Gelfond’s constant

Take the principal branch and compute:

$$(-1)^{-i} \;=\; e^{-i\,\mathrm{Log}(-1)} \;=\; e^{-i\cdot i\pi} \;=\; e^{\pi} \;=\; 23.14069\ldots$$

So $e^\pi$ is an algebraic number raised to an algebraic irrational power. Gelfond proved such powers transcendental — for this very number in 1929, then with Schneider in general (1934), settling Hilbert’s seventh problem: if $\alpha$ is algebraic, not $0$ or $1$, and $\beta$ is algebraic and irrational, then $\alpha^\beta$ is transcendental.1 Hence Gelfond’s constant.

1

The same theorem handles $2^{\sqrt2}$ (Hilbert’s stated example) and $\sqrt2^{\sqrt2}$. It does not touch $\pi^e$: there the base is transcendental, and the theorem’s hypotheses fail.

The contrast across the decimal point is stark. $\pi^e$ is not proven irrational, let alone transcendental. For the pair $e+\pi$ and $e\pi$, consider

$$x^2-(e+\pi)x+e\pi \;=\; (x-e)(x-\pi).$$

If both coefficients were algebraic, the roots would be too — but $e$ is transcendental (Hermite, 1873). So at least one of $e+\pi$, $e\pi$ is transcendental. Which one? Unknown. Number theory can certify the disjunction and cannot, today, certify either disjunct.

Almost an integer

The constant’s most famous relative is

$$e^{\pi\sqrt{163}} \;=\; 262\,537\,412\,640\,768\,743.99999999999925\ldots$$

No coincidence. With $q = -e^{-\pi\sqrt{163}}$, the modular $j$-function expands as $j = 1/q + 744 + 196884\,q + \cdots$; because $\mathbb{Q}(\sqrt{-163})$ has class number one, $j$ evaluates to the integer $(-640320)^3$ there. Rearranging, $e^{\pi\sqrt{163}}$ misses $640320^3+744$ by about $7.5\times10^{-13}$ — the error term is the next coefficient of a modular form.2

2

163 is the largest Heegner number: 1, 2, 3, 7, 11, 19, 43, 67, 163. Each yields a near-integer of this species; 163 yields the best.