  Home | PVAF Programs | Local Samaj | Photo Albums | Forum | Veda | Languages | Donations | Links

 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 twenty-five 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 Web-based 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 self-interest 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 bRHm 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 shktio 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 chtuAR-yug in pRUthivii-lok has number values of in the multiples of 1, 2, 3, 4....sty-yug,  teTR-yug, D'vaapr-yug is  4, 3 and 2 times respectively of the time duration of kli-yug..... -   each of the day and night time the duration of bRHmaa-day in the cyclic creation system in our universe has a  duration in pRUthivii- lok is of 1000 chtuAR-yug...and bRHmaa's life duration is 100 years with each year of 360 ((1+1)x(1+1)x3x3x10) such bRHmaa-day.....Each chtuAR-yug 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) mHaa-puraaAN, 18 up-puraaAN, 18 mHaa-upnishD 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 kRUSHAN had 8 prime queens and 16,000 ((1+1)x8x1000) wives who were liberated but were captive celestial women suffering their paapi kARm-fl of previous life-journeys with time-factored 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 jyotish-shaasTR (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 jyotish-shaasTR 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 prime-numbered life-cycle for insects to save themselves from their predators who evolve at a non-prime 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 award-winning 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 double-checking 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 world-record sprint or an angler's prize-winning 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......
 And now to understand what is a NUMBER please continue reading below followed by the reading on in-depth understanding of the PRIME NUMBER which is the topic of this news item.....

# Number

from WIKIPEDIA - FREE ENCYCLOPEDIA

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.

[hide]

## 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.)

### 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.

### 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 .

### Rational numbers

A rational number is a number that can be expressed as a fraction with an integer numerator and a non-zero 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 .

### 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 one-tenth 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.

### 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.

### 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 µ-recursive 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.

### Other types

Hyperreal and hypercomplex numbers are used in non-standard 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 p-adic 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 non-zero 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 old-fashioned 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:

## 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.

## History

### History of integers

#### 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 place-value (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 place-value was the Mesopotamian base 60 system (ca. 3400 BC) and the earliest known base 10 system dates to 3100 BC in Egypt. 

#### 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 place-value 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. . 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 south-central 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.

#### 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 (Jiu-zhang 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 Brahma-Sphuta-Siddhanta 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 right-most nonzero digit of the corresponding positive number's numeral. 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, 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.

### History of rational, irrational, and real numbers

Further information: History of irrational numbers and History of pi

#### 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 decimal-fraction approximations to pi or the square root of two. Similarly, Babylonian math texts had always used sexagesimal fractions with great frequency.

#### History of irrational numbers

The earliest known use of irrational numbers was in the Indian Sulba Sutras composed between 800-500 BC. 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.

#### Transcendental numbers and reals

The first results concerning transcendental numbers were Lambert's 1761 proof that p cannot be rational, and also that en 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.

### 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 one-to-one 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.

### 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 v-1 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 well-known 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 x2 + 1 = 0). His student, Ferdinand Eisenstein, studied the type a + b?, where ? is a complex root of x3 - 1 = 0. Other such classes (called cyclotomic fields) of complex numbers are derived from the roots of unity xk - 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.

### 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, Adrien-Marie 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 Vallée-Poussin in 1896. The conjectures of Goldbach and Riemann yet remain to be proved or refuted.

## Word alternatives

Some numbers traditionally have alternative words to express them, including the following:

Wikimedia Commons has media related to:
Look up number in Wiktionary, the free dictionary.

 And now please continue reading below the current in-depth understandning of PRIME NUMBERS......

# Prime number

FROM WIKIPEDIA -THE FREE ENCYCLOPEDIA
 Divisibility-based sets of integers Form of factorization: Prime number Composite number Powerful number Square-free number Achilles number Constrained divisor sums: Perfect number Almost perfect number Quasiperfect number Multiply perfect number Hyperperfect number Superperfect number Unitary perfect number Semiperfect number Primitive semiperfect number Practical number Numbers with many divisors: Abundant number Highly abundant number Superabundant number Colossally abundant number Highly composite number Superior highly composite number Other: Deficient number Weird number Amicable number Friendly number Sociable number Solitary number Sublime number Harmonic divisor number Frugal number Equidigital number Extravagant number See also: Divisor function Divisor Prime factor Factorization This box: view • talk • edit

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 twenty-five 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 well-defined 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 non-zero, non-unit 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.

[hide]

## History of prime numbers The Sieve of Eratosthenes is a simple, ancient algorithm for finding all prime numbers up to a specified integer. It is the predecessor to the modern Sieve of Atkin, which is faster but more complex. The eponymous Sieve of Eratosthenes was created in the 3rd century BC by Eratosthenes, an ancient Greek mathematician.

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 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4. However, the very next Fermat number 232+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 2p - 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 2p-1(2p-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 zeta-function 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), and the generalized Lucas–Lehmer test. More recent algorithms like APRT-CL, 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; this changed in the 1970s when the concepts of public-key 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.

### 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, started with 1 as its first prime. Henri Lebesgue is said to be the last professional mathematician to call 1 prime. 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.” 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.

## 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.

## 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 Tn(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.

### 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.

## The number of prime numbers

### 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 non-prime 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. Kummer's is particularly elegant and Harry Furstenberg provides one using general topology.

### 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 20-digit number to be prime?".

The prime-counting 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(1020) can be calculated quickly and accurately with modern computers. Thus, e.g., p(100,000) = 9592, and p(1020) = 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.

## Location of prime numbers

### 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 number-to-be-tested 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.

### 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).

### Formulas yielding prime numbers

Main article: formula for primes

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) = n2 - 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.

#### Special types of primes from formulas for primes

A prime p is called primorial or prime-factorial 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. The largest known factorial prime is 34790! - 1, found by Marchal, Carmody and Kuosa in 2002. It is not known whether there are infinitely many primorial or factorial primes.

Primes of the form 2p - 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.

### 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.

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.



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.

### Gaps between primes

Main article: Prime gap

Let pn denote the nth prime number (i.e. p1 = 2, p2 = 3, etc.). The gap gn between the consecutive primes pn and pn + 1 is the difference between them, i.e.

gn = pn + 1 - pn.

We have g1 = 3 - 2 = 1, g2 = 5 - 3 = 2, g3 = 7 - 5 = 2, g4 = 11 - 7 = 4, and so on. The sequence (gn) 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 gn > N. (Choose n so that pn 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 gn/pn approaches zero as n approaches infinity. Note also that the twin prime conjecture asserts that gn = 2 for infinitely many integers n.

### Location of the largest known prime

Wikinews has related news:
Wikinews has related news:

As of September 2008[update], the largest known prime was discovered by the distributed computing project Great Internet Mersenne Prime Search (GIMPS):

243,112,609 - 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 × 213,018,586 + 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 semi-random binary data, converting it to a number n, multiplying it by 256k for some positive integer k, and searching for possible primes within the interval [256kn + 1, 256k(n + 1) - 1].

## 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.

## 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.

### 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.

### 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).

### 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 p-adic valuations on Q, for every prime number p.

### 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.

## 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 n2 + 1.
• Cramér's conjecture: . This conjecture implies Legendre's, but its status is more unsure.
• 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, n2+1 primes.

## 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 self-interest 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.

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.

### Public-key cryptography

Several public-key cryptography algorithms, such as RSA, are based on large prime numbers (for example with 512 bits).

### 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 five-fold 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. 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. If Magicicadas appeared at a non-prime 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 200-year period, average predator populations during hypothetical outbreaks of 14- and 15-year cicadas would be up to 2% higher than during outbreaks of 13- and 17-year cicadas. Though small, this advantage appears to have been enough to drive natural selection in favour of a prime-numbered life-cycle for these insects.

There is speculation that the zeros of the zeta function are connected to the energy levels of complex quantum systems. 

## 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 (1949-50), 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". 

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 award-winning 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. 

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.