# Euclid's theorem

Euclid's theorem This article is about the theorem on the infinitude of prime numbers. For the theorem on perfect numbers and Mersenne primes, see Euclid–Euler theorem.

Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. It was first proved by Euclid in his work Elements. There are several proofs of the theorem.

Contents 1 Euclid's proof 1.1 Variations 2 Euler's proof 3 Erdős's proof 4 Furstenberg's proof 5 Recent proofs 5.1 Proof using the inclusion-exclusion principle 5.2 Proof using de Polignac's formula 5.3 Proof by construction 5.4 Proof using the incompressibility method 6 Stronger results 6.1 Dirichlet's theorem on arithmetic progressions 6.2 Prime number theorem 6.3 Bertrand–Chebyshev theorem 7 Notes and references 8 External links Euclid's proof 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: 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[3] 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.[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] 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, 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] Euler's proof 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] {displaystyle prod _{pin P_{k}}{frac {1}{1-{frac {1}{p}}}}=sum _{nin N_{k}}{frac {1}{n}},} where {displaystyle P_{k}} denotes the set of the k first prime numbers, and {displaystyle N_{k}} is the set of the positive integers whose prime factors are all in {displaystyle P_{k}.} In order 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).

{displaystyle {begin{aligned}prod _{pin P_{k}}{frac {1}{1-{frac {1}{p}}}}&=prod _{pin P_{k}}sum _{igeq 0}{frac {1}{p^{i}}}\&=left(sum _{igeq 0}{frac {1}{2^{i}}}right)cdot left(sum _{igeq 0}{frac {1}{3^{i}}}right)cdot left(sum _{igeq 0}{frac {1}{5^{i}}}right)cdot left(sum _{igeq 0}{frac {1}{7^{i}}}right)cdots \&=sum _{ell ,m,n,p,ldots geq 0}{frac {1}{2^{ell }3^{m}5^{n}7^{p}cdots }}\&=sum _{nin N_{k}}{frac {1}{n}}.end{aligned}}} 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 {displaystyle infty } the « absolute infinity » and writes that the infinite sum in the statement equals the « value » {displaystyle 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 {displaystyle x} of the harmonic series diverges asymptotically like {displaystyle log x} ). Then in his second corollary Euler notes that the product {displaystyle prod _{ngeq 2}{frac {1}{1-{frac {1}{n^{2}}}}}} 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.[10] Symbol used by Euler to denote infinity 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 {displaystyle sum _{pin P}{frac {1}{p}}} is divergent, where P denotes the set of all prime numbers (Euler writes that the infinite sum {displaystyle =log log infty } , which in modern terminology is equivalent to say that the partial sum up to {displaystyle x} of this series behaves asymptotically like {displaystyle log log x} ).

Erdős's proof 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 and a square number rs2. 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 which is less than or equal to N can then be written in the form {displaystyle left(p_{1}^{e_{1}}p_{2}^{e_{2}}cdots p_{k}^{e_{k}}right)s^{2},} 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, {displaystyle Nleq 2^{k}{sqrt {N}}.} Or, rearranging, k, the number of primes less than or equal to N, is greater than or equal to 1 / 2 log2 N. Since N was arbitrary, k can be as large as desired by choosing N appropriately.

Furstenberg's proof Main article: Furstenberg's proof of the infinitude of primes 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 {displaystyle S(a,b)={an+bmid nin mathbb {Z} }=amathbb {Z} +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 {displaystyle mathbb {Z} setminus {-1,+1}=bigcup _{pmathrm {,prime} }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.[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 {displaystyle {begin{aligned}1+sum _{i}leftlfloor {frac {x}{p_{i}}}rightrfloor -sum _{i

Proof using the incompressibility method Suppose there were only k primes (p1, ..., pk). By the fundamental theorem of arithmetic, any positive integer n could then be represented as {displaystyle n={p_{1}}^{e_{1}}{p_{2}}^{e_{2}}cdots {p_{k}}^{e_{k}},} where the non-negative integer exponents ei together with the finite-sized list of primes are enough to reconstruct the number. Since {displaystyle p_{i}geq 2} for all i, it follows that {displaystyle e_{i}leq lg n} for all i (where {displaystyle lg } denotes the base-2 logarithm). This yields an encoding for n of the following size (using big O notation): {displaystyle O({text{prime list size}}+klg lg n)=O(lg lg n)} bits.

This is a much more efficient encoding than representing n directly in binary, which takes {displaystyle N=O(lg n)} 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 {displaystyle lg lg n=o(lg n)} . Therefore, the number of primes must not be finite.[16] Stronger results The theorems in this section simultaneously imply Euclid's theorem and other results.

Dirichlet's theorem on arithmetic progressions Main article: Dirichlet's theorem on arithmetic progressions 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.

Prime number theorem Main article: Prime number theorem 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: {displaystyle lim _{xrightarrow infty }{frac {pi (x)}{x/ln(x)}}=1.} Using asymptotic notation this result can be restated as {displaystyle pi (x)sim {frac {x}{log x}}.} This yields Euclid's theorem, since {displaystyle lim _{xrightarrow infty }{frac {x}{log x}}=infty .} Bertrand–Chebyshev theorem In number theory, Bertrand's postulate is a theorem stating that for any integer {displaystyle n>1} , there always exists at least one prime number such that {displaystyle n

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