Prime number and Constitution of Canada: Difference between pages

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{{Canadian politics}}
The '''Constitution of Canada''' is the supreme law in [[Canada]]; the country's [[constitution]] is an amalgamation of codified [[Act of Parliament|act]]s and [[uncodified constitution|uncodified]] traditions and [[constitutional convention (political custom)|convention]]s. It outlines Canada's [[system of government]], as well as the [[civil rights]] of all Canadian citizens.
In [[mathematics]], a '''prime number''' (or a '''prime''') is a [[natural number]] which has exactly two '''distinct''' natural number [[divisor]]s: [[1 (number)|1]] and itself. An infinitude of prime numbers exists, as demonstrated by [[Euclid]] around 300 BC. The first twenty-five prime numbers are:<!--Do not add 1 to this list. Its exclusion from the list is addressed in the “History of prime numbers” section below.-->
:[[2 (number)|2]], [[3 (number)|3]], [[5 (number)|5]], [[7 (number)|7]], [[11 (number)|11]], [[13 (number)|13]], [[17 (number)|17]], [[19 (number)|19]], [[23 (number)|23]], [[29 (number)|29]], [[31 (number)|31]], [[37 (number)|37]], [[41 (number)|41]], [[43 (number)|43]], [[47 (number)|47]], [[53 (number)|53]], [[59 (number)|59]], [[61 (number)|61]], [[67 (number)|67]], [[71 (number)|71]], [[73 (number)|73]], [[79 (number)|79]], [[83 (number)|83]], [[89 (number)|89]], [[97 (number)|97]] {{OEIS|id=A000040}}.
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|Primality of one]]''. The [[Set (mathematics)|set]] of prime numbers is sometimes denoted by ℙ.


The composition of the Constitution of Canada is defined in subsection 52(2) of the [[Constitution Act, 1982]] as consisting of the [[Canada Act 1982]] (including the Constitution Act, 1982), all acts and orders referred to in the schedule (including the [[Constitution Act, 1867]], formerly the British North America Act), and any amendments to these documents.<ref>See [[list of Canadian constitutional documents]] for details.</ref> The [[Supreme Court of Canada]] held that the list is not exhaustive and includes unwritten doctrines as well.<ref>''[[New Brunswick Broadcasting Co. v. Nova Scotia (Speaker of the House of Assembly)|New Brunswick Broadcasting Co. v. Nova Scotia]]'' [1993] 1 S.C.R. 319</ref>
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.


== History of the Constitution ==
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's conjecture|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.
{{main|Constitutional history of Canada}}
[[Image:Fathers of Canadian Confederation.jpg|thumb|left|300px|right|A painting depicting negotiations that would lead to the enactment of the [[Constitution Act, 1867|British North America Act, 1867]]]]
The first semblance of a Constitution for Canada was the [[Royal Proclamation of 1763]]. The Act renamed Canada "The Province of Quebec" and redefined its borders and established a British-appointed colonial government. The proclamation was considered the ''de facto'' constitution of Quebec until 1774 when the British government passed the [[Quebec Act]] of 1774 which set out many procedures of governance in the area of Quebec. It extended the boundaries of the colony and adopted the British criminal code among other things.


The colony of Canada received its first full constitution in the [[Constitutional Act of 1791]] which established much of the composition of the government. This was later superseded by the [[British North America Act]] in 1867 which established the Dominion of Canada.
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 [[integer]]s as a [[ring (mathematics)|ring]], &minus;7 is a prime element. Without further specification, however, “prime number” always means a positive integer prime. Among rings of [[complex number|complex]] [[algebraic integer]]s, [[Eisenstein prime]]s and [[Gaussian prime]]s may also be of interest.
* In [[knot theory]], a [[prime knot]] is a [[knot (mathematics)|knot]] which can not be written as the knot sum of two lesser nontrivial knots.


In 1931, the British Parliament passed the [[Statute of Westminster, 1931]] (22 Geo. V, c.4 (UK)). This Act gave all dominion countries equal legislative authority with the United Kingdom. This was followed up in 1982, when the British Parliament passed the '''[[Canada Act 1982|Canada Act, 1982]]''' ([UK] 1982, c.11) giving up all remaining constitutional and legislative authority over Canada. The enactment of the Canada Act is often referred to in Canada as the 'patriation' of the constitution and it was largely due to the work of [[Pierre Trudeau|Pierre Elliot Trudeau]], the [[Prime Minister of Canada]] at the time.
==History==
[[Image:Animation Sieve of Eratosth-2.gif|thumb|300px|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 Greece|ancient Greek]] [[mathematician]].]]
There are hints in the surviving records of the [[ancient Egypt]]ians 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 Greece|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.


With the introduction of the Canada Act and the accompanying Charter, much of Constitutional law in Canada has changed. The Canada Act has entrenched many constitutional conventions and has made amendments significantly more difficult (see [[Amendments to the Constitution of Canada|amendment formula]]). The Charter has shifted the focus of the Constitution to individual and collective rights of the inhabitants of Canada. Before the enactment of the Canadian Charter of Rights and Freedoms in 1982, civil rights and liberties had no solid constitutional protection in Canada. Whenever one level of government passed a law that seemed oppressive to civil rights and liberties, Canadian constitutional lawyers had to argue creatively, such as by saying that the oppressive law violates division of federal and provincial powers or by citing some other technical flaw that had little to do with the concept of civil rights and liberties.{{fact|date=December 2007}} Since 1982, however, the Charter has become the most often cited part of the Constitution and has thus far solidified the protection of rights for people in Canada.
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 [[Gottfried Wilhelm Leibniz|Leibniz]] and [[Leonhard Euler|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<sup>2<sup>''n''</sup></sup> + 1 are prime (they are called [[Fermat number]]s) and he verified this up to ''n'' = 4 (or 2<sup>16</sup>+1). However, the very next Fermat number 2<sup>32</sup>+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<sup>''p''</sup> - 1, with ''p'' a prime. They are called [[Mersenne prime]]s in his honor.


==Constitution Act, 1867==
Euler's work in number theory included many results about primes. He [[Proof that the sum of the reciprocals of the primes diverges|showed]] the [[infinite sum|infinite series]] <sup>1</sup>/<sub>2</sub> + <sup>1</sup>/<sub>3</sub> + <sup>1</sup>/<sub>5</sub> + <sup>1</sup>/<sub>7</sub> + <sup>1</sup>/<sub>11</sub> + … is divergent.
{{main|Constitution Act, 1867}}
In 1747 he showed that the even perfect numbers are precisely the integers of the form 2<sup>''p''-1</sup>(2<sup>''p''</sup>-1) where the second factor is a Mersenne prime. It is believed no odd perfect numbers exist, but there is still no proof.
This was an Act of the British Parliament, originally called the British North America Act 1867,
that created the Dominion of Canada out of three separate provinces in British North America ([[Province of Canada]], [[New Brunswick]], and [[Nova Scotia]]) and allowed for subsequent provinces and colonies to join this union in the future. It outlined Canada's system of government, which combines Britain's Westminster model of parliamentary government with division of sovereignty ([[Canadian federalism|federalism]]). Although it is one of many ''[[British North America Acts]]'' to come, it is still the most famous of these and is understood to be the document of Canadian Confederation (i.e. union of provinces and colonies in British North America). With the patriation of the Constitution in 1982, this Act was renamed ''Constitution Act, 1867''. In recent years, the Constitution Act, 1867 has mainly served as the basis on which the division of powers between the provinces and federal government have been analyzed.


==Constitution Act, 1982==
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 [[Jacques Hadamard|Hadamard]] and [[Charles de la Vallée-Poussin|de la Vallée Poussin]], who independently proved the prime number theorem in 1896.
{{seealso|Constitution Act, 1982}}
<!-- Deleted image removed: [[Image:Ouellet approaches to sign the Constitution.jpg|thumb|left|300px|The [[Constitution Act, 1982]], which included the [[Canadian Charter of Rights and Freedoms]], was brought into force by [[Elizabeth II of the United Kingdom|Queen Elizabeth II]] in [[Ottawa]] on [[April 17]], [[1982]].]] -->
Endorsed by all the provincial governments except Quebec's, this was an Act by the Canadian Parliament requesting full political independence from Britain. Part V of this Act created a constitution-amending formula that did not require an Act by the British Parliament. Further, Part I of this Act is the [[Canadian Charter of Rights and Freedoms]] which outlines the civil rights and liberties of every citizen in Canada, such as freedom of expression, of religion, and of mobility. Part II deals with the rights of Canada's Aboriginal peoples.


===Canadian Charter of Rights and Freedoms===
Proving a number is prime is not done (for large numbers) by trial division. Many mathematicians have worked on [[primality test]]s 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),<ref>[http://primes.utm.edu/notes/by_year.html The Largest Known Prime by Year: A Brief History] [http://primes.utm.edu/curios/page.php?number_id=135 Prime Curios!: 17014…05727 (39-digits)]</ref> and the generalized [[Lucas–Lehmer test]]. More recent algorithms like [[Adleman-Pomerance-Rumely primality test|APRT-CL]], [[Elliptic curve primality proving|ECPP]] and [[AKS primality test|AKS]] work on arbitrary numbers but remain much slower.
{{seealso|Canadian Charter of Rights and Freedoms}}
As noted above, this is Part I of the Constitution Act, 1982. The Charter is the constitutional guarantee of collective and individual rights. It is a relatively short document and written in plain language in order to ensure accessibility to the average citizen. It is said that it is the part of the constitution that has the greatest impact on Canadians' day-to-day lives, and has been the fastest developing area of constitutional law for many years.


===Amending formula ===
For a long time, prime numbers were thought to have no possible application outside of [[pure mathematics]];{{Fact|date=August 2008}} 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.
{{seealso|Amendments to the Constitution of Canada}}
With the Constitution Act, 1982, amendments to the constitution must be done in accordance with Part V of the Constitution Act, 1982 which provides for five different amending formulas. Amendments can be brought forward under section 46(1) by any province or either level of the federal government. The general formula is set out in section 38(1), known as the "7/50 formula", requires: (a) assent from both the House of Commons and the Senate; (b) the approval of two-thirds of the provincial legislatures (at least seven provinces), representing at least 50% of the population (effectively, this would include at least Quebec or Ontario, as they are the most populous provinces). This formula specifically applies to amendments related to the proportionate representation in Parliament, powers, selection, and composition of the Senate, the Supreme Court and the addition of provinces or territories.
The other amendment formulas are for exceptional cases as provided by in the Act:
*In the case of an amendment related to the [[Monarchy in Canada|Office of the Queen]], the number of senators, the use of either official language (subject to section 43), the amending formula, or the composition of the Supreme Court, the amendment must be adopted by unanimous consent of all the provinces in accordance with section 41.
*However, in the case of an amendment related to provincial boundaries or the use of an official language within a province alone, the amendment must be passed by the legislatures affected by the amendment (section 43).
*In the case of an amendment that affects the federal government alone, the amendment does not need approval of the provinces (section 44). The same applies to amendments affecting the provincial government alone (section 45).


===Vandalism of the paper proclamation===
Since 1951 all the [[largest known prime]]s have been found by [[computer]]s. 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.
In 1983, Toronto artist [[Peter Greyson]] entered Ottawa's National Archives (known today as [[Library and Archives Canada]]) and poured red paint over a copy of the proclamation of the 1982 constitutional amendment. He said he was displeased with the federal government's decision to allow U.S. missile testing in Canada, and had wanted to "graphically illustrate to Canadians" how wrong the government was. A grapefruit-sized stain still remains on the original document. Specialists opted to leave most of the paint intact fearing attempts at removing it would only do further damage.<ref>[http://archives.cbc.ca/IDC-1-73-331-1747-10/on_this_day/politics_economy/greyson_protest Missile Protestor defaces Constitution - "On This Day" - CBC Archives<!-- Bot generated title -->]</ref>


===Primality of one===
== Sources of the Constitution ==
{{see|List of Canadian constitutional documents}}


There are three general methods of constitutional entrenchment:
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 [[Moritz Abraham Stern|Stern]] and Zeisel. [[Derrick Norman Lehmer]]'s list of primes up to 10,006,721, reprinted as late as 1956,<ref>Hans Riesel, ''Prime Numbers and Computer Methods for Factorization''. New York: Springer (1994): 36</ref> started with 1 as its first prime.<ref>Richard K. Guy & John Horton Conway, ''The Book of Numbers''. New York: Springer (1996): 129 - 130</ref> [[Henri Lebesgue]] is said to be the last professional mathematician to call 1 prime.{{Fact|date=September 2008}} 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.”<ref>{{cite book|last=Gowers|first=T|authorlink=William Timothy Gowers|year=2002|pages=118|quote=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|title=Mathematics: A Very Short Introduction|publisher=[[Oxford University Press]]|id=ISBN 0-19-285361-9}}</ref><ref>"[http://primes.utm.edu/notes/faq/one.html "Why is the number one not prime?"]". Retrieved 2007-10-02.</ref> 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.<ref>"[http://www.geocities.com/primefan/Prime1ProCon.html "Arguments for and against the primality of 1]".</ref>


* 1. Specific mention as a constitutional document in section 52(2) of the Constitution Act, 1982, such as the Constitution Act, 1867.
==Prime divisors==
* 2. Constitutional entrenchment of an otherwise statutory English, British, or Canadian document because of subject matter provisions in the amending formula of the Constitution Act, 1982, such as provisions with regard to the monarchy in the English [[Bill of Rights 1689]] or the [[Act of Settlement 1701]]. English and British statutes are part of Canadian law because of the Colonial Laws Validity Act, 1865, section 129 of the Constitution Act, 1867, and the [[Statute of Westminster 1931]]. Those laws then became entrenched when the amending formula was made part of the constitution.
[[Image:Prime rectangles.png|thumb|Illustration showing that 11 is a prime number while 12 is not.]]
* 3. Reference by an entrenched document, such as the Preamble of the Constitution Act, 1867's entrenchment of written and unwritten principles from the constitution of the United Kingdom or the [[Constitution Act, 1982]]'s reference of the [[Proclamation of 1763]].


=== Unwritten sources ===
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 [[divisor|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
The existence of an unwritten constitution was reaffirmed by the Supreme Court in ''[[Reference re Secession of Quebec]]''.
<blockquote>
''The Constitution is more than a written text. It embraces the entire global system of rules and principles which govern the exercise of constitutional authority. A superficial reading of selected provisions of the written constitutional enactment, without more, may be misleading.''
</blockquote>


In practice, there have been three sources of unwritten constitutional law:
: <math>23244 = 2^2 \times 3 \times 13 \times 149</math>


'''Conventions:''' [[Constitutional convention (political custom)|Constitutional convention]]s form part of the Constitution, but they are not legally enforceable. They include the existence of the Prime Minister and Parliamentary Cabinet, the fact that the Governor General is required to give assent to Bills, and the requirement that the Prime Minister call an election upon losing a vote of non-confidence.
and any other factorization of 23244 as the product of primes will be identical except for the order of the factors. There are many [[Integer factorization|prime factorization]] algorithms to do this in practice for larger numbers.


'''Royal Prerogative:''' Reserve powers of the [[Monarchy in Canada|Canadian Crown]]; being remnants of the powers once held by the British Crown, reduced over time by the Parliamentary system. Primarily, these are the [[Order-in-Council|Orders-in-Council]] which give the Government the authority to declare war, conclude treaties, issue passports, make appointments, make regulations, incorporate, and receive lands that escheat to the Crown.
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.


'''Unwritten Principles:''' Principles that are incorporated into the Canadian Constitution by reference from the preamble of the ''Constitution Act, 1867''. Unlike conventions, they are legally binding. Amongst the recognized Constitutional principles are federalism, democracy, constitutionalism and the [[rule of law]], and respect for minorities.<ref>these were identified in [[Reference re Secession of Quebec]] [1998] 2 S.C.R. 217</ref> Other principles include [[responsible government]], [[judicial independence]] and an [[Implied Bill of Rights]]. In one case, the ''[[Provincial Judges Reference]]'' (1997), it was found a law can be held invalid for contradicting unwritten principles, in this case judicial independence.
==Properties==
* 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.
* The [[ring (algebra)|ring]] '''Z'''/''n'''''Z''' (see [[modular arithmetic]]) is a [[field (mathematics)|field]] [[if and only if]] ''n'' is a prime. Put another way: ''n'' is prime if and only if [[Euler's totient function|φ(''n'')]] = ''n''&nbsp;&minus;&nbsp;1.
* If ''p'' is prime and ''a'' is any integer, then ''a''<sup>''p''</sup>&nbsp;&minus;&nbsp;''a'' is divisible by ''p'' ([[Fermat's little theorem]]).
* If ''p'' is a prime number other than 2 and 5, <sup>1</sup>/<sub>''p''</sub> is always a [[recurring decimal]], whose period is ''p''&nbsp;&minus;&nbsp;1 or a divisor of ''p''&nbsp;&minus;&nbsp;1. This can be deduced directly from [[Fermat's little theorem]]. <sup>1</sup>/<sub>''p''</sub> 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 decimal]]s shows some of the interesting properties.
* An integer ''p'' > 1 is prime if and only if the [[factorial]] (''p''&nbsp;&minus;&nbsp;1)! + 1 is divisible by ''p'' ([[Wilson's theorem]]). Conversely, an integer ''n'' > 4 is composite if and only if (''n''&nbsp;&minus;&nbsp;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'' < 2''n'' ([[Bertrand's postulate]]).
* Adding the reciprocals of all primes together results in a divergent [[infinite series]] ([[Proof that the sum of the reciprocals of the primes diverges|proof]]). More precisely, if ''S''(''x'') denotes the sum of the reciprocals of all prime numbers ''p'' with ''p'' ≤ ''x'', then ''S''(''x'') = ln&nbsp;ln&nbsp;''x''&nbsp;+&nbsp;[[Big O notation|O]](1) for ''x''&nbsp;→&nbsp;∞.
* In every arithmetic progression ''a'', ''a''&nbsp;+&nbsp;''q'', ''a''&nbsp;+&nbsp;2''q'', ''a''&nbsp;+&nbsp;3''q'',&nbsp;… where the positive integers ''a'' and ''q'' are [[coprime]], there are infinitely many primes ([[Dirichlet's theorem on arithmetic progressions]]).
* The [[Characteristic (algebra)|characteristic]] of every field is either zero or a prime number.
* If ''G'' is a finite [[group (mathematics)|group]] and ''p''<sup>''n''</sup> is the [[p-adic order|highest power of the prime ''p'' which divides]] the order of ''G'', then ''G'' has a subgroup of order ''p''<sup>''n''</sup>. ([[Sylow theorems]].)
* 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's theorem (group theory)|Cauchy Theorem]])
* The [[prime number theorem]] says that the proportion of primes less than ''x'' is asymptotic to <sup>1</sup>/<sub>ln&nbsp;''x''</sub> (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 [[Copeland-Erdős constant]] 0.235711131719232931374143…, obtained by concatenating the prime numbers in [[Decimal|base ten]], is known to be an [[irrational number]].
* 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:
::<math>\zeta(s)=
\sum_{n=1}^\infin \frac{1}{n^s} = \prod_{p} \frac{1}{1-p^{-s}}.</math>
:Evaluating this identity at different integers provides an infinite number of products over the primes whose values can be calculated, the first two being
::<math>\prod_{p} \frac{1}{1-p^{-1}} = \infty</math>
::<math>\prod_{p} \frac{1}{1-p^{-2}}= \frac{\pi^2}{6}.</math>
* If ''p'' > 1, the polynomial <math> x^{p-1}+x^{p-2}+ \cdots + 1 </math> is irreducible over '''Z'''/''p'''''Z''' if and only if ''p'' is prime.
* An integer n is prime if and only if the <math> nth </math> [[Chebyshev polynomial]] of the first kind <math> T_{n}(x) </math>, divided by <math> x </math> is irreducible in <math> Z[x] </math>. Also <math> T_{n}(x) \equiv x^n </math> if and only if <math> n </math> is prime.
* All prime numbers above 3 are of the form 6''n''&nbsp;−&nbsp;1 or 6''n''&nbsp;+&nbsp;1, because all other numbers are divisible by 2 or 3. Generalizing this, all prime numbers above ''q'' are of form [[primorial|''q''#]]·''n''&nbsp;+&nbsp;''m'', where 0 < ''m'' < ''q'', and ''m'' has no prime factor ≤&nbsp;''q''.


== External links ==
===Classification===
{{wikibookspar||Canadian Constitutional Law}}
Two ways of classifying prime numbers, class ''n''+ and class ''n''&minus;, were studied by [[Paul Erdős]] and [[John Selfridge]].
{{wikisourcepar|British North America Act 1867}}
{{wikisourcepar|Canadian Charter of Rights and Freedoms}}


* [http://laws.justice.gc.ca/en/const/index.html Full text of the Constitution]
Determining the class ''n''+ of a prime number ''p'' involves looking at the largest prime factor of ''p''&nbsp;+&nbsp;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 {{OEIS2C|id=A005105}} through {{OEIS2C|id=A005108}} list class 1+ through class 4+ primes.
* [http://www.canadiana.org/citm/ Canada in the Making] - a comprehensive history of the Canadian Constitution with digitized primary sources.
* [http://www.charterofrights.ca/language.php Fundamental Freedoms: The Charter of Rights and Freedoms] - Charter of Rights and Freedoms website with video, audio and the Charter in over 20 languages
* [http://www.solon.org/Constitutions/Canada/English/Proposals/MeechLake.html Meech Lake Accord, 1987]
* [http://www.solon.org/Constitutions/Canada/English/Proposals/CharlottetownConsensus.html Charlottetown Accord, 1992]
* [http://www.solon.org/Constitutions/Canada/English/Proposals/charlottetown-res.html Results of Referendum on the Charlottetown Accord, 1992]
* [http://archives.cbc.ca/IDD-1-73-1092/politics_economy/Patriation/ CBC Digital Archives - Charting the Future: Canada's New Constitution]
* [http://archives.cbc.ca/IDD-1-73-394/politics_economy/constitution_debate_1/ CBC Digital Archives - Canada's Constitutional Debate: What Makes a Nation?]


==References==
The class ''n''&minus; is almost the same as class ''n''+, except that the factorization of ''p''&nbsp;&minus;&nbsp;1 is looked at instead.
{{reflist}}

==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:

<blockquote>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.</blockquote>

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 [[primorial|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 [[Leonhard Euler|Euler]]) shows that [[proof that the sum of the reciprocals of the primes diverges|the sum of the reciprocals of all prime numbers diverges]].
Another [[Fermat number#Basic properties| proof]] based on [[Fermat number]]s was given by [[Christian Goldbach|Goldbach]].<ref>[http://www.math.dartmouth.edu/~euler/correspondence/letters/OO0722.pdf Letter] in [[Latin]] from Goldbach to Euler, July 1730.</ref>
[[Ernst Kummer|Kummer]]'s is particularly elegant<ref>P. Ribenboim: ''The Little Book of Bigger Primes'', second edition, Springer, 2004, p. 4.</ref> and [[Harry Furstenberg]] provides [[Furstenberg's proof of the infinitude of primes|one using general topology]].<ref>{{cite journal|author=Furstenberg, Harry.|url=http://www.jstor.org/stable/2307043|title=On the infinitude of primes|journal=[[American Mathematical Monthly|Amer. Math. Monthly]]|volume=62|issue=5|year=1955|pages=353|doi=10.2307/2307043}}</ref><ref>{{cite web|title=Furstenberg's proof that there are infinitely many prime numbers|url=http://www.everything2.com/index.pl?node_id=1460203|accessdate=2006-11-26|work=[[Everything2]]}}</ref>

===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]] π(''x'') is defined as the number of primes up to ''x''. There are known [[algorithm]]s to compute exact values of π(''x'') faster than it would be possible to compute each prime up to ''x''. Values as large as π(10<sup>20</sup>) can be calculated quickly and accurately with modern computers. Thus, e.g., π(100,000) = 9592, and π(10<sup>20</sup>) = 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: π(''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 test]]s. 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 [[pseudoprime]]s 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

: <math>\frac {N}{\ln N - 1}.</math>

So, to check ''N'' for primality the largest prime factor needed is just less than <math>\scriptstyle\sqrt{N}</math>, and so the number of such prime factor candidates would be close to

: <math>\frac {\sqrt{N}}{\ln\sqrt{N} - 1}.</math>

This increases ever more slowly with ''N'', but, because there is interest in large values for ''N'', the count is large also: for ''N''&nbsp;=&nbsp;10<sup>&nbsp;20</sup> it is 450 million.

===Primality tests===

{{main|primality test}}

A [[primality test|primality test algorithm]] is an algorithm which tests a number for primality, i.e. whether the number is a prime number.

* [[AKS primality test]]
* [[Fermat primality test]]
* [[Lucas-Lehmer test]]
* [[Solovay-Strassen primality test]]
* [[Miller-Rabin primality test]]
* [[Elliptic curve primality proving]]

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 number]]s) are called [[pseudoprime]]s.

In 2002, Indian scientists at [[IIT Kanpur]] discovered a new deterministic algorithm known as the [[AKS primality test|AKS algorithm]]. The amount of time that this algorithm takes to check whether a number N is prime depends on a [[P (complexity)|polynomial function of the number of digits of ''N'']] (i.e. of the logarithm of ''N'').

===Formulas yielding prime numbers===
{{main|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'') = ''n''<sup>2</sup> − ''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 prime|primorial]]'' or ''prime-factorial'' if it has the form ''p'' = ''n''# ± 1 for some number ''n'', where [[primorial|''n''#]] stands for the product 2 · 3 · 5 · 7 · 11 · … of all the primes ≤ n. A prime is called ''[[factorial prime|factorial]]'' if it is of the form [[factorial|''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, … {{OEIS2C | id=A002982}}

: n! + 1 is prime for n = 0, 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, … {{OEIS2C | id=A002981}}

The largest known primorial prime is Π(392113) + 1, found by Heuer in 2001.<ref>[http://primes.utm.edu/top20/page.php?id=5 The Top Twenty: Primorial]</ref> The largest known factorial prime is 34790! − 1, found by Marchal, Carmody and Kuosa in 2002.<ref>[http://primes.utm.edu/top20/page.php?id=30 The Top Twenty: Factorial]</ref> It is not known whether there are infinitely many primorial or factorial primes.

Primes of the form 2<sup>''p''</sup> − 1, where ''p'' is a prime number, are known as [[Mersenne prime]]s, while primes of the form <math>2^{2^n} + 1</math> are known as [[Fermat prime]]s. Prime numbers ''p'' where 2''p'' + 1 is also prime are known as [[Sophie Germain prime]]s. The following list is of other special types of prime numbers that come from formulas:
<div style="-moz-column-count:2; column-count:2;">
* [[Wieferich prime]]s,
* [[Wilson prime]]s,
* [[Wall-Sun-Sun prime]]s,
* [[Wolstenholme prime]]s,
* [[Unique prime]]s,
* [[Newman-Shanks-Williams prime]]s (NSW primes),
* [[Smarandache-Wellin prime]]s,
* [[Wagstaff prime]]s, and
* [[Supersingular prime]]s.
</div>
Some primes are classified according to the properties of their digits in decimal or other bases. For example, numbers whose digits form a [[palindrome|palindromic]] sequence are called [[palindromic prime]]s, 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.
* For a list of special classes of prime numbers‎ see [[List of prime numbers‎]]

===Distribution===
{{further|[[Prime number theorem]]}}
[[Image:PrimeNumbersSmall.png|frame|right|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 number]]s 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.<ref>Julian Havil, ''Gamma: Exploring Euler's Constant (Hardcover)''. Princeton: Princeton University Press (2003): 163</ref>
In a 1975 lecture, [[Don Zagier]] commented
<blockquote>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.</blockquote><ref>Havil (2003): 171</ref>

Additional image with [[:Image:Primenumbers2310inv.png|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|Prime gap}}

Let ''p''<sub>''n''</sub> denote the ''n''th prime number (i.e. ''p''<sub>1</sub> = 2, ''p''<sub>2</sub> = 3, etc.). The ''gap'' ''g''<sub>''n''</sub> between the consecutive primes ''p''<sub>''n''</sub> and ''p''<sub>''n'' + 1</sub> is the difference between them, i.e.

: ''g''<sub>''n''</sub> = ''p''<sub>''n'' + 1</sub> &minus; ''p''<sub>''n''</sub>.

We have ''g''<sub>1</sub> = 3 &minus; 2 = 1, ''g''<sub>2</sub> = 5 &minus; 3 = 2, ''g''<sub>3</sub> = 7 &minus; 5 = 2, ''g''<sub>4</sub> = 11 &minus; 7 = 4, and so on. The sequence (''g''<sub>''n''</sub>) 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'' &minus; 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''<sub>''n''</sub> > ''N''. (Choose ''n'' so that ''p''<sub>''n''</sub> 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''<sub>''n''</sub>/''p''<sub>''n''</sub> [[limit (mathematics)|approaches]] zero as ''n'' approaches infinity. Note also that the [[twin prime conjecture]] asserts that ''g''<sub>''n''</sub> = 2 for infinitely many integers ''n''.

===Location of the largest known prime===
{{main|Largest known prime|Mersenne prime}}
{{wikinews|CMSU computing team discovers another record size prime}}
{{wikinews|Distributed computing discovers largest known prime}}

{{Asof|2008|September}}, the largest known prime was discovered by the [[distributed computing]] project [[Great Internet Mersenne Prime Search]] (GIMPS):

:2<sup>43,112,609</sup> − 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. The 46th known Mersenne prime, 2<sup>37,156,667</sup> − 1, was discovered two weeks later, but it is smaller.

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<sup>13,018,586</sup> + 1 (3,918,990 digits), a [[Proth's theorem|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 number|Sierpiński 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 <var>n</var>, multiplying it by 256<sup><var>k</var></sup> for some positive integer <var>k</var>, and searching for possible primes within the interval [256<sup>''k''</sup>''n'' + 1, 256<sup>''k''</sup>(''n'' + 1) &minus; 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. They may pay out $100,000 to GIMPS and the [[UCLA]] mathematics department for discovering a 13 million digit prime number in August 2008.[http://www.allheadlinenews.com/articles/7012470624][http://www.tgdaily.com/content/view/39527/113/]

The [[RSA Factoring Challenge]] offered prizes up to US$200,000 for finding the prime factors of certain [[semiprime]]s of up to 2048 bits. However, the challenge was closed in 2007 after much smaller prizes for smaller semiprimes had been paid out.<ref>[http://www.rsa.com/rsalabs/node.asp?id=2092 The RSA Factoring Challenge — RSA Laboratories]</ref>

==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 element]]s and [[irreducible element]]s 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 integer]]s '''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 prime]]s. 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 4''k'' + 3 are Gaussian primes, whereas rational primes of the form 4''k'' + 1 are not.

===Prime ideals===

In [[ring theory]], one generally replaces the notion of number with that of [[ideal (ring theory)|ideal]]. ''[[Prime ideal]]s'' are an important tool and object of study in [[commutative algebra]], [[number theory|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 (4''k'' + 3) are ''inert'' (remain prime), and prime ideals of the form (4''k'' + 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 (mathematics)|field]] ''K'', one considers [[valuation]]s on ''K'', certain functions from ''K'' to the real numbers '''R'''. Every such valuation yields a [[topological field|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 number]]s are represented by the standard [[absolute value]] function (known as the [[infinite prime]]) as well as by the [[p-adic number|''p''-adic valuations]] on '''Q''', for every prime number ''p''.

===Prime knots===

In [[knot theory]], a '''prime knot''' is a [[knot (mathematics)|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 number]]s: It is not known whether or not there are an infinite number of prime Euclid numbers.
* [[Goldbach's conjecture|Strong Goldbach conjecture]]: Every even integer greater than 2 can be written as a sum of two primes.
* [[Goldbach's weak conjecture|Weak Goldbach conjecture]]: Every odd integer greater than 5 can be written as a sum of three primes.
* [[Twin prime conjecture]]: There are infinitely many [[twin prime]]s, pairs of primes with difference 2.
* [[Polignac's conjecture]]: For every positive integer n, there are infinitely many pairs of consecutive primes which differ by 2''n''. 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 widely believed there are infinitely many [[Mersenne prime]]s, but not [[Fermat prime]]s.<ref>E.g., see {{citation|last=Guy|first=Richard K.|title=Unsolved Problems in Number Theory|publisher=Springer-Verlag|year=1981}}, problem A3, pp. 7–8.</ref>
* It is conjectured there are infinitely many primes of the form ''n''<sup>2</sup> + 1.<ref>{{MathWorld|urlname=LandausProblems|title=Landau's Problems}}</ref>
* Many well-known conjectures are special cases of the broad [[Schinzel's hypothesis H]].
* It is conjectured that there are infinitely many [[Fibonacci prime]]s.<ref>Caldwell, Chris, [http://primes.utm.edu/top20/page.php?id=48 ''The Top Twenty: Lucas Number''] at The [[Prime Pages]].</ref>
* [[Legendre's conjecture]]: There is a prime number between ''n''<sup>2</sup> and (''n'' + 1)<sup>2</sup> for every positive integer ''n''.
* [[Cramér's conjecture]]: <math>\limsup_{n\rightarrow\infty} \frac{p_{n+1}-p_n}{(\log p_n)^2} = 1</math>. 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, ''n''<sup>2</sup>+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 [[United Kingdom|British]] mathematician [[G. H. Hardy]] prided themselves on doing work that had absolutely no military significance.<ref>{{cite book|quote = 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 | last = Hardy | first = G.H. | authorlink = G. H. Hardy | title = [[A Mathematician's Apology]] | publisher = [[Cambridge University Press]] | year = 1940 | id = ISBN 0-521-42706-1 }}</ref> 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 table]]s and [[pseudorandom number generator]]s.

Some [[rotor machine]]s 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===
{{main|public key cryptography}}

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

===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 [[Symmetry (biology)#Pentamerism|five-fold symmetry]] remains a mystery.

One example of the use of prime numbers in nature is as an evolutionary strategy used by [[cicada]]s of the genus ''[[Magicicada]]''.<ref>Goles, E., Schulz, O. and M. Markus (2001). "Prime number selection of cycles in a predator-prey model", Complexity 6(4): 33-38</ref> These insects spend most of their lives as [[larva|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''.<ref>{{cite journal | author = Paulo R. A. Campos, Viviane M. de Oliveira, Ronaldo Giro, and Douglas S. Galvão. | url = http://link.aps.org/abstract/PRL/v93/e098107 | title = Emergence of Prime Numbers as the Result of Evolutionary Strategy | journal = [[Physical Review Letters|Phys. Rev. Lett.]] | volume = 93 | doi = 10.1103/PhysRevLett.93.098107 | year = 2004 | accessdate = 2006-11-26 | pages = 098107 }}</ref> 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.<ref>{{cite web |work=[[The Economist]]| url=http://economist.com/PrinterFriendly.cfm?Story_ID=2647052 |title=Invasion of the Brood |date=May 6, 2004|accessdate=2006-11-26 }}</ref> 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. <ref>{{cite web |author=Ivars Peterson | work=[[MAA Online]]| url=http://www.maa.org/mathland/mathtrek_6_28_99.html |title=The Return of Zeta |date=June 28, 1999|accessdate=2008-03-14 }}</ref>

==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". <ref> The Messiaen companion', ed. Peter Hill, Amadeus Press, 1994. ISBN 0-931340-95-0 </ref>

In his science fiction novel ''[[Contact (novel)|Contact]]'', later made into a [[Contact (film)|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. <ref>[[Carl Pomerance]], [http://www.math.dartmouth.edu/~carlp/PDF/extraterrestrial.pdf Prime Numbers and the Search for Extraterrestrial Intelligence], Retrieved on December 22, 2007</ref>

[[Tom Stoppard]]'s award-winning 1993 play ''[[Arcadia (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. <ref> Tom Stoppard, Arcadia, Faber and Faber, 1993. ISBN 0-571-16934-1.</ref>
<ref> ''The Cambridge Companion to Tom Stoppard'', ed. Katherine E. Kelly, Cambridge University Press, 2001. ISBN 0521645921 </ref> <ref>
[http://www.msri.org/communications/forsale/arcadia The Mathematics of Arcadia], an event involving Tom Stoppard and [[MSRI]] in the [[University of California, Berkeley]]</ref>

Many films reflect a popular fascination with the mysteries of prime numbers and cryptography: films such as ''[[Cube (film)| Cube]]'', ''[[Sneakers (film)|Sneakers]]'', ''[[The Mirror Has Two Faces]]'' and ''[[A Beautiful Mind (film)|A Beautiful Mind]]'', the latter of which is based on the biography of the mathematician and Nobel laureate [[John Forbes Nash]] by [[Sylvia Nasar]].<ref> [http://www.musicoftheprimes.com/films.htm Music of the Spheres], [[Marcus du Sautoy]]'s selection of films featuring prime numbers</ref> <ref>[http://web.archive.org/web/20020611080521/http://www.nytimes.com/books/first/n/nasar-mind.html A Beautiful Mind]</ref>

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.<ref>[http://math.cofc.edu/kasman/MATHFICT/mfview.php?callnumber=mf476 - A Mathematician reviews PopCo]</ref>


==See also==
==See also==
*[[Law of Canada]].
<div style="-moz-column-count:3; column-count:3;">
*[[Canadian Bill of Rights]].
* [[Full cycle]]
* [[Gödel number]]
* [[Hilbert number]]
* [[Integer factorization]]
* [[Irreducible polynomial]]
* [[Logarithmic integral function]]
* [[Prime power]]
* [[Primon gas]]
* [[Sphenic number]]
* [[List of prime numbers‎]] (list of special classes of prime numbers‎)
</div>

==Distributed computing projects that search for primes==
*[[Great Internet Mersenne Prime Search|GIMPS]] searches for [[Mersenne prime]]s.
*[[PrimeGrid]] searches for [[megaprime]]s.
*[[Seventeen or Bust]] searches for primes which can help prove that 78557 is the smallest [[Sierpinski number]].
*[[Twin Prime Search]] searches for record [[twin prime]]s
*[[Wieferich@Home]] searches for [[Wieferich prime]]s.

==Notes==
{{reflist|2}}

==References==
{{refbegin}}
* 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. Springer-Verlag, 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 0-06-621070-4. [http://www.musicoftheprimes.com/ The Music of Primes website].
* Karl Sabbagh, ''The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics''. Farrar, Straus and Giroux; 340 pages
{{refend}}

==External links==
{{Wikinews|Two largest known prime numbers discovered just two weeks apart, one qualifies for $100k prize}}

* Caldwell, Chris, The [[Prime Pages]] at [http://primes.utm.edu/ primes.utm.edu].
* [http://mathworld.wolfram.com/topics/PrimeNumbers.html Prime Numbers at MathWorld]
* [http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/Prime_numbers.html MacTutor history of prime numbers]
* [http://www.primepuzzles.net/ The prime puzzles]
* [http://aleph0.clarku.edu/~djoyce/java/elements/bookIX/propIX20.html An English translation of Euclid's proof that there are infinitely many primes]
* [http://www.numberspiral.com/index.html Number Spiral with prime patterns]
* [http://www.maths.ex.ac.uk/~mwatkins/zeta/vardi.html An Introduction to Analytic Number Theory, by Ilan Vardi and Cyril Banderier]
* [http://www.eff.org/awards/coop.php EFF Cooperative Computing Awards]
* ''[http://demonstrations.wolfram.com/WhyANumberIsPrime/ Why a Number Is Prime]'' by Enrique Zeleny, [[The Wolfram Demonstrations Project]].
===Prime number generators & calculators===
* [http://www.numberempire.com/primenumbers.php Online Prime Number Generator and Checker] - instantly checks and finds prime numbers up to 128 digits long (does NOT require Java or Javascript)
* [http://www.easycalculation.com/prime-number.php Prime number calculator] — Check prime number, and find next largest and next smallest prime numbers (requires Javascript).
* [http://www.alpertron.com.ar/ECM.HTM Fast Online primality test — Dario Alpern's personal site] – Makes use of the Elliptic Curve Method (up to thousands digits numbers check!, requires Java)
* [http://publicliterature.org/tools/prime_number_generator Prime Number Generator] — Generates a given number of primes above a given start number.
* [http://wims.unice.fr/wims/wims.cgi?module=tool/number/primes.en Primes] from WIMS is an online prime generator.
* [http://www.bigprimes.net/archive/prime.php Huge database of prime numbers]


{{Constitution of Canada}}
[[Category:Integer sequences]]
{{Canada topics}}
[[Category:Prime numbers|*]]
[[Category:Articles containing proofs]]


[[Category:Constitution of Canada| ]]
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[[Category:Constitutions|Canada]]


[[de:Verfassung von Kanada]]
[[af:Priemgetal]]
[[es:Constitución de Canadá]]
[[ang:Frumtæl]]
[[ar:عدد أولي]]
[[fr:Constitution du Canada]]
[[he:חוקת קנדה]]
[[bn:মৌলিক সংখ্যা]]
[[nl:Canadese Grondwet]]
[[zh-min-nan:Sò͘-sò͘]]
[[pl:Konstytucja Kanady]]
[[be-x-old:Просты лік]]
[[pt:Constituição do Canadá]]
[[bs:Prost broj]]
[[br:Niveroù kentael]]
[[bg:Просто число]]
[[ca:Nombre primer]]
[[cs:Prvočíslo]]
[[cy:Rhif cysefin]]
[[da:Primtal]]
[[de:Primzahl]]
[[et:Algarv]]
[[el:Πρώτος αριθμός]]
[[es:Número primo]]
[[eo:Primo]]
[[eu:Zenbaki lehen]]
[[fa:عدد اول]]
[[fr:Nombre premier]]
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[[it:Numero primo]]
[[he:מספר ראשוני]]
[[ka:მარტივი რიცხვი]]
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[[lv:Pirmskaitlis]]
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[[lt:Pirminis skaičius]]
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[[ms:Nombor perdana]]
[[mn:Энгийн тоо]]
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[[no:Primtall]]
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[[pt:Número primo]]
[[ro:Număr prim]]
[[ru:Простое число]]
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[[simple:Prime number]]
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[[yo:Nọ́mbà àkọ́kọ́]]
[[zh-yue:質數]]
[[zh:素数]]

Revision as of 00:28, 13 October 2008

Politics of Canada
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The Constitution of Canada is the supreme law in Canada; the country's constitution is an amalgamation of codified acts and uncodified traditions and conventions. It outlines Canada's system of government, as well as the civil rights of all Canadian citizens.

The composition of the Constitution of Canada is defined in subsection 52(2) of the Constitution Act, 1982 as consisting of the Canada Act 1982 (including the Constitution Act, 1982), all acts and orders referred to in the schedule (including the Constitution Act, 1867, formerly the British North America Act), and any amendments to these documents.[1] The Supreme Court of Canada held that the list is not exhaustive and includes unwritten doctrines as well.[2]

History of the Constitution

Main article: Constitutional history of Canada
File:Fathers of Canadian Confederation.jpg
A painting depicting negotiations that would lead to the enactment of the British North America Act, 1867

The first semblance of a Constitution for Canada was the Royal Proclamation of 1763. The Act renamed Canada "The Province of Quebec" and redefined its borders and established a British-appointed colonial government. The proclamation was considered the de facto constitution of Quebec until 1774 when the British government passed the Quebec Act of 1774 which set out many procedures of governance in the area of Quebec. It extended the boundaries of the colony and adopted the British criminal code among other things.

The colony of Canada received its first full constitution in the Constitutional Act of 1791 which established much of the composition of the government. This was later superseded by the British North America Act in 1867 which established the Dominion of Canada.

In 1931, the British Parliament passed the Statute of Westminster, 1931 (22 Geo. V, c.4 (UK)). This Act gave all dominion countries equal legislative authority with the United Kingdom. This was followed up in 1982, when the British Parliament passed the Canada Act, 1982 ([UK] 1982, c.11) giving up all remaining constitutional and legislative authority over Canada. The enactment of the Canada Act is often referred to in Canada as the 'patriation' of the constitution and it was largely due to the work of Pierre Elliot Trudeau, the Prime Minister of Canada at the time.

With the introduction of the Canada Act and the accompanying Charter, much of Constitutional law in Canada has changed. The Canada Act has entrenched many constitutional conventions and has made amendments significantly more difficult (see amendment formula). The Charter has shifted the focus of the Constitution to individual and collective rights of the inhabitants of Canada. Before the enactment of the Canadian Charter of Rights and Freedoms in 1982, civil rights and liberties had no solid constitutional protection in Canada. Whenever one level of government passed a law that seemed oppressive to civil rights and liberties, Canadian constitutional lawyers had to argue creatively, such as by saying that the oppressive law violates division of federal and provincial powers or by citing some other technical flaw that had little to do with the concept of civil rights and liberties.[citation needed] Since 1982, however, the Charter has become the most often cited part of the Constitution and has thus far solidified the protection of rights for people in Canada.

Constitution Act, 1867

Main article: Constitution Act, 1867

This was an Act of the British Parliament, originally called the British North America Act 1867, that created the Dominion of Canada out of three separate provinces in British North America (Province of Canada, New Brunswick, and Nova Scotia) and allowed for subsequent provinces and colonies to join this union in the future. It outlined Canada's system of government, which combines Britain's Westminster model of parliamentary government with division of sovereignty (federalism). Although it is one of many British North America Acts to come, it is still the most famous of these and is understood to be the document of Canadian Confederation (i.e. union of provinces and colonies in British North America). With the patriation of the Constitution in 1982, this Act was renamed Constitution Act, 1867. In recent years, the Constitution Act, 1867 has mainly served as the basis on which the division of powers between the provinces and federal government have been analyzed.

Constitution Act, 1982

See also: Constitution Act, 1982

Endorsed by all the provincial governments except Quebec's, this was an Act by the Canadian Parliament requesting full political independence from Britain. Part V of this Act created a constitution-amending formula that did not require an Act by the British Parliament. Further, Part I of this Act is the Canadian Charter of Rights and Freedoms which outlines the civil rights and liberties of every citizen in Canada, such as freedom of expression, of religion, and of mobility. Part II deals with the rights of Canada's Aboriginal peoples.

Canadian Charter of Rights and Freedoms

See also: Canadian Charter of Rights and Freedoms

As noted above, this is Part I of the Constitution Act, 1982. The Charter is the constitutional guarantee of collective and individual rights. It is a relatively short document and written in plain language in order to ensure accessibility to the average citizen. It is said that it is the part of the constitution that has the greatest impact on Canadians' day-to-day lives, and has been the fastest developing area of constitutional law for many years.

Amending formula

See also: Amendments to the Constitution of Canada

With the Constitution Act, 1982, amendments to the constitution must be done in accordance with Part V of the Constitution Act, 1982 which provides for five different amending formulas. Amendments can be brought forward under section 46(1) by any province or either level of the federal government. The general formula is set out in section 38(1), known as the "7/50 formula", requires: (a) assent from both the House of Commons and the Senate; (b) the approval of two-thirds of the provincial legislatures (at least seven provinces), representing at least 50% of the population (effectively, this would include at least Quebec or Ontario, as they are the most populous provinces). This formula specifically applies to amendments related to the proportionate representation in Parliament, powers, selection, and composition of the Senate, the Supreme Court and the addition of provinces or territories. The other amendment formulas are for exceptional cases as provided by in the Act:

  • In the case of an amendment related to the Office of the Queen, the number of senators, the use of either official language (subject to section 43), the amending formula, or the composition of the Supreme Court, the amendment must be adopted by unanimous consent of all the provinces in accordance with section 41.
  • However, in the case of an amendment related to provincial boundaries or the use of an official language within a province alone, the amendment must be passed by the legislatures affected by the amendment (section 43).
  • In the case of an amendment that affects the federal government alone, the amendment does not need approval of the provinces (section 44). The same applies to amendments affecting the provincial government alone (section 45).

Vandalism of the paper proclamation

In 1983, Toronto artist Peter Greyson entered Ottawa's National Archives (known today as Library and Archives Canada) and poured red paint over a copy of the proclamation of the 1982 constitutional amendment. He said he was displeased with the federal government's decision to allow U.S. missile testing in Canada, and had wanted to "graphically illustrate to Canadians" how wrong the government was. A grapefruit-sized stain still remains on the original document. Specialists opted to leave most of the paint intact fearing attempts at removing it would only do further damage.[3]

Sources of the Constitution

Further information: List of Canadian constitutional documents

There are three general methods of constitutional entrenchment:

  • 1. Specific mention as a constitutional document in section 52(2) of the Constitution Act, 1982, such as the Constitution Act, 1867.
  • 2. Constitutional entrenchment of an otherwise statutory English, British, or Canadian document because of subject matter provisions in the amending formula of the Constitution Act, 1982, such as provisions with regard to the monarchy in the English Bill of Rights 1689 or the Act of Settlement 1701. English and British statutes are part of Canadian law because of the Colonial Laws Validity Act, 1865, section 129 of the Constitution Act, 1867, and the Statute of Westminster 1931. Those laws then became entrenched when the amending formula was made part of the constitution.
  • 3. Reference by an entrenched document, such as the Preamble of the Constitution Act, 1867's entrenchment of written and unwritten principles from the constitution of the United Kingdom or the Constitution Act, 1982's reference of the Proclamation of 1763.

Unwritten sources

The existence of an unwritten constitution was reaffirmed by the Supreme Court in Reference re Secession of Quebec.

The Constitution is more than a written text. It embraces the entire global system of rules and principles which govern the exercise of constitutional authority. A superficial reading of selected provisions of the written constitutional enactment, without more, may be misleading.

In practice, there have been three sources of unwritten constitutional law:

Conventions: Constitutional conventions form part of the Constitution, but they are not legally enforceable. They include the existence of the Prime Minister and Parliamentary Cabinet, the fact that the Governor General is required to give assent to Bills, and the requirement that the Prime Minister call an election upon losing a vote of non-confidence.

Royal Prerogative: Reserve powers of the Canadian Crown; being remnants of the powers once held by the British Crown, reduced over time by the Parliamentary system. Primarily, these are the Orders-in-Council which give the Government the authority to declare war, conclude treaties, issue passports, make appointments, make regulations, incorporate, and receive lands that escheat to the Crown.

Unwritten Principles: Principles that are incorporated into the Canadian Constitution by reference from the preamble of the Constitution Act, 1867. Unlike conventions, they are legally binding. Amongst the recognized Constitutional principles are federalism, democracy, constitutionalism and the rule of law, and respect for minorities.[4] Other principles include responsible government, judicial independence and an Implied Bill of Rights. In one case, the Provincial Judges Reference (1997), it was found a law can be held invalid for contradicting unwritten principles, in this case judicial independence.

External links

The Wikibook [[wikibooks:|]] has a page on the topic of: Canadian Constitutional Law
Wikisource has original text related to this article:
Wikisource has original text related to this article:

References

  1. ^ See list of Canadian constitutional documents for details.
  2. ^ New Brunswick Broadcasting Co. v. Nova Scotia [1993] 1 S.C.R. 319
  3. ^ Missile Protestor defaces Constitution - "On This Day" - CBC Archives
  4. ^ these were identified in Reference re Secession of Quebec [1998] 2 S.C.R. 217

See also

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