163 (number)

163 is the natural number following one hundred sixty-two and preceding one hundred sixty-four.

163 is a strong prime in the sense that it is greater than the arithmetic mean of its two neighboring primes. 163 is a lucky prime.

Given 163, the Mertens function returns 0.

163 figures in an approximation of π, in which Failed to parse (Missing texvc executable; please see math/README to configure.): \pi \approx {2^9 \over 163} \approx 3.1411 .

163 is a strictly non-palindromic number.

163 is a Heegner number. That is, the ring of integers of the field Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{Q}(\sqrt{-a})

has unique factorization for Failed to parse (Missing texvc executable; please see math/README to configure.): a=163

. The only other such integers are Failed to parse (Missing texvc executable; please see math/README to configure.): a = 1, 2, 3, 7, 11, 19, 43, 67 .

Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{163}

The square root of 163 occurs in several interesting pieces of mathematics.

The function Failed to parse (Missing texvc executable; please see math/README to configure.): f(n) = n^2 + n + 41

gives prime values for all values of Failed to parse (Missing texvc executable; please see math/README to configure.): n
between 0 and 39, and for Failed to parse (Missing texvc executable; please see math/README to configure.): n < 10^7
approximately half of all values are prime. 163 appears as a result of solving Failed to parse (Missing texvc executable; please see math/README to configure.): f(n)=0

, which gives Failed to parse (Missing texvc executable; please see math/README to configure.): n = (-1+ \sqrt{-163} ) / 2 .

Failed to parse (Missing texvc executable; please see math/README to configure.): \sqrt{163}

appears in the Ramanujan constant, in which Failed to parse (Missing texvc executable; please see math/README to configure.): e^{\pi \sqrt{163}}

almost equals the integer 262537412640768744 = 640320^3 + 744.   Martin Gardner famously asserted that this identity was exact in a 1975 April Fools' hoax in Scientific American; in fact the value is 262537412640768743.99999999999925007259... Failed to parse (Missing texvc executable; please see math/README to configure.): 32 ln(163)
is also nearly integer, being equal to 163.000006425...

Other occurrences

One hundred sixty-three is also:

• The first number that cannot be shot with three darts on a standard dart board.
• The year AD 163 or 163 BC.

References

• Wells, D. (1987). The Penguin Dictionary of Curious and Interesting Numbers (pp. 141 - 142). London: Penguin Group.

See also

• 163 (disambiguation)