View source for 163 (number)

{{About|the number|the internet services company|NetEase}}

{{Infobox number
| number = 163
| factorization=[[prime number|prime]]
| prime=38th
| divisor=1, 163
}}

'''163''' ('''one hundred [and] sixty-three''') is the [[natural number]] following [[162 (number)|162]] and preceding [[164 (number)|164]].

==In mathematics==
163 is the 38th prime number and a [[strong prime]] in the sense that it is greater than the arithmetic mean of its two neighboring primes.

163 is a [[lucky prime]]<ref>{{Cite OEIS|1=A031157|2=Numbers that are both lucky and prime|access-date=2016-05-28}}</ref> and a [[fortunate number]].<ref>{{Cite OEIS|1=A005235|2=Fortunate numbers|access-date=2016-05-28}}</ref>

163 is a strictly non-palindromic number, since it is not palindromic in any [[radix|base]] between [[binary number|base 2]] and base 161.

Given 163, the [[Mertens function]] returns 0, it is the fourth prime with this property, the first three such primes are [[2 (number)|2]], [[101 (number)|101]] and [[149 (number)|149]].<ref>{{Cite OEIS|1=A100669|2=Zeros of the Mertens function that are also prime|access-date=2016-05-29}}</ref>

As approximations, <math>\pi \approx {2^9 \over 163} \approx 3.1411...</math>, and <math>e \approx {163 \over 3\cdot4\cdot5} \approx 2.7166\dots</math>

163 is a [[permutable prime]] in [[duodecimal|base 12]], which it is written as 117, the [[permutation]]s of its [[Numerical digit|digits]] are 171 and 711, the two numbers in base 12 are [[229 (number)|229]] and [[1021 (number)|1021]] in base 10, both of which are prime.

The function <math>f(n) = n^2 - n + 41</math> gives prime values for all values of <math>n</math> between 0 and 39, while for <math>n < 3000</math> approximately half of all values are prime. 163 appears as a result of solving <math>f(n)=0</math>, which gives <math>n = (-1+ \sqrt{-163} ) / 2</math>.

163 is a [[Heegner number]], the largest of the nine such numbers. That is, the [[ring of integers]] of the [[field (mathematics)|field]] <math>\mathbb{Q}(\sqrt{-a})</math> has [[unique factorization domain|unique factorization]] for <math>a=163</math>. The only other such integers are
<math>a = 1, 2, 3, 7, 11, 19, 43, 67</math>. {{OEIS|id=A003173}}

163 is the number of linearly {{math|'''Z'''}}-independent McKay-Thompson series for the [[monster group]], which also represent their collective maximum [[dimension]]al representation. This fact about 163 might be a clue for understanding [[monstrous moonshine]].<ref>{{cite arXiv|author=He, Yang-Hui|author2=McKay, John|title=Sporadic and exceptional|eprint=1505.06742|year=2015|pages=1–49|class=math.AG}} (See p. 13)</ref>

<math>\sqrt{163}</math>
appears in the [[Ramanujan constant]], since -163 is a [[quadratic nonresidue]] to modulo all the primes 3, 5, 7, ..., 37. In which <math>e^{\pi \sqrt{163}}</math> almost equals the integer 262537412640768744 = 640320<sup>3</sup> + 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...

It also satisfies <math>\frac{163}{\log(163)}=32.99999873884...</math>.<ref>{{MathWorld|id=AlmostInteger|title=Almost integer|access-date=August 5, 2024}}</ref>

==In other fields==
'''163''' is also:
* The number of days following the first day of [[Passover]] (Pesach), used to calculate the date of [[Rosh Hashanah]],  the Jewish New Year.

==See also==
* [[163 (disambiguation)]]

==References==
{{reflist}}
* Wells, D. (1987). ''[[The Penguin Dictionary of Curious and Interesting Numbers]]'' (pp.&nbsp;141–142). London: Penguin Group.

==External links==
{{Commons category}}
* {{cite web|last=Clark|first=Alex|title=163 and Ramanujan Constant|url=http://www.numberphile.com/videos/163.html|work=Numberphile|publisher=[[Brady Haran]]|access-date=2013-04-02|archive-url=https://web.archive.org/web/20130516045906/http://www.numberphile.com/videos/163.html|archive-date=2013-05-16|url-status=dead}}
* [http://www.numdic.com/163 The Number 163]
* [http://www.positiveintegers.org/163 The Positive Integer 163]
* [http://primes.utm.edu/curios/page.php/163.html Prime curiosities: 163]
* [http://www.163.com 163.com] website from [[NetEase]]
* [http://homepages.umflint.edu/~mclemanc/index-4.html#Coolest The coolest number is 163]
* [https://web.archive.org/web/20110927094617/http://www.math.uic.edu/~ronan/163 163, the Monster and Number Theory]
* [http://www.virtuescience.com/163.html VirtueScience: 163] {{Webarchive|url=https://web.archive.org/web/20201127162707/https://www.virtuescience.com/163.html |date=2020-11-27 }}
* [https://web.archive.org/web/20081129024026/http://www.ibew.org/IBEW/directory/SearchDirectory/detailLU.asp?LocalUnion=163 International Brotherhood of Electrical Workers Local Union 163]
* [http://www.untraveledroad.com/Categories/Highways/USHighway163.htm US Highway 163]
* [https://math.stackexchange.com/q/609760 Other interesting consequences of ''d''=163?], from [[Stack Exchange|Mathematics Stack Exchange]]

{{Integers|1}}

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[[Category:Integers]]