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 proved 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, which is paraphrased here.
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. 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.
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. If P is the set of all prime numbers, Euler wrote that:
The first equality is given by the formula for a geometric series in each term of the product. The second equality is a special case of the Euler product formula for the Riemann zeta function. To show this, distribute the product over the sum:
In the result, every product of primes appears exactly once and so by the fundamental theorem of arithmetic the sum is equal to the sum over all integers.
The sum on the right is the harmonic series, which diverges. Thus the product on the left must also diverge. Since each term of the product is finite, the number of terms must be infinite; therefore, there is an infinite number of primes.

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 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, 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 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.

Some recent proofs

Proof using the inclusion-exclusion principle

Juan Pablo Pinasco has written the following proof.
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 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, ..., pN.

Proof using de Polignac's formula

In 2010, Junho Peter Whang published the following proof by contradiction. Let k be any positive integer. Then according to de Polignac'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 has at least one prime factor, and two successive numbers n and have no 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 irrationality of

Representing the Leibniz formula for as an Euler product gives
The numerators of this product are the odd prime numbers, and each denominator is the multiple of four nearest to the numerator.
If there were finitely many primes this formula would show that is a rational number whose denominator is the product of all multiples of 4 that are one more or less than a prime number, contradicting the fact that is irrational.

Proof using information theory

Alexander Shen and others have presented a proof that uses incompressibility:
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 ei together with the finite-sized list of primes are enough to reconstruct the number. Since for all i, it follows that all .
This yields an encoding for n of the following size :
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 less 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.

A generalization: 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. Euclid's theorem is a special case of Dirichlet's theorem for a = d = 1. Every case of Dirichlet's theorem yields Euclid's theorem.

A stronger result: the 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