Euclid's theorem

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 at least 200 proofs of the theorem.

Euclid's proof

Euclid offered a proof in his work Elements, which is paraphrased here.
Consider any finite list of prime numbers p₁, p₂, ..., p_n. It will be shown that there exists at least one additional prime number not included in this list. Let P be the product of all the prime numbers in the list: P = p₁p₂⋅⋅⋅p_n. Let q = P + 1. Since 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 also divide P. If p divides P and q, then p must also divide the difference of the two numbers, which is − 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 that is not 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, Euclid denoted the arbitrary finite set of prime numbers as A, B, Γ.
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, 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 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. 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, relies on the fundamental theorem of arithmetic: that every integer has a unique prime factorization. What Euler wrote is equivalent to the statement that
where denotes the set of the 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.
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. Then in his second corollary, Euler notes that the product
converges to the finite value 2, and there are consequently more primes than squares. This proves Euclid's theorem.
In the same paper Euler in fact used the above equality to prove a much stronger theorem that was unknown before him, namely that the series
is Divergence of [the sum of the reciprocals of the primes|divergent], where denotes the set of all prime numbers.

Erdős's proof

gave a proof 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. 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
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,
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 introduced a proof by contradiction using point-set topology.
Define a topology on the integers, called the evenly spaced integer topology, by declaring a subset to be an open set if and only if it is either the empty set,, or it is a union of arithmetic sequences , where
Then a contradiction follows from the property that a finite set of integers cannot be open and the property that the basis sets 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.

Recent proofs

Proof using the inclusion–exclusion principle

Juan Pablo Pinasco has written the following proof.
Let p₁, ..., p_N 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, ..., p_N exist, then the expression in is equal to and the expression in is equal to 1, but clearly the expression in is not equal to 1. Therefore, there must be more primes than p₁, ..., p_N.

Proof using Legendre's formula

In 2010, Junho Peter Whang published the following proof by contradiction. Let k be any positive integer. Then according to Legendre's formula
where
But if only finitely many primes exist, then
,
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, which does not use reductio ad absurdum or Euclid's lemma.
Since each natural number greater than 1 has at least one prime factor, and two successive numbers n and have no prime factor in common, the product n 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. By the fundamental theorem of arithmetic, any positive integer n could then be represented as
where the non-negative integer exponents e_i together with the finite-sized list of primes are enough to reconstruct the number. Since for all i, it follows that for all i. This yields an encoding for n of the following size :
bits.
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.

Proof using an even–odd argument

Romeo Meštrović used an even-odd argument to show that if the number of primes is not infinite then 3 is the largest prime, a contradiction.
Suppose that are all the prime numbers. Consider and note that by assumption all positive integers relatively prime to it are in the set. In particular, is relatively prime to and so is. However, this means that is an odd number in the set, so, or. This means that must be the largest prime number which is a contradiction.
The above proof continues to work if is replaced by any prime with, the product becomes and even vs. odd argument is replaced with a divisible vs. not divisible by argument. The resulting contradiction is that must, simultaneously, equal and be greater than, which is impossible.

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 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

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:
Using asymptotic notation this result can be restated as
This yields Euclid's theorem, since

Bertrand–Chebyshev theorem

In number theory, Bertrand's postulate is a theorem stating that for any integer, there always exists at least one prime number such that
Equivalently, writing for the prime-counting function, the theorem asserts that for all.
This statement was first conjectured in 1845 by Joseph Bertrand. Bertrand himself verified his statement for all numbers in the interval
His conjecture was completely proved by Chebyshev in 1852 and so the postulate is also called the Bertrand–Chebyshev theorem or Chebyshev's theorem.