163 (number)

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

List of numbers — Integers

<< 160 161 162 163 164 165 166 167 168 169 >>

Cardinal

One hundred [and]
sixty-three

163rd

CLXIII

10100011

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 $\pi \approx {2^9 \over 163} \approx 3.1411$ .

163 figures in an approximation of e, in which $e \approx {163 \over 3\cdot4\cdot5} \approx 2.7166\dots$ .

163 is a strictly non-palindromic number.

163 is a Heegner number. That is, the ring of integers of the field $\mathbb{Q}(\sqrt{-a})$ has unique factorization for $a = 163$ . The only other such integers are $a = 1,2,3,7,11,19,43,67$ .

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2 Other occurrences
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[edit] $\sqrt{163}$

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

The function $f (n) = n 2 + n + 41$ gives prime values for all values of $n$ between 0 and 39, and for $n < 10 7$ approximately half of all values are prime. 163 appears as a result of solving $f (n) = 0$ , which gives $n = (-1+ \sqrt{-163} ) / 2$ .

$\sqrt{163}$ appears in the Ramanujan constant, in which $e^{\pi \sqrt{163}}$ 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... $32 l n (163)$ is also nearly integer, being equal to 163.000006425...