In
mathematics, particularly 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, "Mathe ...
,
Hillel Furstenberg's proof of the infinitude of primes is a
topological
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
proof
Proof most often refers to:
* Proof (truth), argument or sufficient evidence for the truth of a proposition
* Alcohol proof, a measure of an alcoholic drink's strength
Proof may also refer to:
Mathematics and formal logic
* Formal proof, a co ...
that the
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 contain
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 number
A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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 ...
s. When examined closely, the proof is less a statement about topology than a statement about certain properties of
arithmetic sequence
An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...
s.
[ See discussion immediately prior to Lemma 3.2 or see Section 3.5.] Unlike Euclid's classical proof, Furstenberg's proof is a proof by contradiction. The proof was published in 1955 in the ''American Mathematical Monthly
''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America.
The ''American Mathematical Monthly'' is an ...
'' while Furstenberg was still an undergraduate student
Undergraduate education is education conducted after secondary education and before postgraduate education. It typically includes all postsecondary programs up to the level of a bachelor's degree. For example, in the United States, an entry-lev ...
at Yeshiva University
Yeshiva University is a private Orthodox Jewish university with four campuses in New York City.["About YU]
on the Yeshiva Universi ...
.
Furstenberg's proof
Define a topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
on the integers , called the evenly spaced integer topology, by declaring a subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
''U'' ⊆ 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 ...
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 bic ...
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
** ''U ...
of arithmetic sequences ''S''(''a'', ''b'') for ''a'' ≠ 0, or is empty (which can be seen as a nullary union (empty union) of arithmetic sequences), where
:
Equivalently, ''U'' is open if and only if for every ''x'' in ''U'' there is some non-zero integer ''a'' such that ''S''(''a'', ''x'') ⊆ ''U''. The axioms for a topology are easily verified:
* ∅ is open by definition, and is just the sequence ''S''(1, 0), and so is open as well.
* Any union of open sets is open: for any collection of open sets ''U''''i'' and ''x'' in their union ''U'', any of the numbers ''a''''i'' for which ''S''(''a''''i'', ''x'') ⊆ ''U''''i'' also shows that ''S''(''a''''i'', ''x'') ⊆ ''U''.
* The intersection of two (and hence finitely many) open sets is open: let ''U''1 and ''U''2 be open sets and let ''x'' ∈ ''U''1 ∩ ''U''2 (with numbers ''a''1 and ''a''2 establishing membership). Set ''a'' to be the least common multiple
In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), is the smallest positive integer that is divisible by bo ...
of ''a''1 and ''a''2. Then ''S''(''a'', ''x'') ⊆ ''S''(''a''''i'', ''x'') ⊆ ''U''i.
This topology has two notable properties:
# Since any non-empty open set contains an infinite sequence, a finite set cannot be open; put another way, the complement of a finite set cannot be a closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
.
# The basis sets ''S''(''a'', ''b'') are both open and closed: they are open by definition, and we can write ''S''(''a'', ''b'') as the complement of an open set as follows:
:::
The only integers that are not integer multiples of prime numbers are −1 and +1, i.e.
:::
Now, by the first topological property, the set on the left-hand side cannot be closed. On the other hand, by the second topological property, the sets ''S''(''p'', 0) are closed. So, if there were only finitely many prime numbers, then the set on the right-hand side would be a finite union of closed sets, and hence closed. This would be a contradiction
In traditional logic, a contradiction occurs when a proposition conflicts either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle' ...
, so there must be infinitely many prime numbers.
Topological properties
The evenly spaced integer topology on is the topology induced by the inclusion , where is the profinite integer In mathematics, a profinite integer is an element of the ring (sometimes pronounced as zee-hat or zed-hat)
:\widehat = \varprojlim \mathbb/n\mathbb = \prod_p \mathbb_p
where
:\varprojlim \mathbb/n\mathbb
indicates the profinite completion of \mathb ...
ring with its profinite topology.
It is homeomorphic
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorp ...
to the rational numbers
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
with the subspace topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced t ...
inherited from the real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
, which makes it clear that any finite subset of it, such as , cannot be open.
Notes
References
*
*
*
* Keith conrod https://kconrad.math.uconn.edu/blurbs/ugradnumthy/primestopology.pdf
*
External links
Furstenberg's proof that there are infinitely many prime numbers
at Everything2
Everything2 (styled Everything2 or E2 for short) is a collaborative Web-based community consisting of a database of interlinked user-submitted written material. E2 is moderated for quality, but has no formal policy on subject matter. Writing o ...
*
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