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one hundred sixty-three (Feature)

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,

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

$\displaystyle \sum_{i = 0}^4 {8 \choose i} = 163.$

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).



"one hundred sixty-three" is owned by PrimeFan.
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Cross-references: strictly non-palindromic number, binary, base, palindromic prime, Chen prime, Gaussian prime, complex plane, prime number, real, number, unique factorization, imaginary quadratic field, Kurt Heegner, digits, approximation, integer, Srinivasa Ramanujan
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This is version 2 of one hundred sixty-three, born on 2007-04-07, modified 2007-04-10.
Object id is 9163, canonical name is OneHundredSixtyThree.
Accessed 1024 times total.

Classification:
AMS MSC11A99 (Number theory :: Elementary number theory :: Miscellaneous)

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