Definition:Hilbert 23/7/Historical Note

Historical Note on Problem $7$ of the Hilbert $23$

David Hilbert believed that determining whether certain numbers are irrational or transcendental would be a problem more difficult than the Riemann Hypothesis or Fermat's Last Theorem.

Two specific cases, however, were resolved relatively quickly:

$e^\pi$ (Gelfond's constant) was proved transcendental in $1929$

$2^{\sqrt 2}$ (the Gelfond-Schneider constant) was proved transcendental in $1930$.

Subsequently they were seen as special cases of the Gelfond-Schneider Theorem, which was proved in $\text {1934}$ – $\text {1935}$.

It is sometimes pointed out that, in view of the fact that Fermat's Last Theorem held out till $1994$ and the Riemann Hypothesis still refuses to yield, in this case Hilbert's pessimism was unfounded.

However, note that certain other numbers of a similar type: it has still not been established whether $2^e$, $\pi^e$ and $2^\pi$, for example, are transcendental or not.

So, in general it may be accepted that Hilbert was not altogether incorrect.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $2 \cdotp 665 \, 144 \ldots$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $2 \cdotp 66514 \, 4 \ldots$