Non-Integral Value of Göbel's Sequence

Theorem

Consider Göbel's sequence defined recursively as:

$x_n = \begin{cases}

1 & : n = 0 \\ \ds \paren {1 + \sum_{k \mathop = 0}^{n - 1} {x_k}^2} / n & : n > 0 \end{cases}$

The smallest $n$ such that $x_n$ is not an integer is $43$.

Proof

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Sources

• 1994: Richard K. Guy: Unsolved Problems in Number Theory (2nd ed.)

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $43$

• Weisstein, Eric W. "Göbel's Sequence." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GoebelsSequence.html