
NUMBERS  LIFE IS GOVERNED BY NUMBERS YOU CREATE, WIN AND LOSE....because life is created and operated out of numbers..... Posted by Vishva News Reporter on September 23, 2008 
NUMBERS 
WITHOUT 0 & 1 THE ENTIRE COMPUTER SCIENCE WILL COLLAPSE.......
A number is an abstract object, tokens of which are symbols used in
counting and measuring. A symbol which represents a number is called a
numeral, but in common usage the word number is used for both the
abstract object and the symbol.
In mathematics, the definition of number has been extended over the
years to include such numbers as zero, negative numbers, rational
numbers, irrational numbers, and complex numbers. As a result, there is
no one encompassing definition of number and the concept of number is
open for further development.
The branch of mathematics that studies structures of number systems such
as groups, rings and fields is called abstract algebra.
The study of numerical operations is called arithmetic.
In mathematics, a prime number
(or a prime) is a natural number which has exactly two distinct natural
number divisors: 1 and itself.
An infinitude of prime numbers exists, as demonstrated by Euclid around
300 BC. The first twentyfive prime numbers are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67,
71, 73, 79, 83, 89, 97
There are hints in the surviving records of the ancient Egyptians that
they had some knowledge of prime numbers:
However, the earliest surviving records of the explicit study of prime
numbers come from the Ancient Greeks. Euclid's Elements (circa 300 BC)
contain important theorems about primes, including the infinitude of
primes and the fundamental theorem of arithmetic.

TODYAS NEWS STORY:
BIGGEST PRIME NUMBER DISCOVERY...12.9
million digits long...
If you read about 200 digits each minute, it would take 45 days to read
the number aloud.
This discovery of the biggest prime number was don by Webbased number
hunt run by a global group of math geeks, who've all downloaded software
on their computers that quietly and efficiently searches for massive,
"undiscovered" numbers. There are about 100,000 computers running on the
system, PrimeNet, combining for 30 trillion calculations per second.
The new largest prim number is bigger than its discovery because
prime numbers are useful "building blocks" to many equations, said
Cameron Stewart, a University of Waterloo professor and the Canada
Research Chair in Number Theory. There are some practical implications
in computer and security encryptions are based on prime numbers.....
For a long time, number theory in general, and the study of prime
numbers in particular, was seen as the canonical example of pure
mathematics, with no applications outside of the selfinterest of
studying the topic.
In particular, number theorists such as British mathematician G. H.
Hardy prided themselves on doing work that had absolutely no military
significance.
However, this vision was shattered in the 1970s, when it was publicly
announced that prime numbers could be used as the basis for the creation
of public key cryptography algorithms. Prime numbers are also used for
hash tables and pseudorandom number generators.

vED
OF NUMBERS....
NUMBERS ARE THE BASIC UNITS OF
GREED AND JELOUSY, JOY AND HATRED....
(contributed by Champaklal Dajibhai
Mistry of Edmonton, Alberta, Canada from his study and library of
SCIENCES OF LIFE AND CREATION called vED...) 
During the study of sciences of
vED through the simplest source of
vED knowledge
contained in the
sNskRUt language texts called
puraaAN
and
itihaas, one will find the following information
which shows the importance of numbers and prime numbers in life and
creation per se....
1. Creation of all life and time controlled life activities
...animate and inanimate.... has numbers as its basic ingredient similar
to numbers 0 and 1 in computer science....Numbering in
vED is a
metric system of 0 and 1 to 9 numbers and their compounding and
fractions based on mathematical rules....
2. Examples of above TRUTH are:
 Creator bRH`m
is describe as 1 or ONE (also in most belief system of today)....with
Creator's transformations into primary natural forces of creation,
sustenance of creation and cyclic absorption into the Creator and cyclic
recreations called shk`tio
called DEvtaao
(gods and goddesses of all belief systems) who number 3 prime, 33
subprime, and 330 million operational....
 Creator's creation cycle time of
chtuARyug in
pRUthiviilok
has number values of in the multiples of 1, 2, 3, 4....styyug,
teTRyug, D'vaapryug is 4, 3 and 2 times
respectively of the time duration of
kliyug.....
 each of the day and night time the duration of
bRHmaaday
in the cyclic creation system in our universe has a duration in
pRUthivii lok
is of 1000 chtuARyug...and
bRH`maa's
life duration is 100 years with each year of 360 ((1+1)x(1+1)x3x3x10)
such bRHmaaday.....Each
chtuARyug
translates to 4.320 million human years with each human years of 360
human days grouped into 12 (3x(1+1)x(1+1)) human months each of
which has 30 (3x10) human days each of which has 30 (3x10)
muHARt of
48 minutes each......
 mHaaBHaart war
was fought for 18 (3x3x(1+1)) days and
BHgvat giitaa which came about on the
first day of this war has 18 (3x3x(1+1)) chapters....
 In vED,
there are 18 (3x3x (1+1)
mHaapuraaAN, 18
uppuraaAN,
18 mHaaupnishD
of the total of 108 (3x3x3x(1+1)x(1+1)
upnishD
 3, 5, 7, 8 are numbers repeated in a creation processes
and to make whole processes from start to finish....SHRii
kRUSH`AN had 8 prime queens and 16,000
((1+1)x8x1000) wives who were liberated but were captive celestial women
suffering their paapi kARmfl of previous lifejourneys with
timefactored pardons .....SHRii
kRUSH`AN lived for 125 (5x5x5) years....SHRii
raam lived for 11,000 years....These numbers and
their multiples are seen in the life processes such as
bhaav, raag,
music, fasting periods, cyclic happening in solar and lunar time cycles
in jyotishshaasTR
(astrology and astronomy) ......
The above is just a sample which can be expanded into fascinating
numbers systems....with a hypothesis that life and creation is based on
mathematical equations of numbers....and/or anything in life can be
interpreted in mathematical equations similar to the reckoning that the
entire jyotishshaasTR
has mathematics as its foundation of forecasting of all
that moves and happens in life....

PRIME NUMBERS IN
NATURE AND USE BY HUAMITY 
Many numbers occur in nature including prime
numbers...but prime numbers appears have to draw more attention of
humans as researched below:
 prime numbers is used in nature is as an evolutionary strategy to
drive natural selection in favour of a primenumbered lifecycle for
insects to save themselves from their predators who evolve at a
nonprime number intervals.
 There is speculation that the zeros of the zeta function are connected
to the energy levels of complex quantum systems.
 Prime numbers have influenced many artists and writers. The French
composer Olivier Messiaen used prime numbers to create ametrical music
through "natural phenomena". According to Messiaen this way of composing
was "inspired by the movements of nature, movements of free and unequal
durations".

 In his science fiction novel Contact, later made into a film of
the same name, the NASA scientist Carl Sagan suggested that prime
numbers could be used as a means of communicating with aliens, an idea
that he had first developed informally with American astronomer Frank
Drake in 1975.
 Tom Stoppard's awardwinning 1993 play Arcadia was a conscious attempt
to discuss mathematical ideas on the stage. In the opening scene, the 13
year old heroine puzzles over Fermat's Last Theorem, a theorem involving
prime numbers.
 Many films reflect a popular fascination with the mysteries of prime
numbers and cryptography: films such as Cube, Sneakers, The Mirror Has
Two Faces and A Beautiful Mind, based on the biography of the
mathematician and Nobel laureate John Forbes Nash by Sylvia Nasar. 
To increase your knowledge of NUMBERS
AND PRIME NUMBERS please click on the next line through today's
news about discovery of BIGGEST PRIME
NUMBER and the backgrouders on numbers and prime numbers.......

A quick prime primer for the news story (from
canadian globe and mail):
What's a
prime?
Prime numbers are divisible only by themselves and the number one. Basic
examples include the numbers three, five and seven.
What's a
Mersenne prime?
Named for Marin Mersenne, Mersenne primes are found by the formula 2^P,
minus one, where P is another prime. All Mersenne primes are prime, but
not all primes are Mersenne primes.
What
number was found & WHEN?
2^43,112,609, minus one. It's 12.9 million digits long. If you read
about 200 digits each minute, it would take 45 days to read the number
aloud.
Discovered Aug. 23. Canadian Jeff Gilchrist began doublechecking it on
Aug. 26, a process that took 16 days.
What's
the big deal?
It's the biggest prime ever found, a sort of mathematical equivalent of
a worldrecord sprint or an angler's prizewinning catch.
What
now?
They qualify for a $100,000 prize. Half will go to UCLA (which owns the
computers that found it); a quarter to charity; and a quarter to other
discoverers. Mr. Gilchrist's take: $0. The same group won a $50,000
prize after discovering a smaller prime in 1999.
What
next?
The search continues. There's $150,000 waiting for anyone that can
discover a Mersenne more than 100 million digits long.
And now continue
reading the largest prime number discovery reported in Canadian Globe
and Mail......

Experts
cheer math geeks' primal scheme....
Historic prime number found by
global group of number enthusiasts
......is a digital dream for the math set....
Canadian
Globe and Mail:
September 23, 2008: Josh Wingrove
At the top of the heap sit the algorithmists, who've got the most street
cred among math aficionados.
Below them, you'll find the computer programmers and the professors, the
engineers and the retirees, and the generally numberliterate. But for
more than a decade, they've each suited up for an aroundtheclock bit
of "recreational math"  the Great Internet Mersenne Prime Search.
It's a broad, Webbased number hunt run by a global group of math geeks,
who've all downloaded software on their computers that quietly and
efficiently searches for massive, "undiscovered" numbers. There are
about 100,000 computers running on the system, PrimeNet, combining for
30 trillion calculations per second.
It may not be sexy, but it's fruitful. Last week, the hunters announced
they'd bagged a big one: a prime number 12.9 million digits long, the
largest ever proven. And it was the ragtag group of anonymous math
hobbyists who, number by number, inched toward the academic achievement.
"There's a very tightknit crowd of the few experts, and these are the
algorithm folks," explains San Diegobased Scott Kurowski, who developed
the PrimeNet server. "Then there's the very significant larger crowd of
people who are just enthusiasts."
The algorithmists set up the software and guide the Web process, Mr.
Kurowski said, like division commanders of sorts. Then, the enthusiasts
chip in where they can, even if it means simply downloading the software
on an underused work computer.
"The idea is you take a very difficult problem to solve, you break it up
into smaller pieces, and hand it out to people around the world,"
explains Jeff Gilchrist, a 32yearold Canadian whose duty it was to
verify the prime number discovery.

It's that distributive computing, not the find itself, that catches
the eyes of math professors. "The actual number isn't as exciting as the
process that went into finding it," said Kevin Hare, assistant professor
of pure mathematics at the University of Waterloo, calling the find an
"incredible achievement."
"Being on the moon isn't as impressive as getting there."
While perhaps not math experts, the hobbyists who chip in know prime
from perfect. Mr. Gilchrist is completing his PhD in systems and
computer engineering at Carleton University, looking at how to crunch
patient data to spot outbreaks at an Ottawaarea children's hospital.
Add to that two young boys of his own, aged one and three, and he's got
plenty keeping him busy. The prime number stuff, as with thousands of
his PrimeNet colleagues, is pure sport  their slogan is "Serious
research. Totally for fun."
"It's more like shaving a second off a world record," Mr. Gilchrist
said. "It's sort of nice to have bragging rights, that your achievements
are larger than anyone else's."
The new number is little more than that. Prime numbers are useful
"building blocks" to many equations, but using existing algorithms to
find new, large primes won't likely affect ongoing research, said
Cameron Stewart, a University of Waterloo professor and the Canada
Research Chair in Number Theory. There are some practical implications
(computer and security encryptions are based on prime numbers), but the
find is more sport.
"They're a good tool. They're also mysterious; they're subtle objects
..." Prof. Stewart said of prime numbers.
It was Edson Smith, a computer resource manager at the University of
California Los Angeles, who found the number on Aug. 23, but he had to
keep his mouth shut until Mr. Gilchrist confirmed it. He didn't have a
problem with the gag order. "It's pretty easy not to tell someone a 12.9
million digit number," he said. "Plus, they don't know what I'm talking
about, anyway."
So, the celebrations from last week's find will be confined to the math
set.
"Maybe some day it can be used for something that can be used for
something else, but for now it's just pure knowledge," said Mr. Smith.
"People do it because they like it." 
And now to
understand what is a NUMBER please continue reading below followed by
the reading on indepth understanding of the PRIME NUMBER which is the
topic of this news item..... 
Number
from
WIKIPEDIA
 FREE ENCYCLOPEDIA
From Wikipedia, the free encyclopedia
For other uses of "number", see
numbers.
A number is an
abstract object,
tokens of which are
symbols
used in
counting and
measuring. A symbol which represents a number is called a
numeral, but in common usage the word number is used for both
the abstract object and the symbol. In addition to their use in
counting and measuring, numerals are often used for labels (telephone
numbers), for ordering (serial
numbers), and for codes (ISBNs).
In
mathematics, the definition of number has been extended over the
years to include such numbers as
zero,
negative numbers,
rational numbers,
irrational numbers, and
complex numbers. As a result, there is no one encompassing
definition of number and the concept of number is open for further
development.
Certain procedures which input one or more numbers and output a
number are called numerical
operations.
Unary operations input a single number and output a single
number. For example, the successor operation adds one to an integer:
the successor of 4 is 5. More common are
binary operations which input two numbers and output a single
number. Examples of binary operations include
addition,
subtraction,
multiplication,
division, and
exponentiation. The study of numerical operations is called
arithmetic.
The branch of
mathematics that studies structures of number systems such as
groups,
rings and
fields is called
abstract algebra.
[edit]
Types of numbers
Numbers can be classified into
sets, called
number systems. (For different methods of expressing numbers
with symbols, such as the
Roman numerals, see
numeral systems.)
[edit]
Natural numbers
The most familiar numbers are the
natural numbers or counting numbers: one, two, three, ... .
In the
base ten number system, in almost universal use today for
arithmetic operations, the symbols for natural numbers are written
using ten
digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. In this base ten
system, the rightmost digit of a natural number has a place value of
one, and every other digit has a place value ten times that of the
place value of the digit to its right. The symbol for the set of all
natural numbers is N, also written
.
In
set theory, which is capable of acting as an axiomatic
foundation for modern mathematics, natural numbers can be
represented by classes of equivalent sets. For instance, the number
3 can be represented as the class of all sets that have exactly
three elements. Alternatively, in
Peano Arithmetic, the number 3 is represented as sss0, where s
is the "successor" function. Many different representations are
possible; all that is needed to formally represent 3 is to inscribe
a certain symbol or pattern of symbols 3 times.
[edit]
Integers
Negative numbers are numbers that are less than zero. They
are the opposite of positive numbers. For example, if a positive
number indicates a bank deposit, then a negative number indicates a
withdrawal of the same amount. Negative numbers are usually written
by writing a negative sign (also called a minus sign) in front of
the number they are the opposite of. Thus the opposite of 7 is
written 7. When the
set of negative numbers is combined with the natural numbers and
zero, the result is the set of integer numbers, also called
integers, Z (German Zahl, plural Zahlen),
also written
.
[edit]
Rational numbers
A
rational number is a number that can be expressed as a
fraction with an integer
numerator and a nonzero natural number
denominator. The fraction m/n or

represents m equal parts, where n equal parts of
that size make up one whole. Two different fractions may correspond
to the same rational number; for example 1/2 and 2/4 are equal, that
is:

If the
absolute value of m is greater than n, then the
absolute value of the fraction is greater than 1. Fractions can be
greater than, less than, or equal to 1 and can also be positive,
negative, or zero. The set of all rational numbers includes the
integers, since every integer can be written as a fraction with
denominator 1. For example 7 can be written 7/1. The symbol for
the rational numbers is Q (for
quotient), also written
.
[edit]
Real numbers
The
real numbers include all of the measuring numbers. Real
numbers are usually written using
decimal numerals, in which a decimal point is placed to the
right of the digit with place value one. Each digit to the right of
the decimal point has a place value onetenth of the place value of
the digit to its left. Thus

represents 1 hundred, 2 tens, 3 ones, 4 tenths, 5 hundredths, and
6 thousandths. In saying the number, the decimal is read "point",
thus: "one two three point four five six". In the US and UK and a
number of other countries, the decimal point is represented by a
period, whereas in continental Europe and certain other
countries the decimal point is represented by a
comma. Zero is often written as 0.0 when necessary to indicate
that it is to be treated as a real number rather than as an integer.
Negative real numbers are written with a preceding
minus sign:

.
Every rational number is also a real number. To write a fraction
as a decimal, divide the numerator by the denominator. It is not the
case, however, that every real number is rational. If a real number
cannot be written as a fraction of two integers, it is called
irrational. A decimal that can be written as a fraction either
ends (terminates) or forever repeats, because it is the answer to a
problem in division. Thus the real number 0.5 can be written as 1/2
and the real number 0.333... (forever repeating threes) can be
written as 1/3. On the other hand, the real number p (pi),
the ratio of the
circumference of any circle to its
diameter, is

.
Since the decimal neither ends nor forever repeats, it cannot be
written as a fraction, and is an example of an irrational number.
Other irrational numbers include

(the
square root of 2, that is, the positive number whose square is
2).
Just as fractions can be written in more than one way, so too can
decimals. For example, if we multiply both sides of the equation

by three, we discover that

.
Thus 1.0 and
0.999... are two different decimal numerals representing the
natural number 1. There are infinitely many other ways of
representing the number 1, for example 2/2, 3/3, 1.00, 1.000, and so
on.
Every real number is either rational or irrational. Every real
number corresponds to a point on the
number line. The real numbers also have an important but highly
technical property called the
least upper bound property. The symbol for the real numbers is
R or
.
When a real number represents a
measurement, there is always a
margin of error. This is often indicated by
rounding or
truncating a decimal, so that digits that suggest a greater
accuracy than the measurement itself are removed. The remaining
digits are called
significant digits. For example, measurements with a ruler can
seldom be made without a margin of error of at least 0.01 meters. If
the sides of a
rectangle are measured as 1.23 meters and 4.56 meters, then
multiplication gives an area for the rectangle of 5.6088 square
meters. Since only the first two digits after the decimal place are
significant, this is usually rounded to 5.61.
In
abstract algebra, the real numbers are up to isomorphism
uniquely characterized by being the only
complete
ordered field. They are not, however, an
algebraically closed field.
[edit]
Complex numbers
Moving to a greater level of abstraction, the real numbers can be
extended to the
complex numbers. This set of numbers arose, historically,
from the question of whether a negative number can have a
square root. This led to the invention of a new number: the
square root of negative one, denoted by
i, a symbol assigned by
Leonhard Euler, and called the
imaginary unit. The complex numbers consist of all numbers of
the form

where a and b are real numbers. In the expression
a + bi, the real number a is called the real
part and bi is called the imaginary part. If the
real part of a complex number is zero, then the number is called an
imaginary number or is referred to as purely imaginary;
if the imaginary part is zero, then the number is a real number.
Thus the real numbers are a
subset
of the complex numbers. If the real and imaginary parts of a complex
number are both integers, then the number is called a
Gaussian integer. The symbol for the complex numbers is C
or
.
In
abstract algebra, the complex numbers are an example of an
algebraically closed field, meaning that every
polynomial with complex
coefficients can be
factored into linear factors. Like the real number system, the
complex number system is a
field and is
complete, but unlike the real numbers it is not
ordered. That is, there is no meaning in saying that i is
greater than 1, nor is there any meaning in saying that that i
is less than 1. In technical terms, the complex numbers lack the
trichotomy property.
Complex numbers correspond to points on the
complex plane, sometimes called the Argand plane.
Each of the number systems mentioned above is a
proper subset of the next number system. Symbolically, N
? Z ? Q ? R ? C.
[edit]
Computable numbers
Moving to problems of computation, the
computable numbers are determined in the set of the real
numbers. The computable numbers, also known as the recursive
numbers or the computable reals, are the
real numbers that can be computed to within any desired
precision by a finite, terminating
algorithm. Equivalent definitions can be given using
µrecursive functions,
Turing machines or
?calculus as the formal representation of algorithms. The
computable numbers form a
real closed field and can be used in the place of real numbers
for many, but not all, mathematical purposes.
[edit]
Other types
Hyperreal and hypercomplex numbers are used in
nonstandard analysis. The hyperreals, or nonstandard reals
(usually denoted as *R), denote an
ordered field which is a proper
extension of the ordered field of
real numbers R and which satisfies the
transfer principle. This principle allows true
first order statements about R to be reinterpreted as
true first order statements about *R.
Superreal and
surreal numbers extend the real numbers by adding
infinitesimally small numbers and infinitely large numbers, but
still form
fields.
The idea behind
padic numbers is this: While real numbers may have infinitely
long expansions to the right of the decimal point, these numbers
allow for infinitely long expansions to the left. The number system
which results depends on what
base
is used for the digits: any base is possible, but a system with the
best mathematical properties is obtained when the base is a
prime number.
For dealing with infinite collections, the natural numbers have
been generalized to the
ordinal numbers and to the
cardinal numbers. The former gives the ordering of the
collection, while the latter gives its size. For the finite set, the
ordinal and cardinal numbers are equivalent, but they differ in the
infinite case.
There are also other sets of numbers with specialized uses. Some
are subsets of the complex numbers. For example,
algebraic numbers are the roots of
polynomials with rational
coefficients. Complex numbers that are not algebraic are called
transcendental numbers.
Sets of numbers that are not subsets of the complex numbers are
sometimes called
hypercomplex numbers. They include the
quaternions H, invented by Sir
William Rowan Hamilton, in which multiplication is not
commutative, and the
octonions, in which multiplication is not
associative. Elements of
function fields of nonzero
characteristic behave in some ways like numbers and are often
regarded as numbers by number theorists.
In addition, various specific kinds of numbers are studied in
sets of
natural and
integer numbers.
An even number is an integer that is "evenly divisible" by
2, i.e., divisible by 2 without remainder; an odd number is
an integer that is not evenly divisible by 2. (The oldfashioned
term "evenly divisible" is now almost always shortened to "divisible".)
A formal definition of an odd number is that it is an integer of the
form n = 2k + 1, where k is an integer. An even
number has the form n = 2k where k is an
integer.
A perfect number is defined as a
positive integer which is the sum of its proper positive
divisors, that is, the sum of the positive divisors not
including the number itself. Equivalently, a perfect number is a
number that is half the sum of all of its positive divisors, or
s(n) = 2 n. The first perfect number is
6, because 1, 2, and 3 are its proper positive divisors and
1 + 2 + 3 = 6. The next perfect number is
28 = 1 + 2 + 4 + 7 + 14. The next perfect numbers are
496 and
8128 (sequence
A000396 in
OEIS). These first four perfect numbers were the only ones known
to early
Greek mathematics.
A figurate number is a number that can be represented as a
regular and discrete
geometric pattern (e.g. dots). If the pattern is
polytopic, the figurate is labeled a polytopic number,
and may be a
polygonal number or a polyhedral number. Polytopic
numbers for r = 2, 3, and 4 are:
[edit]
Numerals
Numbers should be distinguished from
numerals, the symbols used to represent numbers. Boyer
showed that Egyptians created the first ciphered numeral system.
Greeks followed by mapping their counting numbers onto Ionian and
Doric alpabets. The number five can be represented by both the base
ten numeral '5', by the
Roman numeral 'V' and ciphered letters. Notations used to
represent numbers are discussed in the article
numeral systems. An important development in the history of
numerals was the development of a positional system, like modern
decimals, which can represent very large numbers. The Roman numerals
require extra symbols for larger numbers.
[edit]
History
[edit]
History of integers
[edit]
The first use of numbers
It is speculated that the first known use of numbers dates back
to around 30,000 BC. Bones and other artifacts have been discovered
with marks cut into them which many consider to be
tally marks. The uses of these tally marks may have been for
counting elapsed time, such as numbers of days, or keeping records
of quantities, such as of animals.
Tallying systems have no concept of placevalue (such as in the
currently used decimal notation), which limit its representation of
large numbers and as such is often considered that this is the first
kind of abstract system that would be used, and could be considered
a Numeral System.
The first known system with placevalue was the
Mesopotamian base 60 system (ca. 3400 BC) and the earliest known
base 10 system dates to 3100 BC in
Egypt.
[1]
[edit]
History of zero
 Further information:
History of zero
The use of zero as a number should be distinguished from its use
as a placeholder numeral in
placevalue systems. Many ancient texts used zero. Babylonians
and Egyptian texts used it. Egyptians used the word nfr to denote
zero balance in double entry accounting entries. Indian texts used a
Sanskrit word Shunya to refer to the concept of void;
in mathematics texts this word would often be used to refer to the
number zero.
[2]. In a similar vein,
Pa?ini
(5th century BC) used the null (zero) operator (ie a
lambda production) in the
Ashtadhyayi, his
algebraic grammar for the
Sanskrit language. (also see
Pingala)
Records show that the
Ancient Greeks seemed unsure about the status of zero as a
number: they asked themselves "how can 'nothing' be something?"
leading to interesting
philosophical and, by the Medieval period, religious arguments
about the nature and existence of zero and the
vacuum.
The
paradoxes of
Zeno of Elea depend in large part on the uncertain
interpretation of zero. (The ancient Greeks even questioned if
1 was a number.)
The late
Olmec
people of southcentral
Mexico
began to use a true zero (a shell glyph) in the New World possibly
by the 4th century BC but certainly by 40 BC, which became an
integral part of
Maya numerals and the
Maya calendar. Mayan arithmetic used base 4 and base 5 written
as base 20. Sanchez in 1961 reported a base 4, base 5 'finger'
abacus.
By 130,
Ptolemy, influenced by
Hipparchus and the Babylonians, was using a symbol for zero (a
small circle with a long overbar) within a sexagesimal numeral
system otherwise using alphabetic
Greek numerals. Because it was used alone, not as just a
placeholder, this
Hellenistic zero was the first documented use of a true
zero in the Old World. In later
Byzantine manuscripts of his Syntaxis Mathematica (Almagest),
the Hellenistic zero had morphed into the
Greek letter
omicron (otherwise meaning 70).
Another true zero was used in tables alongside
Roman numerals by 525 (first known use by
Dionysius Exiguus), but as a word, nulla meaning
nothing, not as a symbol. When division produced zero as a
remainder, nihil, also meaning nothing, was used.
These medieval zeros were used by all future medieval
computists (calculators of
Easter).
An isolated use of their initial, N, was used in a table of Roman
numerals by
Bede or a colleague about 725, a true zero symbol.
An early documented use of the zero by
Brahmagupta (in the
Brahmasphutasiddhanta) dates to 628. He treated zero as a number
and discussed operations involving it, including
division. By this time (7th century) the concept had clearly
reached
Cambodia, and documentation shows the idea later spreading to
China
and the
Islamic world.
[edit]
History of negative numbers
 Further information:
First usage of negative numbers
The abstract concept of negative numbers was recognised as early
as 100 BC  50 BC. The
Chinese
”Nine
Chapters on the Mathematical Art” (Jiuzhang Suanshu)
contains methods for finding the areas of figures; red rods were
used to denote positive
coefficients, black for negative. This is the earliest known
mention of negative numbers in the East; the first reference in a
western work was in the 3rd century in
Greece.
Diophantus referred to the equation equivalent to
4x + 20 = 0 (the
solution would be negative) in
Arithmetica, saying that the equation gave an absurd result.
During the 600s, negative numbers were in use in
India
to represent debts.
Diophantus’ previous reference was discussed more explicitly by
Indian mathematician
Brahmagupta, in
BrahmaSphutaSiddhanta 628, who used negative numbers to
produce the general form
quadratic formula that remains in use today. However, in the
12th century in India,
Bhaskara gives negative roots for quadratic equations but says
the negative value "is in this case not to be taken, for it is
inadequate; people do not approve of negative roots."
European mathematicians, for the most part, resisted the concept
of negative numbers until the 17th century, although
Fibonacci allowed negative solutions in financial problems where
they could be interpreted as debits (chapter 13 of
Liber Abaci, 1202) and later as losses (in Flos). At
the same time, the
Chinese
were indicating negative numbers by drawing a diagonal stroke
through the rightmost nonzero digit of the corresponding positive
number's numeral^{[citation
needed]}. The first use of negative numbers in a
European work was by
Chuquet during the 15th century. He used them as
exponents, but referred to them as “absurd numbers”.
As recently as the 18th century, the
Swiss mathematician
Leonhard Euler believed that negative numbers were greater than
infinity^{[citation
needed]}, and it was common practice to ignore
any negative results returned by equations on the assumption that
they were meaningless, just as
René Descartes did with negative solutions in a
cartesian coordinate system.
[edit]
History of rational, irrational, and real
numbers
 Further information:
History of irrational numbers and
History of pi
[edit]
History of rational numbers
It is likely that the concept of fractional numbers dates to
prehistoric times. Even the
Ancient Egyptians wrote math texts describing how to convert
general
fractions into their
special notation. The
RMP 2/n table and the
Kahun Papyrus wrote out unit fraction series by using least
common multiples. Classical Greek and Indian mathematicians made
studies of the theory of rational numbers, as part of the general
study of
number theory. The best known of these is
Euclid's Elements, dating to roughly 300 BC. Of the Indian
texts, the most relevant is the
Sthananga Sutra, which also covers number theory as part of a
general study of mathematics.
The concept of
decimal fractions is closely linked with decimal place value
notation; the two seem to have developed in tandem. For example, it
is common for the Jain math sutras to include calculations of
decimalfraction approximations to
pi or the
square root of two. Similarly, Babylonian math texts had always
used sexagesimal fractions with great frequency.
[edit]
History of irrational numbers
The earliest known use of irrational numbers was in the
Indian
Sulba Sutras composed between 800500 BC.^{[citation
needed]} The first existence proofs of
irrational numbers is usually attributed to
Pythagoras, more specifically to the
Pythagorean
Hippasus of Metapontum, who produced a (most likely geometrical)
proof of the irrationality of the
square root of 2. The story goes that Hippasus discovered
irrational numbers when trying to represent the square root of 2 as
a fraction. However
Pythagoras believed in the absoluteness of numbers, and could
not accept the existence of irrational numbers. He could not
disprove their existence through logic, but his beliefs would not
accept the existence of irrational numbers and so he sentenced
Hippasus to death by drowning.
The sixteenth century saw the final acceptance by Europeans of
negative, integral and
fractional numbers. The seventeenth century saw decimal
fractions with the modern notation quite generally used by
mathematicians. But it was not until the nineteenth century that the
irrationals were separated into algebraic and transcendental parts,
and a scientific study of theory of irrationals was taken once more.
It had remained almost dormant since
Euclid.
The year 1872 saw the publication of the theories of
Karl Weierstrass (by his pupil
Kossak),
Heine (Crelle,
74),
Georg Cantor (Annalen, 5), and
Richard Dedekind.
Méray had taken in 1869 the same point of departure as
Heine, but the theory is generally referred to the year 1872.
Weierstrass's method has been completely set forth by
Salvatore Pincherle (1880), and Dedekind's has received
additional prominence through the author's later work (1888) and the
recent endorsement by
Paul Tannery (1894). Weierstrass, Cantor, and Heine base their
theories on infinite series, while Dedekind founds his on the idea
of a
cut (Schnitt) in the system of
real numbers, separating all
rational numbers into two groups having certain characteristic
properties. The subject has received later contributions at the
hands of Weierstrass,
Kronecker (Crelle, 101), and Méray.
Continued fractions, closely related to irrational numbers (and
due to Cataldi, 1613), received attention at the hands of
Euler, and at the opening of the nineteenth century were brought
into prominence through the writings of
Joseph Louis Lagrange. Other noteworthy contributions have been
made by
Druckenmüller (1837),
Kunze
(1857),
Lemke (1870), and
Günther (1872).
Ramus
(1855) first connected the subject with
determinants, resulting, with the subsequent contributions of
Heine,
Möbius, and
Günther, in the theory of Kettenbruchdeterminanten. Dirichlet
also added to the general theory, as have numerous contributors to
the applications of the subject.
[edit]
Transcendental numbers and reals
The first results concerning transcendental numbers were
Lambert's 1761 proof that p cannot be rational, and also that
e^{n} is irrational if n is rational
(unless n = 0). (The constant
e was first referred to in
Napier's 1618 work on
logarithms.)
Legendre extended this proof to showed that p is not the square
root of a rational number. The search for roots of
quintic and higher degree equations was an important
development, the
Abel–Ruffini theorem (Ruffini
1799,
Abel 1824) showed that they could not be solved by
radicals (formula involving only arithmetical operations and
roots). Hence it was necessary to consider the wider set of
algebraic numbers (all solutions to polynomial equations).
Galois (1832) linked polynomial equations to
group theory giving rise to the field of
Galois theory.
Even the set of algebraic numbers was not sufficient and the full
set of real number includes
transcendental numbers. The existence of which was first
established by
Liouville (1844, 1851).
Hermite proved in 1873 that
e is transcendental and
Lindemann proved in 1882 that p is transcendental. Finally
Cantor shows that the set of all
real numbers is
uncountably infinite but the set of all
algebraic numbers is
countably infinite, so there is an uncountably infinite number
of transcendental numbers.
[edit]
Infinity
 Further information:
History of infinity
The earliest known conception of mathematical
infinity appears in the
Yajur Veda  an ancient script in India, which at one point
states "if you remove a part from infinity or add a part to
infinity, still what remains is infinity". Infinity was a popular
topic of philosophical study among the
Jain mathematicians circa 400 BC. They distinguished between
five types of infinity: infinite in one and two directions, infinite
in area, infinite everywhere, and infinite perpetually.
In the West, the traditional notion of mathematical infinity was
defined by
Aristotle, who distinguished between
actual infinity and
potential infinity; the general consensus being that only the
latter had true value.
Galileo's
Two New Sciences discussed the idea of
onetoone correspondences between infinite sets. But the next
major advance in the theory was made by
Georg Cantor; in 1895 he published a book about his new
set theory, introducing, among other things,
transfinite numbers and formulating the
continuum hypothesis. This was the first mathematical model that
represented infinity by numbers and gave rules for operating with
these infinite numbers.
In the 1960s,
Abraham Robinson showed how infinitely large and infinitesimal
numbers can be rigorously defined and used to develop the field of
nonstandard analysis. The system of hyperreal numbers
represents a rigorous method of treating the ideas about
infinite and
infinitesimal numbers that had been used casually by
mathematicians, scientists, and engineers ever since the invention
of
calculus by
Newton and
Leibniz.
A modern geometrical version of infinity is given by
projective geometry, which introduces "ideal points at
infinity," one for each spatial direction. Each family of parallel
lines in a given direction is postulated to converge to the
corresponding ideal point. This is closely related to the idea of
vanishing points in
perspective drawing.
[edit]
Complex numbers
 Further information:
History of complex numbers
The earliest fleeting reference to square roots of negative
numbers occurred in the work of the mathematician and inventor
Heron of Alexandria in the 1st century AD, when he considered
the volume of an impossible
frustum of a
pyramid. They became more prominent when in the 16th century
closed formulas for the roots of third and fourth degree polynomials
were discovered by Italian mathematicians (see
Niccolo Fontana Tartaglia,
Gerolamo Cardano). It was soon realized that these formulas,
even if one was only interested in real solutions, sometimes
required the manipulation of square roots of negative numbers.
This was doubly unsettling since they did not even consider
negative numbers to be on firm ground at the time. The term
"imaginary" for these quantities was coined by
René Descartes in 1637 and was meant to be derogatory (see
imaginary number for a discussion of the "reality" of complex
numbers). A further source of confusion was that the equation

seemed to be capriciously inconsistent with the algebraic
identity

which is valid for positive real numbers a and b,
and which was also used in complex number calculations with one of
a, b positive and the other negative. The incorrect
use of this identity, and the related identity

in the case when both a and b are negative even
bedeviled
Euler. This difficulty eventually led him to the convention of
using the special symbol i in place of v1 to guard against
this mistake.
The 18th century saw the labors of
Abraham de Moivre and
Leonhard Euler. To De Moivre is due (1730) the wellknown
formula which bears his name,
de Moivre's formula:

and to Euler (1748)
Euler's formula of
complex analysis:

The existence of complex numbers was not completely accepted
until the geometrical interpretation had been described by
Caspar Wessel in 1799; it was rediscovered several years later
and popularized by
Carl Friedrich Gauss, and as a result the theory of complex
numbers received a notable expansion. The idea of the graphic
representation of complex numbers had appeared, however, as early as
1685, in
Wallis's De Algebra tractatus.
Also in 1799, Gauss provided the first generally accepted proof
of the
fundamental theorem of algebra, showing that every polynomial
over the complex numbers has a full set of solutions in that realm.
The general acceptance of the theory of complex numbers is not a
little due to the labors of
Augustin Louis Cauchy and
Niels Henrik Abel, and especially the latter, who was the first
to boldly use complex numbers with a success that is well known.
Gauss studied
complex numbers of the form a + bi, where a
and b are integral, or rational (and i is one of the
two roots of x^{2} + 1 = 0). His student,
Ferdinand Eisenstein, studied the type a + b?,
where ? is a complex root of x^{3}  1 = 0.
Other such classes (called
cyclotomic fields) of complex numbers are derived from the
roots of unity x^{k}  1 = 0 for higher
values of k. This generalization is largely due to
Ernst Kummer, who also invented
ideal numbers, which were expressed as geometrical entities by
Felix Klein in 1893. The general theory of fields was created by
Évariste Galois, who studied the fields generated by the roots
of any polynomial equation F(x) = 0.
In 1850
Victor Alexandre Puiseux took the key step of distinguishing
between poles and branch points, and introduced the concept of
essential singular points; this would eventually lead to the
concept of the
extended complex plane.
[edit]
Prime numbers
Prime numbers have been studied throughout recorded history.
Euclid devoted one book of the Elements to the theory of
primes; in it he proved the infinitude of the primes and the
fundamental theorem of arithmetic, and presented the
Euclidean algorithm for finding the
greatest common divisor of two numbers.
In 240 BC,
Eratosthenes used the
Sieve of Eratosthenes to quickly isolate prime numbers. But most
further development of the theory of primes in Europe dates to the
Renaissance and later eras.
In 1796,
AdrienMarie Legendre conjectured the
prime number theorem, describing the asymptotic distribution of
primes. Other results concerning the distribution of the primes
include Euler's proof that the sum of the reciprocals of the primes
diverges, and the
Goldbach conjecture which claims that any sufficiently large
even number is the sum of two primes. Yet another conjecture related
to the distribution of prime numbers is the
Riemann hypothesis, formulated by
Bernhard Riemann in 1859. The prime number theorem was finally
proved by
Jacques Hadamard and
Charles de la ValléePoussin in 1896. The conjectures of
Goldbach and Riemann yet remain to be proved or refuted.
[edit]
Word alternatives
Some numbers traditionally have alternative words to express
them, including the following:
[edit]
See also
[edit]
References

Tobias Dantzig, Number, the language of science; a
critical survey written for the cultured nonmathematician,
New York, The Macmillan company, 1930.

Erich Friedman,
What's special about this number?

Steven Galovich, Introduction to
Mathematical Structures, Harcourt Brace Javanovich,
23 January
1989,
ISBN 0155434683.

Paul Halmos, Naive Set Theory, Springer, 1974,
ISBN 0387900926.

Morris Kline, Mathematical Thought from Ancient to Modern
Times, Oxford University Press, 1972.

Alfred North Whitehead and
Bertrand Russell,
Principia Mathematica to *56, Cambridge University
Press, 1910.
 George I. Sanchez, Arithmetic in Maya,AustinTexas, 1961.

What's a Number? at
cuttheknot
[edit]
External links

And now please
continue reading below the current indepth understandning of PRIME
NUMBERS...... 
Prime number
From Wikipedia, the free encyclopedia
FROM
WIKIPEDIA
THE FREE ENCYCLOPEDIA
In
mathematics, a prime number (or a prime) is a
natural number which has exactly two distinct natural
number
divisors:
1 and itself. An infinitude of prime numbers exists, as
demonstrated by
Euclid
around
300 BC. The first twentyfive prime numbers are:

2,
3,
5,
7,
11,
13,
17,
19,
23,
29,
31,
37,
41,
43,
47,
53,
59,
61,
67,
71,
73,
79,
83,
89,
97 (sequence
A000040 in
OEIS).
See the
list of prime numbers for a longer list. The number one
is by definition not a prime number; see the discussion below under
Primality of one.
The property of being a prime is called primality, and the
word prime is also used as an adjective. Since two is the
only even prime number, the term odd prime refers to any
prime number greater than two.
The study of prime numbers is part of
number theory, the branch of mathematics which encompasses the
study of natural numbers. Prime numbers have been the subject of
intense research, yet some fundamental questions, such as the
Riemann hypothesis and the
Goldbach conjecture, have been unresolved for more than a
century. The problem of modelling the distribution of prime numbers
is a popular subject of investigation for number theorists: when
looking at individual numbers, the primes seem to be randomly
distributed, but the “global” distribution of primes follows
welldefined laws.
The notion of prime number has been generalized in many different
branches of mathematics.
 In
ring theory, a branch of
abstract algebra, the term “prime
element” has a specific meaning. Here, a nonzero, nonunit
ring element a is defined to be prime if whenever a
divides bc for ring elements b and c, then
a divides at least one of b or c. With this
meaning, the additive inverse of any prime number is also prime.
In other words, when considering the set of
integers as a
ring, 7 is a prime element. Without further specification,
however, “prime number” always means a positive integer prime.
Among rings of
complex
algebraic integers,
Eisenstein primes and
Gaussian primes may also be of interest.
 In
knot theory, a
prime knot is a
knot which can not be written as the knot sum of two lesser
nontrivial knots.
[edit]
History of prime numbers
There are hints in the surviving records of the
ancient Egyptians that they had some knowledge of prime numbers:
the
Egyptian fraction expansions in the
Rhind papyrus, for instance, have quite different forms for
primes and for composites. However, the earliest surviving records
of the explicit study of prime numbers come from the
Ancient Greeks.
Euclid's Elements (circa
300 BC)
contain important theorems about primes, including the infinitude of
primes and the
fundamental theorem of arithmetic. Euclid also showed how to
construct a
perfect number from a
Mersenne prime. The
Sieve of Eratosthenes, attributed to
Eratosthenes, is a simple method to compute primes, although the
large primes found today with computers are not generated this way.
After the Greeks, little happened with the study of prime numbers
until the 17th century. In 1640
Pierre de Fermat stated (without proof)
Fermat's little theorem (later proved by
Leibniz and
Euler). A special case of Fermat's theorem may have been known
much earlier by the Chinese. Fermat conjectured that all numbers of
the form 2^{2n} + 1 are prime (they are
called
Fermat numbers) and he verified this up to n = 4.
However, the very next Fermat number 2^{32}+1 is composite
(one of its prime factors is 641), as Euler discovered later, and in
fact no further Fermat numbers are known to be prime. The French
monk
Marin Mersenne looked at primes of the form 2^{p}
 1, with p a prime. They are called
Mersenne primes in his honor.
Euler's work in number theory included many results about primes.
He
showed the
infinite series ^{1}/_{2} + ^{1}/_{3}
+ ^{1}/_{5} + ^{1}/_{7} + ^{1}/_{11}
+ … is divergent. In 1747 he showed that the even perfect numbers
are precisely the integers of the form 2^{p1}(2^{p}1)
where the second factor is a Mersenne prime. It is believed no odd
perfect numbers exist, but there is still no proof.
At the start of the 19th century, Legendre and Gauss
independently conjectured that as x tends to infinity, the
number of primes up to x is asymptotic to x/log(x),
where log(x) is the natural logarithm of x. Ideas of
Riemann in his 1859 paper on the zetafunction sketched a program
which would lead to a proof of the prime number theorem. This
outline was completed by
Hadamard and
de la Vallée Poussin, who independently proved the prime number
theorem in 1896.
Proving a number is prime is not done (for large numbers) by
trial division. Many mathematicians have worked on
primality tests for large numbers, often restricted to specific
number forms. This includes
Pépin's test for Fermat numbers (1877),
Proth's theorem (around 1878), the
Lucas–Lehmer test for Mersenne numbers (originated 1856),^{[1]}
and the generalized
Lucas–Lehmer test. More recent algorithms like
APRTCL,
ECPP and
AKS work on arbitrary numbers but remain much slower.
For a long time, prime numbers were thought as having no possible
application outside of
pure mathematics;^{[citation
needed]} this changed in the 1970s when the
concepts of
publickey cryptography were invented, in which prime numbers
formed the basis of the first algorithms such as the
RSA
cryptosystem algorithm.
Since 1951 all the
largest known primes have been found by
computers. The search for ever larger primes has generated
interest outside mathematical circles. The
Great Internet Mersenne Prime Search and other
distributed computing projects to find large primes have become
popular in the last ten to fifteen years, while mathematicians
continue to struggle with the theory of primes.
[edit]
Primality of one
Until the 19th century, most mathematicians considered the number
1 a prime, with the definition being just that a prime is divisible
only by 1 and itself but not requiring a specific number of distinct
divisors. There is still a large body of mathematical work that is
valid despite labelling 1 a prime, such as the work of
Stern and Zeisel.
Derrick Norman Lehmer's list of primes up to 10,006,721,
reprinted as late as 1956,^{[2]}
started with 1 as its first prime.^{[3]}
Henri Lebesgue is said to be the last professional mathematician
to call 1 prime.^{[citation
needed]} The change in label occurred so that
the
fundamental theorem of arithmetic, as stated, is valid, i.e.,
“each number has a unique factorization into primes.”^{[4]}^{[5]}
Furthermore, the prime numbers have several properties that the
number 1 lacks, such as the relationship of the number to its
corresponding value of
Euler's totient function or the sum of divisors function.^{[6]}
[edit]
Prime divisors
Illustration showing that 11 is a prime number while 12
is not.
The
fundamental theorem of arithmetic states that every positive
integer larger than 1 can be written as a product of one or more
primes in a way which is
unique except possibly for the order of the prime
factors. The same prime factor may occur multiple times. Primes
can thus be considered the “basic building blocks” of the natural
numbers. For example, we can write

and any other factorization of 23244 as the product of primes
will be identical except for the order of the factors. There are
many
prime factorization algorithms to do this in practice for larger
numbers.
The importance of this theorem is one of the reasons for the
exclusion of 1 from the set of prime numbers. If 1 were admitted as
a prime, the precise statement of the theorem would require
additional qualifications.
[edit]
Properties of primes
 When written in
base 10, all prime numbers except 2 and 5 end in 1, 3, 7 or
9. (Numbers ending in 0, 2, 4, 6 or 8 represent multiples of 2
and numbers ending in 0 or 5 represent multiples of 5.)
 If p is a prime number and p divides a product
ab of integers, then p divides a or p
divides b. This proposition was proved by Euclid and is
known as
Euclid's lemma. It is used in some proofs of the uniqueness
of prime factorizations.
 If p is a prime number other than 2 and 5, ^{1}/_{p}
is always a
recurring decimal, whose period is p  1 or a divisor
of p  1. This can be deduced directly from
Fermat's little theorem. ^{1}/_{p}
expressed likewise in base q (other than base 10) has
similar effect, provided that p is not a prime factor of
q. The article on
recurring decimals shows some of the interesting properties.
 An integer p > 1 is prime if and only if the
factorial (p  1)! + 1 is divisible by p (Wilson's
theorem). Conversely, an integer n > 4 is composite
if and only if (n  1)! is divisible by n.
 If n is a positive integer greater than 1, then there
is always a prime number p with n < p < 2n
(Bertrand's
postulate).
 Adding the reciprocals of all primes together results in a
divergent
infinite series (proof).
More precisely, if S(x) denotes the sum of the
reciprocals of all prime numbers p with p = x,
then S(x) = ln ln x + O(1)
for x ? 8.
 If G is a finite group and p is a prime number
dividing the order of G, then G contains an
element of order p. (Cauchy
Theorem)
 The
prime number theorem says that the proportion of primes less
than x is asymptotic to ^{1}/_{ln x}
(in other words, as x gets very large, the likelihood
that a number less than x is prime is inversely
proportional to the number of digits in x).
 The value of the
Riemann zeta function at each point in the complex plane is
given as a meromorphic continuation of a function, defined by a
product over the set of all primes for Re(s) > 1:


 Evaluating this identity at different integers provides an
infinite number of products over the primes whose values can be
calculated, the first two being


 If p > 1, the polynomial
is irreducible over Z/pZ if and only if
p is prime.
 An integer n is prime if and only if the
nth
Chebyshev polynomial of the first kind
T_{n}(x),
divided by x is
irreducible in Z[x].
Also
if and only if n
is prime.
 All prime numbers above 3 are of the form 6n  1 or 6n + 1,
because all other numbers are divisible by 2 or 3. Generalizing
this, all prime numbers above q are of form
q#·n + m, where 0 < m < q,
and m has no prime factor = q.
[edit]
Classification of prime numbers
Two ways of classifying prime numbers, class n+ and class
n, were studied by
Paul Erdos and
John Selfridge.
Determining the class n+ of a prime number p
involves looking at the largest prime factor of p + 1. If
that largest prime factor is 2 or 3, then p is class 1+. But
if that largest prime factor is another prime q, then the
class n+ of p is one more than the class n+ of
q. Sequences
A005105 through
A005108 list class 1+ through class 4+ primes.
The class n is almost the same as class n+, except
that the factorization of p  1 is looked at instead.
[edit]
The number of prime numbers
[edit]
There are infinitely many prime numbers
The oldest known proof for the statement that there are
infinitely many prime numbers is given by the Greek
mathematician Euclid in his Elements (Book IX, Proposition
20). Euclid states the result as "there are more than any given
[finite] number of primes", and his proof is essentially the
following:
Consider any finite set of primes. Multiply all of them
together and add one (see
Euclid number). The resulting number is not divisible by any
of the primes in the finite set we considered, because dividing
by any of these would give a remainder of one. Because all
nonprime numbers can be decomposed into a product of underlying
primes, then either this resultant number is prime itself, or
there is a prime number or prime numbers which the resultant
number could be decomposed into but are not in the original
finite set of primes. Either way, there is at least one more
prime that was not in the finite set we started with. This
argument applies no matter what finite set we began with. So
there are more primes than any given finite number.
This previous argument explains why the product P of
finitely many primes plus 1 must be divisible by some prime not
among those finitely many primes (possibly itself).
The proof is sometimes phrased in a way that falsely leads some
readers to think that P + 1 must itself be prime, and think
that Euclid's proof says the prime product plus 1 is always prime.
This confusion especially arises when P is assumed to be the
product of the first primes. The smallest counterexample with
composite P + 1 is (2 × 3 × 5 × 7 × 11 × 13) + 1 = 30,031 =
59 × 509 (both primes). See also
Euclid's theorem.
Other mathematicians have given other proofs. One of these (due
to
Euler) shows that
the sum of the reciprocals of all prime numbers diverges.
Another
proof based on
Fermat numbers was given by
Goldbach.^{[7]}
Kummer's is particularly elegant^{[8]}
and
Harry Furstenberg provides
one using general topology.^{[9]}^{[10]}
[edit]
Counting the number of prime numbers below
a given number
Even though the total number of primes is infinite, one could
still ask "Approximately how many primes are there below 100,000?",
or "How likely is a random 20digit number to be prime?".
The
primecounting function p(x) is defined as the number of
primes up to x. There are known
algorithms to compute exact values of p(x) faster than it
would be possible to compute each prime up to x. Values as
large as p(10^{20}) can be calculated quickly and accurately
with modern computers. Thus, e.g., p(100,000) = 9592, and p(10^{20})
= 2,220,819,602,560,918,840.
For larger values of x, beyond the reach of modern
equipment, the
prime number theorem provides a good estimate: p(x) is
approximately x/ln(x). Even better estimates are
known.
[edit]
Location of prime numbers
[edit]
Finding prime numbers
The ancient
sieve of Eratosthenes is a simple way to compute all prime
numbers up to a given limit, by making a list of all integers and
repeatedly striking out multiples of already found primes. The
modern
sieve of Atkin is more complicated, but faster when properly
optimized.
In practice one often wants to check whether a given number is
prime, rather than generate a list of primes. Further, it is often
satisfactory to know the answer with a high
probability. It is possible to quickly check whether a given
large number (say, up to a few thousand digits) is prime using
probabilistic
primality tests. These typically pick a random number called a
"witness" and check some formula involving the witness and the
potential prime N. After several iterations, they declare
N to be "definitely composite" or "probably prime". Some of
these tests are not perfect: there may be some composite numbers,
called
pseudoprimes for the respective test, that will be declared
"probably prime" no matter what witness is chosen. However, the most
popular probabilistic tests do not suffer from this drawback.
One method for determining whether a number is prime is to divide
by all primes less than or equal to the square root of that number.
If any of the divisions come out as an integer, then the original
number is not a prime. Otherwise, it is a prime. One need not
actually calculate the square root; once one sees that the
quotient is less than the divisor, one can stop. More precisely,
the last prime factor possibility for some number N would be
Prime(m) where Prime(m + 1) squared exceeds N.
This is known as trial division; it is the simplest primality test
and it quickly becomes impractical for testing large integers
because the number of possible factors grows too rapidly as the
numbertobetested increases.
The number of prime numbers less than N is near

So, to check N for primality the largest prime factor
needed is just less than
,
and so the number of such prime factor candidates would be close to

This increases ever more slowly with N, but, because there
is interest in large values for N, the count is large also:
for N = 10^{ 20} it is 450 million.
[edit]
Primality tests

Main article:
primality test
A
primality test algorithm is an algorithm which tests a number
for primality, i.e. whether the number is a prime number.
A
probable prime is an integer which, by virtue of having passed a
certain test, is considered to be probably prime. Probable primes
which are in fact composite (such as
Carmichael numbers) are called
pseudoprimes.
In 2002, Indian scientists at
IIT Kanpur discovered a new deterministic algorithm known as the
AKS algorithm. The amount of time that this algorithm takes to
check whether a number N is prime depends on a
polynomial function of the number of digits of N (i.e. of
the logarithm of N).
[edit]
Formulas yielding prime numbers

There is no known
formula for primes which is more efficient at finding primes
than the methods mentioned above under “Finding prime numbers”.
There is a set of
Diophantine equations in 9 variables and one parameter with the
following property: the parameter is prime if and only if the
resulting system of equations has a solution over the natural
numbers. This can be used to obtain a single formula with the
property that all its positive values are prime.
There is no
polynomial, even in several variables, that takes only prime
values. For example, the curious polynomial in one variable f(n)
= n^{2}  n + 41 yields primes for n =
0,…, 40,43 but f(41) and f(42) are composite. However,
there are polynomials in several variables, whose positive values
(as the variables take all positive integer values) are exactly the
primes.
Another formula is based on Wilson's theorem mentioned above, and
generates the number two many times and all other primes exactly
once. There are other similar formulas which also produce primes.
[edit]
Special types of primes from formulas for
primes
A prime p is called
primorial or primefactorial if it has the form p
= n# ± 1 for some number n, where
n# stands for the product 2 · 3 · 5 · 7 · 11 · … of all
the primes = n. A prime is called
factorial if it is of the form
n! ± 1. The first factorial primes are:
 n!  1 is prime for n = 3, 4, 6, 7, 12, 14, 30, 32, 33, 38,
94, 166, 324, … (sequence
A002982 in
OEIS)
 n! + 1 is prime for n = 0, 1, 2, 3, 11, 27, 37, 41, 73, 77,
116, 154, 320, … (sequence
A002981 in
OEIS)
The largest known primorial prime is ?(392113) + 1, found by
Heuer in 2001.^{[11]}
The largest known factorial prime is 34790!  1, found by Marchal,
Carmody and Kuosa in 2002.^{[12]}
It is not known whether there are infinitely many primorial or
factorial primes.
Primes of the form 2^{p}  1, where p is a
prime number, are known as
Mersenne primes, while primes of the form
are known as
Fermat primes. Prime numbers p where 2p + 1 is
also prime are known as
Sophie Germain primes. The following list is of other special
types of prime numbers that come from formulas:
Some primes are classified according to the properties of their
digits in decimal or other bases. For example, numbers whose digits
form a
palindromic sequence are called
palindromic primes, and a prime number is called a
truncatable prime if successively removing the first digit at
the left or the right yields only new prime numbers.
[edit]
The distribution of the prime numbers
 Further information:
Prime number theorem
The distribution of all the prime numbers in the range
of 1 to 76,800, from left to right and top to bottom,
where each pixel represents a number. Black pixels mean
that number is prime and white means it is not prime.
The problem of modelling the distribution of prime numbers is a
popular subject of investigation for number theorists. The
occurrence of individual prime numbers among the
natural numbers is (so far) unpredictable, even though there are
laws (such as the
prime number theorem and
Bertrand's postulate) that govern their average distribution.
Leonhard Euler commented
 Mathematicians have tried in vain to this day to discover
some order in the sequence of prime numbers, and we have reason
to believe that it is a mystery into which the mind will never
penetrate.^{[13]}
In a 1975 lecture,
Don Zagier commented
There are two facts about the distribution of prime numbers
of which I hope to convince you so overwhelmingly that they will
be permanently engraved in your hearts. The first is that,
despite their simple definition and role as the building blocks
of the natural numbers, the prime numbers grow like weeds among
the natural numbers, seeming to obey no other law than that of
chance, and nobody can predict where the next one will sprout.
The second fact is even more astonishing, for it states just the
opposite: that the prime numbers exhibit stunning regularity,
that there are laws governing their behavior, and that they obey
these laws with almost military precision.
^{
[14]}
Additional image with
2310 columns is linked here, preserving multiples of 2, 3, 5, 7,
11 in respective columns. Predictably, prime numbers fall into
columns if the numbers are arranged from left to right and the width
is a multiple of a prime number. More surprisingly, when arranged in
a spiral such as the
Ulam spiral, prime numbers cluster on certain diagonals and not
others.
[edit]
Gaps between primes

Let p_{n} denote the nth prime
number (i.e. p_{1} = 2, p_{2} = 3,
etc.). The gap g_{n} between the
consecutive primes p_{n} and p_{n
+ 1} is the difference between them, i.e.
 g_{n} = p_{n + 1}
 p_{n}.
We have g_{1} = 3  2 = 1, g_{2} =
5  3 = 2, g_{3} = 7  5 = 2, g_{4} =
11  7 = 4, and so on. The sequence (g_{n}) of
prime gaps has been extensively studied.
For any natural number N larger than 1, the sequence (for
the notation N! read
factorial)
 N! + 2, N! + 3, …, N! + N
is a sequence of N  1 consecutive composite integers.
Therefore, there exist gaps between primes which are arbitrarily
large, i.e. for any natural number N, there is an integer
n with g_{n} > N. (Choose n
so that p_{n} is the greatest prime number
less than N! + 2.)
On the other hand, the gaps get arbitrarily small in proportion
to the primes: the quotient g_{n}/p_{n}
approaches zero as n approaches infinity. Note also that
the
twin prime conjecture asserts that g_{n} =
2 for infinitely many integers n.
[edit]
Location of the largest known prime

As of September 2008^{[update]},
the largest known prime was discovered by the
distributed computing project
Great Internet Mersenne Prime Search (GIMPS):
 2^{43,112,609}  1.
This was found to be a prime number on August 23, 2008. This
number is 12,978,189 digits long and is (chronologically) the 45th
known Mersenne prime.
Historically, the largest known prime has almost always been a
Mersenne prime since the dawn of electronic computers, because there
exists a particularly fast primality test for numbers of this form,
the
Lucas–Lehmer test for Mersenne numbers.
The largest known prime that is not a Mersenne prime is
19,249 × 2^{13,018,586} + 1 (3,918,990 digits), a
Proth number. This is also the seventh largest known prime of
any form. It was found on March 26, 2007 by the
Seventeen or Bust project and it brings them one step closer to
solving the
Sierpinski problem.
Some of the largest primes not known to have any particular form
(that is, no simple formula such as that of Mersenne primes) have
been found by taking a piece of semirandom binary data, converting
it to a number n, multiplying it by 256^{k}
for some positive integer k, and searching for possible
primes within the interval [256^{k}n + 1, 256^{k}(n
+ 1)  1].
[edit]
Awards for finding primes
The
Electronic Frontier Foundation (EFF) has offered a US$100,000
prize to the first discoverers of a prime with at least 10 million
digits. They also offer $150,000 for 100 million digits, and
$250,000 for 1 billion digits. In 2000 they paid out $50,000 for 1
million digits.
The
RSA Factoring Challenge offered prizes up to US$200,000 for
finding the prime factors of certain
semiprimes of up to 2048 bits. However, the challenge was closed
in 2007 after much smaller prizes for smaller semiprimes had been
paid out.^{[15]}
[edit]
Generalizations of the prime concept
The concept of prime number is so important that it has been
generalized in different ways in various branches of mathematics.
[edit]
Prime elements in rings
One can define
prime elements and
irreducible elements in any
integral domain. For any
unique factorization domain, such as the ring Z of
integers, the set of prime elements equals the set of irreducible
elements, which for Z is {…, 11, 7, 5, 3, 2, 2, 3, 5, 7,
11, …}.
As an example, we consider the
Gaussian integers Z[i], that is, complex numbers
of the form a + bi with a and b in Z.
This is an integral domain, and its prime elements are the
Gaussian primes. Note that 2 is not a Gaussian prime,
because it factors into the product of the two Gaussian primes (1 +
i) and (1  i). The element 3, however, remains prime
in the Gaussian integers. In general, rational primes (i.e. prime
elements in the ring Z of integers) of the form 4k + 3
are Gaussian primes, whereas rational primes of the form 4k +
1 are not.
[edit]
Prime ideals
In
ring theory, one generally replaces the notion of number with
that of
ideal.
Prime ideals are an important tool and object of study in
commutative algebra,
algebraic number theory and
algebraic geometry. The prime ideals of the ring of integers are
the ideals (0), (2), (3), (5), (7), (11), …
A central problem in algebraic number theory is how a prime ideal
factors when it is lifted to an extension field. For example,
in the Gaussian integer example above, (2) ramifies into a
prime power (1 + i and 1  i generate the same prime
ideal), prime ideals of the form (4k + 3) are inert
(remain prime), and prime ideals of the form (4k + 1)
split (are the product of 2 distinct prime ideals).
[edit]
Primes in valuation theory
In algebraic number theory, yet another generalization is used.
Given an arbitrary
field K, one considers
valuations on K, certain functions from K to the
real numbers R. Every such valuation yields a
topology on K, and two valuations are called
equivalent if they yield the same topology. A prime of K
(sometimes called a place of K) is an
equivalence class of valuations. With this definition, the
primes of the field Q of
rational numbers are represented by the standard
absolute value function (known as the
infinite prime) as well as by the
padic valuations on Q, for every prime number
p.
[edit]
Prime knots
In
knot theory, a prime knot is a
knot which is, in a certain sense, indecomposable. Specifically,
it is one which cannot be written as the
knot sum of two nontrivial knots.
[edit]
Open questions
There are many open questions about prime numbers. A very
significant one is the
Riemann hypothesis, which essentially says that the primes are
as regularly distributed as possible. From a physical viewpoint, it
roughly states that the irregularity in the distribution of primes
only comes from random noise. From a mathematical viewpoint, it
roughly states that the asymptotic distribution of primes (about 1/
log x of numbers less than x are primes, the
prime number theorem) also holds for much shorter intervals of
length about the square root of x (for intervals near x).
This hypothesis is generally believed to be correct, in particular,
the simplest assumption is that primes should have no significant
irregularities without good reason.
Many famous conjectures appear to have a very high probability of
being true (in a formal sense, many of them follow from simple
heuristic probabilistic arguments):
 Prime
Euclid numbers: It is not known whether or not there are an
infinite number of prime Euclid numbers.

Polignac's conjecture: For every positive integer n, there
are infinitely many pairs of consecutive primes which differ by
2n. When n = 1 this is the twin prime conjecture.
 A weaker form of Polignac's conjecture: Every
even number is the difference of two primes.
 It is conjectured there are infinitely many primes of the
form n^{2} + 1.^{[17]}

Legendre's conjecture: There is a prime number between n^{2}
and (n + 1)^{2} for every positive integer n.

Brocard's conjecture: There are always at least four primes
between the squares of consecutive primes greater than 2.
All four of
Landau's problems from 1912 are listed above and still unsolved:
Goldbach, twin primes, Legendre, n^{2}+1 primes.
[edit]
Applications
For a long time, number theory in general, and the study of prime
numbers in particular, was seen as the canonical example of pure
mathematics, with no applications outside of the selfinterest of
studying the topic. In particular, number theorists such as
British mathematician
G. H. Hardy prided themselves on doing work that had absolutely
no military significance.^{[19]}
However, this vision was shattered in the 1970s, when it was
publicly announced that prime numbers could be used as the basis for
the creation of
public key cryptography algorithms. Prime numbers are also used
for
hash tables and
pseudorandom number generators.
Some
rotor machines were designed with a different number of pins on
each rotor, with the number of pins on any one rotor either prime,
or
coprime to the number of pins on any other rotor. This helped
generate the
full cycle of possible rotor positions before repeating any
position.
[edit]
Publickey cryptography

Several publickey cryptography algorithms, such as
RSA, are
based on large prime numbers (for example with 512
bits).
[edit]
Prime numbers in nature
Many numbers occur in nature, and inevitably some of these are
prime. There are, however, relatively few examples of numbers that
appear in nature because they are prime. For example, most
starfish have 5 arms, and 5 is a prime number. However there is
no evidence to suggest that starfish have 5 arms because 5 is
a prime number. Indeed, some starfish have different numbers of
arms. Echinaster luzonicus normally has six arms, Luidia
senegalensis has nine arms, and Solaster endeca can have
as many as twenty arms. Why the majority of starfish (and most other
echinoderms) have
fivefold symmetry remains a mystery.
One example of the use of prime numbers in nature is as an
evolutionary strategy used by
cicadas
of the genus
Magicicada.^{[20]}
These insects spend most of their lives as
grubs
underground. They only pupate and then emerge from their burrows
after 13 or 17 years, at which point they fly about, breed, and then
die after a few weeks at most. The logic for this is believed to be
that the prime number intervals between emergences makes it very
difficult for predators to evolve that could specialise as predators
on Magicicadas.^{[21]}
If Magicicadas appeared at a nonprime number intervals, say
every 12 years, then predators appearing every 2, 3, 4, 6, or 12
years would be sure to meet them. Over a 200year period, average
predator populations during hypothetical outbreaks of 14 and
15year cicadas would be up to 2% higher than during outbreaks of
13 and 17year cicadas.^{[22]}
Though small, this advantage appears to have been enough to drive
natural selection in favour of a primenumbered lifecycle for these
insects.
There is speculation that the zeros of the
zeta function are connected to the energy levels of complex
quantum systems. ^{
[23]}
[edit]
Prime numbers in the arts and literature
Prime numbers have influenced many artists and writers. The
French
composer
Olivier Messiaen used prime numbers to create ametrical music
through "natural phenomena". In works such as La Nativité du
Seigneur (1935) and Quatre études de rythme (194950), he
simultaneously employs motifs with lengths given by different prime
numbers to create unpredictable rhythms: the primes 41, 43, 47 and
53 appear in one of the études. According to Messiaen this way of
composing was "inspired by the movements of nature, movements of
free and unequal durations". ^{
[24]}
In his science fiction novel
Contact, later made into a
film of the same name, the
NASA
scientist
Carl Sagan suggested that prime numbers could be used as a means
of communicating with aliens, an idea that he had first developed
informally with American astronomer
Frank Drake in 1975. ^{
[25]}
Tom Stoppard's awardwinning 1993 play
Arcadia was a conscious attempt to discuss mathematical
ideas on the stage. In the opening scene, the 13 year old heroine
puzzles over
Fermat's Last Theorem, a theorem involving prime numbers.
^{
[26]}
^{
[27]}
^{
[28]}
Many films reflect a popular fascination with the mysteries of
prime numbers and cryptography: films such as
Cube,
Sneakers,
The Mirror Has Two Faces and
A Beautiful Mind, based on the biography of the
mathematician and Nobel laureate
John Forbes Nash by
Sylvia Nasar.^{[29]}
^{
[30]}
In the novel
PopCo
by
Scarlett Thomas the main character, Alice Butler's grandmother
works on proving the
Riemann Hypothesis. In the book, a table of the first 1000 prime
numbers is displayed.^{[31]}
[edit]
See also
[edit]
Distributed computing projects that search
for primes
 ^
The Largest Known Prime by Year: A Brief History
Prime Curios!: 17014…05727 (39digits)
 ^
Hans Riesel, Prime Numbers and Computer Methods for
Factorization. New York: Springer (1994): 36
 ^
Richard K. Guy & John Horton Conway, The Book of Numbers.
New York: Springer (1996): 129  130
 ^
Gowers, T (2002). Mathematics: A Very Short
Introduction.
Oxford University Press, 118.
ISBN 0192853619. “The seemingly arbitrary exclusion
of 1 from the definition of a prime … does not express some
deep fact about numbers: it just happens to be a useful
convention, adopted so there is only one way of factorizing
any given number into primes”
 ^
""Why
is the number one not prime?"". Retrieved 20071002.
 ^
""Arguments
for and against the primality of 1".
 ^
Letter in
Latin from Goldbach to Euler, July 1730.
 ^
P. Ribenboim: The Little Book of Bigger Primes,
second edition, Springer, 2004, p. 4.
 ^
Furstenberg, Harry. (1955).
"On
the infinitude of primes".
Amer. Math. Monthly 62 (5): 353.
doi:10.2307/2307043.
 ^
"Furstenberg's
proof that there are infinitely many prime numbers".
Everything2. Retrieved on
20061126.
 ^
The Top Twenty: Primorial
 ^
The Top Twenty: Factorial
 ^
Julian Havil, Gamma: Exploring Euler's Constant
(Hardcover). Princeton: Princeton University Press
(2003): 163
 ^
Havil (2003): 171
 ^
The RSA Factoring Challenge — RSA Laboratories
 ^
E.g., see
Guy,
Richard K. (1981), Unsolved Problems in Number Theory,
SpringerVerlag ,
problem A3, pp. 7–8.
 ^
Eric W. Weisstein,
Landau's Problems at
MathWorld.
 ^
Caldwell, Chris,
The Top Twenty: Lucas Number at The
Prime Pages.
 ^
Hardy, G.H. (1940).
A Mathematician's Apology.
Cambridge University Press.
ISBN 0521427061. “No one has yet discovered any
warlike purpose to be served by the theory of numbers or
relativity, and it seems unlikely that anyone will do so for
many years”
 ^
Goles, E., Schulz, O. and M. Markus (2001). "Prime number
selection of cycles in a predatorprey model", Complexity
6(4): 3338
 ^
Paulo R. A. Campos, Viviane
M. de Oliveira, Ronaldo Giro, and Douglas S. Galvão. (2004).
"Emergence
of Prime Numbers as the Result of Evolutionary Strategy".
Phys. Rev. Lett. 93: 098107.
doi:10.1103/PhysRevLett.93.098107.
Retrieved on
20061126.
 ^
"Invasion
of the Brood".
The Economist (May
6,
2004). Retrieved on
20061126.
 ^
Ivars Peterson (June
28,
1999). "The
Return of Zeta".
MAA Online. Retrieved on
20080314.
 ^
The Messiaen companion', ed. Peter Hill, Amadeus Press,
1994.
ISBN 0931340950
 ^
Carl Pomerance,
Prime Numbers and the Search for Extraterrestrial
Intelligence, Retrieved on
December 22,
2007
 ^
Tom Stoppard, Arcadia, Faber and Faber, 1993.
ISBN 0571169341.
 ^
The Cambridge Companion to Tom Stoppard, ed.
Katherine E. Kelly, Cambridge University Press, 2001.
ISBN 0521645921
 ^
The Mathematics of Arcadia, an event involving Tom
Stoppard and
MSRI in the
University of California, Berkeley
 ^
Music of the Spheres,
Marcus du Sautoy's selection of films featuring prime
numbers
 ^
A Beautiful Mind
 ^
 A Mathematician reviews PopCo
[edit]
References
 John Derbyshire, Prime Obsession: Bernhard Riemann
and the Greatest Unsolved Problem in Mathematics. Joseph
Henry Press; 448 pages
 Wladyslaw Narkiewicz, The development of prime number
theory. From Euclid to Hardy and Littlewood. Springer
Monographs in Mathematics. SpringerVerlag, Berlin, 2000.
 H. Riesel, Prime Numbers and Computer Methods for
Factorization, 2nd ed., Birkhäuser 1994.
 Marcus du Sautoy, The Music of the Primes: Searching
to Solve the Greatest Mystery in Mathematics.
HarperCollins; 352 pages.
ISBN 0066210704.
The Music of Primes website.
 Karl Sabbagh, The Riemann Hypothesis: The Greatest
Unsolved Problem in Mathematics. Farrar, Straus and
Giroux; 340 pages
[edit]
External links
[edit]
Prime number generators & calculators


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