In 1975, the Dutch mathematician

Hendrik Lenstra Hendrik Willem Lenstra Jr. (born 16 April 1949, Zaandam) is a Dutch mathematician. Biography Lenstra received his doctorate from the University of Amsterdam in 1977 and became a professor there in 1978. In 1987, he was appointed to the faculty o ...

showed that the 43rd term is not an integer.

Generalization

Göbel's sequence can be generalized to ''k''th powers by :

x_n = \frac.

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, ... Regardless of the value chosen for ''k'', the initial 19 terms are always integers.

References

External links

Göbel's Sequence
{{DEFAULTSORT:Gobel's sequence Integer sequences Recurrence relations

History

Generalization

See also

References

External links