163 (one hundred
ndsixty-three) is the
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
following
162 and preceding
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 and a
fortunate number.
163 is a strictly non-palindromic number, since it is not palindromic in any
base between
base 2 and base 161.
Given 163, the
Mertens function
In number theory, the Mertens function is defined for all positive integers ''n'' as
: M(n) = \sum_^n \mu(k),
where \mu(k) is the Möbius function. The function is named in honour of Franz Mertens. This definition can be extended to positive r ...
returns 0, it is the fourth prime with this property, the first three such primes are
2,
101 and
149.
As approximations,
, and
163 is a
permutable prime
A permutable prime, also known as anagrammatic prime, is a prime number which, in a given radix, base, can have its digits' positions switched through any permutation and still be a prime number. H. E. Richert, who is supposedly the first to stu ...
in
base 12
The duodecimal system, also known as base twelve or dozenal, is a positional notation, positional numeral system using 12 (number), twelve as its radix, base. In duodecimal, the number twelve is denoted "10", meaning 1 twelve and 0 1, units; ...
, which it is written as 117, the
permutation
In mathematics, a permutation of a set can mean one of two different things:
* an arrangement of its members in a sequence or linear order, or
* the act or process of changing the linear order of an ordered set.
An example of the first mean ...
s of its
digits are 171 and 711, the two numbers in base 12 are
229
__NOTOC__
Year 229 ( CCXXIX) was a common year starting on Thursday of the Julian calendar. At the time, it was known as the Year of the Consulship of Severus and Cassius (or, less frequently, year 982 ''Ab urbe condita''). The denomination 22 ...
and
1021 in base 10, both of which are prime.
The function
gives prime values for all values of
between 0 and 39, while for
approximately half of all values are prime. 163 appears as a result of solving
, which gives
.
163 is a
Heegner number
In number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from int ...
, the largest of the nine such numbers. That is, the
ring of integers
In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often de ...
of the
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
has
unique factorization
In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is ...
for
. The only other such integers are
.
163 is the number of linearly -independent McKay-Thompson series for the
monster group
In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group; it has order
:
: = 2463205976112133171923293 ...
, which also represent their collective maximum
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
al representation. This fact about 163 might be a clue for understanding
monstrous moonshine
In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular the ''j'' function. The initial numerical observation was made by John McKay in 1978, ...
.
appears in the
Ramanujan constant, since -163 is a
quadratic nonresidue
In number theory, an integer ''q'' is a quadratic residue modulo ''n'' if it is congruent to a perfect square modulo ''n''; that is, if there exists an integer ''x'' such that
:x^2\equiv q \pmod.
Otherwise, ''q'' is a quadratic nonresidue modu ...
to modulo all the primes 3, 5, 7, ..., 37. In which
almost equals the integer 262537412640768744 = 640320
3 + 744.
Martin Gardner
Martin Gardner (October 21, 1914May 22, 2010) was an American popular mathematics and popular science writer with interests also encompassing magic, scientific skepticism, micromagic, philosophy, religion, and literatureespecially the writin ...
famously asserted that this identity was exact in a 1975 April Fools' hoax in ''
Scientific American
''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it, with more than 150 Nobel Pri ...
''; in fact the value is 262537412640768743.99999999999925007259...
It also satisfies
.
In other fields
163 is also:
* The number of days following the first day of
Passover
Passover, also called Pesach (; ), is a major Jewish holidays, Jewish holiday and one of the Three Pilgrimage Festivals. It celebrates the Exodus of the Israelites from slavery in Biblical Egypt, Egypt.
According to the Book of Exodus, God in ...
(Pesach), used to calculate the date of
Rosh Hashanah
Rosh Hashanah (, , ) is the New Year in Judaism. The Hebrew Bible, biblical name for this holiday is Yom Teruah (, , ). It is the first of the High Holy Days (, , 'Days of Awe"), as specified by Leviticus 23:23–25, that occur in the late summe ...
, the Jewish New Year.
See also
*
163 (disambiguation)
References
* Wells, D. (1987). ''
The Penguin Dictionary of Curious and Interesting Numbers
''The Penguin Dictionary of Curious and Interesting Numbers'' is a reference book for recreational mathematics and elementary number theory written by David Wells. The first edition was published in paperback by Penguin Books in 1986 in the UK, a ...
'' (pp. 141–142). London: Penguin Group.
External links
*
The Number 163The Positive Integer 163163.comwebsite from
NetEase
NetEase, Inc. () is a Chinese Internet technology company founded by Ding Lei in June 1997. It provides online services with content, community, communications, and commerce. The company develops and operates online PC and mobile games, adverti ...
The coolest number is 163163, the Monster and Number Theory
International Brotherhood of Electrical Workers Local Union 163Other interesting consequences of ''d''=163? from
Mathematics Stack Exchange
{{DEFAULTSORT:163 (Number)
Integers