Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. It was first proven by Euclid in his work Elements . There are several proofs of the theorem.
Euclid offered a proof published in his work Elements (Book IX, Proposition 20), [1] which is paraphrased here. [2]
Consider any finite list of prime numbers p1, p2, ..., pn. 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 = p1p2...pn. Let q = P + 1. Then q is either prime or not:
This proves that for every finite list of prime numbers there is a prime number not in the list. [4] 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. [5]
Euclid is often erroneously reported to have proved this result by contradiction beginning with the assumption that the finite set initially considered contains all prime numbers, [6] though it is actually a proof by cases, a direct proof method. The philosopher Torkel Franzén, 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 q1, ... qn are all the primes'. However, since this assumption isn't even used in the proof, the reformulation is pointless." [7]
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, n! + 1 is not divisible by any of the integers from 2 to n, inclusive (it gives a remainder of 1 when divided by each). Hence n! + 1 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. [8]
Another proof, by the Swiss mathematician Leonhard Euler, relies on the fundamental theorem of arithmetic: 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 [9]
where denotes the set of the k first prime numbers, and is the set of the positive integers whose prime factors are all in
To show this, one expands each factor in the product as a geometric series, and distributes the product over the sum (this is a special case of the Euler product formula for the Riemann zeta function).
In the penultimate sum, every product of primes appears exactly once, 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 the « absolute infinity » and writes that the infinite sum in the statement equals the « value » , to which the infinite product is thus also equal (in modern terminology this is equivalent to say that the partial sum up to of the harmonic series diverges asymptotically like ). Then in his second corollary, Euler notes that the product
converges to the finite value 2, and there are consequently more primes than squares (« sequitur infinities plures esse numeros primos »). This proves Euclid's Theorem. [10]
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
is divergent, where P denotes the set of all prime numbers (Euler writes that the infinite sum , which in modern terminology is equivalent to say that the partial sum up to of this series behaves asymptotically like ).
Paul Erdős gave a proof [11] that also relies on the fundamental theorem of arithmetic. Every positive integer has a unique factorization into a square-free number r and a square number s2. For example, 75,600 = 24 33 52 71 = 21 ⋅ 602.
Let N be a positive integer, and let k be the number of primes less than or equal to N. Call those primes p1, ... , pk. Any positive integer a which is less than or equal to N can then be written in the form
where each ei is either 0 or 1. There are 2k ways of forming the square-free part of a. And s2 can be at most N, so s≤√N. Thus, at most 2k√N numbers can be written in this form. In other words,
Or, rearranging, k, the number of primes less than or equal to N, is greater than or equal to 1/2log2N. Since N was arbitrary, k can be as large as desired by choosing N appropriately.
In the 1950s, Hillel Furstenberg introduced a proof by contradiction using point-set topology. [12]
Define a topology on the integers Z, called the evenly spaced integer topology, by declaring a subset U ⊆ Z to be an open set if and only if it is either the empty set, ∅, or it is a union of arithmetic sequences S(a, b) (for a ≠ 0), where
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
cannot be closed because its complement is finite, but is closed since it is a finite union of closed sets.
Juan Pablo Pinasco has written the following proof. [13]
Let p1, ..., pN be the smallest N primes. Then by the inclusion–exclusion principle, the number of positive integers less than or equal to x that are divisible by one of those primes is
Dividing by x and letting x → ∞ gives
This can be written as
If no other primes than p1, ..., pN exist, then the expression in (1) is equal to 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 p1, ..., pN.
In 2010, Junho Peter Whang published the following proof by contradiction. [14] Let k be any positive integer. Then according to Legendre's formula (sometimes attributed to de Polignac)
where
But if only finitely many primes exist, then
(the numerator of the fraction would grow singly exponentially while by Stirling's approximation 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.
Filip Saidak gave the following proof by construction, which does not use reductio ad absurdum [15] or Euclid's lemma (that if a prime p divides ab then it must divide a or b).
Since each natural number greater than 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}, 2×3 = 6 {2, 3}, 6×7 = 42 {2, 3, 7}, 42×43 = 1806 {2, 3, 7, 43}, 1806×1807 = 3263442 {2, 3, 7, 43, 13, 139}, · · ·
provides a sequence of unlimited growing sets of primes.
Suppose there were only k primes (p1, ..., pk). By the fundamental theorem of arithmetic, any positive integer n could then be represented as
where the non-negative integer exponents ei together with the finite-sized list of primes are enough to reconstruct the number. Since for all i, it follows that for all i (where denotes the base-2 logarithm). This yields an encoding for n of the following size (using big O notation):
This is a much more efficient encoding than representing n directly in binary, which takes bits. An established result in lossless data compression 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 . Therefore, the number of primes must not be finite. [16]
The theorems in this section simultaneously imply Euclid's theorem and other results.
Dirichlet's theorem states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n is also a positive integer. In other words, there are infinitely many primes that are congruent to a modulo d.
Let π(x) be the prime-counting function that gives the number of primes less than or equal to x, for any real number x. The prime number theorem then states that x / log x is a good approximation to π(x), in the sense that the limit of the quotient of the two functions π(x) and x / log x as x increases without bound is 1:
Using asymptotic notation this result can be restated as
This yields Euclid's theorem, since
In number theory, Bertrand's postulate is a theorem stating that for any integer , there always exists at least one prime number such that
Bertrand–Chebyshev theorem can also be stated as a relationship with , where is the prime-counting function (number of primes less than or equal to ):
This statement was first conjectured in 1845 by Joseph Bertrand [17] (1822–1900). Bertrand himself verified his statement for all numbers in the interval [2, 3 × 106]. His conjecture was completely proved by Chebyshev (1821–1894) in 1852 [18] and so the postulate is also called the Bertrand–Chebyshev theorem or Chebyshev's theorem.
A prime number 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 writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann.
In number theory, Waring's problem asks whether each natural number k has an associated positive integer s such that every natural number is the sum of at most s natural numbers raised to the power k. For example, every natural number is the sum of at most 4 squares, 9 cubes, or 19 fourth powers. Waring's problem was proposed in 1770 by Edward Waring, after whom it is named. Its affirmative answer, known as the Hilbert–Waring theorem, was provided by Hilbert in 1909. Waring's problem has its own Mathematics Subject Classification, 11P05, "Waring's problem and variants".
In number theory, Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. It is written using the Greek letter phi as or , and may also be called Euler's phi function. In other words, it is the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd(n, k) is equal to 1. The integers k of this form are sometimes referred to as totatives of n.
In mathematics, the floor function (or greatest integer function) is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted ⌊x⌋ or floor(x). Similarly, the ceiling function maps x to the smallest integer greater than or equal to x, denoted ⌈x⌉ or ceil(x).
In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions:
In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers:
The sum of the reciprocals of all prime numbers diverges; that is:
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x. It is denoted by π(x) (unrelated to the number π).
In number theory, the Mertens function is defined for all positive integers n as
The Basel problem is a problem in mathematical analysis with relevance to number theory, concerning an infinite sum of inverse squares. It was first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734, and read on 5 December 1735 in The Saint Petersburg Academy of Sciences. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate fame when he was twenty-eight. Euler generalised the problem considerably, and his ideas were taken up more than a century later by Bernhard Riemann in his seminal 1859 paper "On the Number of Primes Less Than a Given Magnitude", in which he defined his zeta function and proved its basic properties. The problem is named after Basel, hometown of Euler as well as of the Bernoulli family who unsuccessfully attacked the problem.
In analytic number theory, Mertens' theorems are three 1874 results related to the density of prime numbers proved by Franz Mertens.
In mathematics, the Legendre sieve, named after Adrien-Marie Legendre, is the simplest method in modern sieve theory. It applies the concept of the Sieve of Eratosthenes to find upper or lower bounds on the number of primes within a given set of integers. Because it is a simple extension of Eratosthenes' idea, it is sometimes called the Legendre–Eratosthenes sieve.
In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles.
The Engel expansion of a positive real number x is the unique non-decreasing sequence of positive integers such that
In number theory, the law of quadratic reciprocity, like the Pythagorean theorem, has lent itself to an unusually large number of proofs. Several hundred proofs of the law of quadratic reciprocity have been published.
A decimal representation of a non-negative real number r is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator:
In mathematics, a Beatty sequence is the sequence of integers found by taking the floor of the positive multiples of a positive irrational number. Beatty sequences are named after Samuel Beatty, who wrote about them in 1926.
In number theory, the prime omega functions and count the number of prime factors of a natural number Thereby counts each distinct prime factor, whereas the related function counts the total number of prime factors of honoring their multiplicity. That is, if we have a prime factorization of of the form for distinct primes , then the respective prime omega functions are given by and . These prime factor counting functions have many important number theoretic relations.
The purpose of this page is to catalog new, interesting, and useful identities related to number-theoretic divisor sums, i.e., sums of an arithmetic function over the divisors of a natural number , or equivalently the Dirichlet convolution of an arithmetic function with one: