# 163 (number)

 ← 162 163 164 →
Cardinalone hundred and sixty-three
Ordinal163rd
(one hundred and sixty-third)
Factorizationprime
Prime38th
Divisors1, 163
Roman numeralCLXIII
Binary101000112
Ternary200013
Quaternary22034
Quinary11235
Senary4316
Octal2438
Duodecimal11712
Vigesimal8320
Base 364J36

 ← 162 163 164 →
Cardinalone hundred and sixty-three
Ordinal163rd
(one hundred and sixty-third)
Factorizationprime
Prime38th
Divisors1, 163
Roman numeralCLXIII
Binary101000112
Ternary200013
Quaternary22034
Quinary11235
Senary4316
Octal2438
Duodecimal11712
Vigesimal8320
Base 364J36

163 is the natural number following 162 and preceding 164.

## In mathematics

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 and a fortunate number.

163 is a strictly non-palindromic number. 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 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 $a = 1, 2, 3, 7, 11, 19, 43, 67$.

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 = 6403203 + 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...