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,

\pi \approx  \approx 3.1411...

, and

e \approx  \approx 2.7166\dots

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 ...

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

f(n) = n^2 - n + 41

gives prime values for all values of

n

between 0 and 39, while for

n < 3000

approximately half of all values are prime. 163 appears as a result of solving

f(n)=0

, which gives

n = (-1+ \sqrt ) / 2

. 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 ...

\mathbb(\sqrt)

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

a=163

. The only other such integers are

a = 1, 2, 3, 7, 11, 19, 43, 67

. 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, ...

\sqrt

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

e^

almost equals the integer 262537412640768744 = 640320³ + 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

\frac=31.999998738490...

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.

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 163

The Positive Integer 163

163.com
website 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 163

163, the Monster and Number Theory

International Brotherhood of Electrical Workers Local Union 163

Other interesting consequences of ''d''=163?
from Mathematics Stack Exchange {{DEFAULTSORT:163 (Number) Integers

In mathematics

In other fields

See also

References

External links