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.
Consider any finite list of prime numbers p1, p2, ..., pn. 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...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.[3] 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,[4] 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."
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 (it gives a remainder of 1 when divided by each). 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.[5]
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[6]
\prod | |
p\inPk |
1 | |||
|
=\sum | |
n\inNk |
1 | |
n |
,
Pk
Nk
Pk.
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).
\begin{align} \prod | |
p\inPk |
1 | |||
|
&
=\prod | |
p\inPk |
\sumi\geq
1 | |
pi |
\ &=\left(\sumi\geq
1 | |
2i |
\right) ⋅ \left(\sumi\geq
1 | |
3i |
\right) ⋅ \left(\sumi\geq
1 | |
5i |
\right) ⋅ \left(\sumi\geq
1 | |
7i |
\right) … \ &=\sum\ell,m,n,p,\ldots
1 | |
2\ell3m5n7p … |
\ &
=\sum | |
n\inNk |
1 | |
n |
. \end{align}
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
infty
loginfty
x
logx
\prodn\ge2
1 | |||
|
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
\sump\in
1p | |
=logloginfty
x
loglogx
Paul Erdős gave a proof[8] 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
a=\left(
e1 | |
p | |
1 |
e2 | |
p | |
2 |
…
ek | |
p | |
k |
\right)s2,
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,
N\leq2k\sqrt{N}.
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.
See main article: Furstenberg's proof of the infinitude of primes.
In the 1950s, Hillel Furstenberg introduced a proof by contradiction using point-set topology.[9]
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
S(a,b)=\{an+b\midn\inZ\}=aZ+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
Z\setminus\{-1,+1\}=cuppS(p,0)
Juan Pablo Pinasco has written the following proof.[10]
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
\begin{align} 1+\sumi\left\lfloor
x | |
pi |
\right\rfloor-\sumi\left\lfloor
x | |
pipj |
\right\rfloor&+\sumi\left\lfloor
x | |
pipjpk |
\right\rfloor- … \\ & … \pm(-1)N+1\left\lfloor
x | |
p1 … pN |
\right\rfloor. (1) \end{align}
Dividing by x and letting x → ∞ gives
\sumi
1 | |
pi |
-\sumi
1 | |
pipj |
+\sumi
1 | |
pipjpk |
- … \pm(-1)N+1
1 | |
p1 … pN |
. (2)
This can be written as
1-
N | |
\prod | |
i=1 |
\left(1-
1 | |
pi |
\right). (3)
If no other primes than p1, ..., pN exist, then the expression in (1) is equal to
\lfloorx\rfloor
In 2010, Junho Peter Whang published the following proof by contradiction.[11] Let k be any positive integer. Then according to Legendre's formula (sometimes attributed to de Polignac)
k!=\prodpprimepf(p,k)
where
f(p,k)=\left\lfloor
k | |
p |
\right\rfloor+\left\lfloor
k | |
p2 |
\right\rfloor+ … .
f(p,k)<
k | |
p |
+
k | |
p2 |
+ … =
k | |
p-1 |
\lek.
But if only finitely many primes exist, then
\limk\toinfty
\left(\prodpp\right)k | |
k! |
=0,
(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[12] 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, 2×3 = 6, 6×7 = 42, 42×43 = 1806, 1806×1807 = 3263442, · · ·
provides a sequence of unlimited growing sets of primes.
See also: incompressibility method. Suppose there were only k primes (p1, ..., pk). By the fundamental theorem of arithmetic, any positive integer n could then be represented aswhere the non-negative integer exponents ei together with the finite-sized list of primes are enough to reconstruct the number. Since
pi\geq2
ei\leqlgn
lg
O(primelistsize+klglgn)=O(lglgn)
N=O(lgn)
lglgn=o(lgn)
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.[13]
Suppose that
p1=2<p2=3<p3< … <pk
P=3p3p4 … pk
S=\{1,2,22,23,...\}
2
P
P-2
P-2
S
P-2=1
P=3
3
Remark: The above proof continues to work if
2
pj
j\in\{1,2,...,k-1\}
P
p1p2 … pj-1 ⋅ pj+1 … pk
pj
P-pj
1
1
The theorems in this section simultaneously imply Euclid's theorem and other results.
See main article: article and 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.
See main article: article and 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:
\limx → infty
\pi(x) | |
x/log(x) |
=1.
Using asymptotic notation this result can be restated as
\pi(x)\sim
x | |
logx |
.
This yields Euclid's theorem, since
\limx → infty
x | |
logx |
=infty.
n>1
n<p<2n.
Bertrand–Chebyshev theorem can also be stated as a relationship with
\pi(x)
\pi(x)
x
\pi(x)-\pi(\tfrac{x}{2})\ge1,
x\ge2.
This statement was first conjectured in 1845 by Joseph Bertrand[14] (1822–1900). Bertrand himself verified his statement for all numbers in the interval His conjecture was completely proved by Chebyshev (1821–1894) in 1852[15] and so the postulate is also called the Bertrand–Chebyshev theorem or Chebyshev's theorem.
a\midb
a\midc
a\mid(b-c)