infinitude of the prime numbers
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Euclid's theorem is a fundamental statement in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
that asserts that there are
infinitely Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
many
prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
numbers. It was first proved by
Euclid Euclid (; grc-gre, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of ...
in his work '' Elements''. There are several proofs of the theorem.


Euclid's proof

Euclid offered a proof published in his work ''Elements'' (Book IX, Proposition 20), which is paraphrased here. Consider any finite list of prime numbers ''p''1, ''p''2, ..., ''p''''n''. It will be shown that at least one additional prime number not in this list exists. Let ''P'' be the product of all the prime numbers in the list: ''P'' = ''p''1''p''2...''p''''n''. Let ''q'' = ''P'' + 1. Then ''q'' is either prime or not: *If ''q'' is prime, then there is at least one more prime that is not in the list, namely, ''q'' itself. *If ''q'' is not prime, then some prime factor ''p'' divides ''q''. If this factor ''p'' were in our list, then it would divide ''P'' (since ''P'' is the product of every number in the list); but ''p'' also divides ''P'' + 1 = ''q'', as just stated. If ''p'' divides ''P'' and also ''q,'' then ''p'' must also divide the difference of the two numbers, which is (''P'' + 1) − ''P'' or just 1. Since no prime number divides 1, ''p'' cannot be in the list. This means that at least one more prime number exists beyond those in the list. This proves that for every finite list of prime numbers there is a prime number not in the list. In the original work, as Euclid had no way of writing an arbitrary list of primes, he used a method that he frequently applied, that is, the method of generalizable example. Namely, he picks just three primes and using the general method outlined above, proves that he can always find an additional prime. Euclid presumably assumes that his readers are convinced that a similar proof will work, no matter how many primes are originally picked. Euclid is often erroneously reported to have proved this result by contradiction beginning with the assumption that the
finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. T ...
initially considered contains all prime numbers, though it is actually a proof by cases, a direct proof method. The philosopher
Torkel Franzén Torkel Franzén (1 April 1950, Norrbotten County – 19 April 2006, Stockholm) was a Swedish academic. Biography Franzén worked at the Department of Computer Science and Electrical Engineering at Luleå University of Technology, Sweden, in the f ...
, in a book on logic, states, "Euclid's proof that there are infinitely many primes is not an indirect proof ..The argument is sometimes formulated as an indirect proof by replacing it with the assumption 'Suppose are all the primes'. However, since this assumption isn't even used in the proof, the reformulation is pointless."


Variations

Several variations on Euclid's proof exist, including the following: The factorial ''n''! of a positive integer ''n'' is divisible by every integer from 2 to ''n'', as it is the product of all of them. Hence, is not divisible by any of the integers from 2 to ''n'', inclusive (it gives a remainder of 1 when divided by each). Hence is either prime or divisible by a prime larger than ''n''. In either case, for every positive integer ''n'', there is at least one prime bigger than ''n''. The conclusion is that the number of primes is infinite.


Euler's proof

Another proof, by the Swiss mathematician
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
, relies on the
fundamental theorem of arithmetic In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the ord ...
: that every integer has a unique prime factorization. What Euler wrote (not with this modern notation and, unlike modern standards, not restricting the arguments in sums and products to any finite sets of integers) is equivalent to the statement that we have : \prod_ \frac=\sum_\frac, where P_k denotes the set of the first prime numbers, and N_k is the set of the positive integers whose prime factors are all in P_k. In order to show this, one expands each factor in the product as a
geometric series In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series :\frac \,+\, \frac \,+\, \frac \,+\, \frac \,+\, \cdots is geometric, because each suc ...
, and distributes the product over the sum (this is a special case of the
Euler product In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard Eu ...
formula for the Riemann zeta function). : \begin \prod_ \frac & =\prod_ \sum_ \frac\\ & = \left(\sum_ \frac\right) \cdot \left(\sum_ \frac\right) \cdot \left(\sum_ \frac\right) \cdot \left(\sum_ \frac\right)\cdots \\ & =\sum_ \frac \\ & =\sum_\frac. \end In the penultimate sum every product of primes appears exactly once, and so the last equality is true by the fundamental theorem of arithmetic. In his first corollary to this result Euler denotes by a symbol similar to \infty the « absolute infinity » and writes that the infinite sum in the statement equals the « value » \log\infty, to which the infinite product is thus also equal (in modern terminology this is equivalent to say that the partial sum up to x of the harmonic series diverges asymptotically like \log x). Then in his second corollary Euler notes that the product : \prod_ \frac converges to the finite value 2, and that there are consequently more primes than squares (« sequitur infinities plures esse numeros primos »). This proves Euclid Theorem. In the same paper (Theorem 19) Euler in fact used the above equality to prove a much stronger theorem that was unknown before him, namely that the series :\sum_\frac 1p is divergent, where denotes the set of all prime numbers (Euler writes that the infinite sum =\log\log\infty, which in modern terminology is equivalent to say that the partial sum up to x of this series behaves asymptotically like \log\log x).


Erdős's proof

Paul Erdős gave a proof that also relies on the fundamental theorem of arithmetic. Every positive integer has a unique factorization into a
square-free {{no footnotes, date=December 2015 In mathematics, a square-free element is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that s^2\mid r is a unit of ''R''. A ...
number and a square number . For example, . Let be a positive integer, and let be the number of primes less than or equal to . Call those primes . Any positive integer which is less than or equal to can then be written in the form :\left( p_1^ p_2^ \cdots p_k^ \right) s^2, where each is either or . There are ways of forming the square-free part of . And can be at most , so . Thus, at most numbers can be written in this form. In other words, :N \leq 2^k \sqrt. Or, rearranging, , the number of primes less than or equal to , is greater than or equal to . Since was arbitrary, can be as large as desired by choosing appropriately.


Furstenberg's proof

In the 1950s,
Hillel Furstenberg Hillel (Harry) Furstenberg ( he, הלל (הארי) פורסטנברג) (born September 29, 1935) is a German-born American-Israeli mathematician and professor emeritus at the Hebrew University of Jerusalem. He is a member of the Israel Academy o ...
introduced a proof by contradiction using
point-set topology In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...
. Define a topology on the integers Z, called the
evenly spaced integer topology In general topology and number theory, branches of mathematics, one can define various topologies on the set \mathbb of integers or the set \mathbb_ of positive integers by taking as a base a suitable collection of arithmetic progressions, sequenc ...
, by declaring a subset ''U'' ⊆ Z to be an
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is b ...
it is either the empty set, ∅, or it is a
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...
of arithmetic sequences ''S''(''a'', ''b'') (for ''a'' ≠ 0), where :S(a, b) = \ = a \mathbb + b. Then a contradiction follows from the property that a finite set of integers cannot be open and the property that the basis sets ''S''(''a'', ''b'') are both open and closed, since :\mathbb \setminus \ = \bigcup_ S(p, 0) cannot be closed because its complement is finite, but is closed since it is a finite union of closed sets.


Recent proofs


Proof using the inclusion-exclusion principle

Juan Pablo Pinasco has written the following proof. Let ''p''1, ..., ''p''''N'' be the smallest ''N'' primes. Then by the
inclusion–exclusion principle In combinatorics, a branch of mathematics, the inclusion–exclusion principle is a counting technique which generalizes the familiar method of obtaining the number of elements in the union of two finite sets; symbolically expressed as : , A \cu ...
, the number of positive integers less than or equal to ''x'' that are divisible by one of those primes is : \begin 1 + \sum_ \left\lfloor \frac \right\rfloor - \sum_ \left\lfloor \frac \right\rfloor & + \sum_ \left\lfloor \frac \right\rfloor - \cdots \\ & \cdots \pm (-1)^ \left\lfloor \frac \right\rfloor. \qquad (1) \end Dividing by ''x'' and letting ''x'' → ∞ gives : \sum_ \frac - \sum_ \frac + \sum_ \frac - \cdots \pm (-1)^ \frac. \qquad (2) This can be written as : 1 - \prod_^N \left( 1 - \frac \right). \qquad (3) If no other primes than ''p''1, ..., ''p''''N'' exist, then the expression in (1) is equal to \lfloor x \rfloor and the expression in (2) is equal to 1, but clearly the expression in (3) is not equal to 1. Therefore, there must be more primes than  ''p''1, ..., ''p''''N''.


Proof using de Polignac's formula

In 2010, Junho Peter Whang published the following proof by contradiction. Let ''k'' be any positive integer. Then according to
de Polignac's formula In mathematics, Legendre's formula gives an expression for the exponent of the largest power of a prime ''p'' that divides the factorial ''n''!. It is named after Adrien-Marie Legendre. It is also sometimes known as de Polignac's formula, aft ...
(actually due to Legendre) : k! = \prod_ p^ where : f(p,k) = \left\lfloor \frac \right\rfloor + \left\lfloor \frac \right\rfloor + \cdots. : f(p,k) < \frac + \frac + \cdots = \frac \le k. But if only finitely many primes exist, then : \lim_ \frac = 0, (the numerator of the fraction would grow singly exponentially while by
Stirling's approximation In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of n. It is named after James Stirling, though a related but less p ...
the denominator grows more quickly than singly exponentially), contradicting the fact that for each ''k'' the numerator is greater than or equal to the denominator.


Proof by construction

Filip Saidak gave the following
proof by construction In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existenc ...
, which does not use
reductio ad absurdum In logic, (Latin for "reduction to absurdity"), also known as (Latin for "argument to absurdity") or ''apagogical arguments'', is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absu ...
or Euclid's lemma (that if a prime ''p'' divides ''ab'' then it must divide ''a'' or ''b''). Since each natural number (> 1) has at least one prime factor, and two successive numbers ''n'' and (''n'' + 1) have no factor in common, the product ''n''(''n'' + 1) has more different prime factors than the number ''n'' itself.  So the chain of pronic numbers:
1×2 = 2 ,    2×3 = 6 ,    6×7 = 42 ,    42×43 = 1806 ,    1806×1807 = 3263442 , · · ·
provides a sequence of unlimited growing sets of primes.


Proof using the incompressibility method

Suppose there were only ''k'' primes (''p''1, ..., ''p''''k''). By the
fundamental theorem of arithmetic In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the ord ...
, any positive integer ''n'' could then be represented as n = ^ ^ \cdots ^, where the non-negative integer exponents ''e''''i'' together with the finite-sized list of primes are enough to reconstruct the number. Since p_i \geq 2 for all ''i'', it follows that e_i \leq \lg n for all ''i'' (where \lg denotes the base-2 logarithm). This yields an encoding for ''n'' of the following size (using big O notation): :O(\text + k \lg \lg n) = O(\lg \lg n) bits. This is a much more efficient encoding than representing ''n'' directly in binary, which takes N = O(\lg n) bits. An established result in
lossless data compression Lossless compression is a class of data compression that allows the original data to be perfectly reconstructed from the compressed data with no loss of information. Lossless compression is possible because most real-world data exhibits statistic ...
states that one cannot generally compress ''N'' bits of information into fewer than ''N'' bits. The representation above violates this by far when ''n'' is large enough since \lg \lg n = o(\lg n). Therefore, the number of primes must not be finite.


Stronger results

The theorems in this section simultaneously imply Euclid's theorem and other results.


Dirichlet's theorem on arithmetic progressions

Dirichlet's theorem states that for any two positive
coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s ''a'' and ''d'', there are infinitely many
primes A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
of the form ''a'' + ''nd'', where ''n'' is also a positive integer. In other words, there are infinitely many primes that are
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...
to ''a'' modulo ''d''.


Prime number theorem

Let be the prime-counting function that gives the number of primes less than or equal to , for any real number . The prime number theorem then states that is a good approximation to , in the sense that the limit of the ''quotient'' of the two functions and as increases without bound is 1: :\lim_ \frac=1. Using
asymptotic notation Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Lan ...
this result can be restated as :\pi(x)\sim \frac. This yields Euclid's theorem, since \lim_ \frac=\infty.


Bertrand–Chebyshev theorem

In
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mat ...
, Bertrand's postulate is a
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of t ...
stating that for any
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
n > 1, there always exists at least one
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
such that :n < p < 2n. Bertrand–Chebyshev theorem can also be stated as a relationship with \pi(x), where \pi(x) is the prime-counting function (number of primes less than or equal to x \,): :\pi(x) - \pi(\tfrac) \ge 1, for all x \ge 2. This statement was first conjectured in 1845 by
Joseph Bertrand Joseph Louis François Bertrand (; 11 March 1822 – 5 April 1900) was a French mathematician who worked in the fields of number theory, differential geometry, probability theory, economics and thermodynamics. Biography Joseph Bertrand was ...
(1822–1900). Bertrand himself verified his statement for all numbers in the interval His conjecture was completely proved by Chebyshev (1821–1894) in 1852. (Proof of the postulate: 371–382). Also see Mémoires de l'Académie Impériale des Sciences de St. Pétersbourg, vol. 7, pp. 15–33, 1854 and so the postulate is also called the Bertrand–Chebyshev theorem or Chebyshev's theorem.


Notes and references


External links

*
Euclid's Elements, Book IX, Prop. 20
(Euclid's proof, on David Joyce's website at Clark University) {{Ancient Greek mathematics Articles containing proofs Theorems about prime numbers