one hundred sixty-three

Of Martin Gardner’s April Fool’s hoaxes, perhaps the most famous comes from the April 1975 issue of Scientific American, in which he claimed that Srinivasa Ramanujan had proven that $e^{\pi \sqrt{163}}$ is exactly equal to an integer. The truth is that it is not, but it comes surprisingly close, being approximately .0000000000007499274028018143 short of the nearest integer.

One hundred sixty-three also appears in an approximation of $\pi$, namely,

$$\frac{2^9}{163}\approx \pi ,$$

but this is barely correct to three decimal digits.

There are other qualities of 163 that are somewhat more exact, such as the fact that

$$\sum _{i=0}^4\left(\genfrac{}{}{0pt}{}{8}{i}\right)=163.$$

Kurt Heegner proved that $n=163$ is the largest value for which the imaginary quadratic field $\mathbb{Q}(\sqrt{-n})$ has a unique factorization (thus 163 is the largest Heegner number).

Among the real integers, 163 is a prime number. As $163+0i$, it is also a prime on the complex plane, that is, a Gaussian prime.

163 is the eighth prime that is not a Chen prime. Nor is it a palindromic prime in any base from binary to base 161 (hence it’s a strictly non-palindromic number).