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'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, ..., 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 = p₁p₂...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^[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, ... q_n 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]

${\displaystyle \prod _{p\in P_{k}}{\frac {1}{1-{\frac {1}{p}}}}=\sum _{n\in N_{k}}{\frac {1}{n}},}$

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 ${\displaystyle P_{k}.}$

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).

${\displaystyle {\begin{aligned}\prod _{p\in P_{k}}{\frac {1}{1-{\frac {1}{p}}}}&=\prod _{p\in P_{k}}\sum _{i\geq 0}{\frac {1}{p^{i}}}\\&=\left(\sum _{i\geq 0}{\frac {1}{2^{i}}}\right)\cdot \left(\sum _{i\geq 0}{\frac {1}{3^{i}}}\right)\cdot \left(\sum _{i\geq 0}{\frac {1}{5^{i}}}\right)\cdot \left(\sum _{i\geq 0}{\frac {1}{7^{i}}}\right)\cdots \\&=\sum _{\ell ,m,n,p,\ldots \geq 0}{\frac {1}{2^{\ell }3^{m}5^{n}7^{p}\cdots }}\\&=\sum _{n\in N_{k}}{\frac {1}{n}}.\end{aligned}}}$

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 ${\displaystyle \infty }$ 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

${\displaystyle \prod _{n\geq 2}{\frac {1}{1-{\frac {1}{n^{2}}}}}}$

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

${\displaystyle \sum _{p\in P}{\frac {1}{p}}}$

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 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, ... , p_k. Any positive integer a which is less than or equal to N can then be written in the form

${\displaystyle a=\left(p_{1}^{e_{1}}p_{2}^{e_{2}}\cdots p_{k}^{e_{k}}\right)s^{2},}$

where each e_i is either 0 or 1. There are 2^k ways of forming the square-free part of a. And s² can be at most N, so s ≤ √N. Thus, at most 2^k √N numbers can be written in this form. In other words,

${\displaystyle N\leq 2^{k}{\sqrt {N}}.}$

Or, rearranging, k, the number of primes less than or equal to N, is greater than or equal to 1/2log₂ N. Since N was arbitrary, k can be as large as desired by choosing N appropriately.

Furstenberg's proof

Zdroj:https://en.wikipedia.org?pojem=Infinitude_of_the_prime_numbers
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