Proof of Bertrand's postulate

Short description: Solved prime-number problem

In mathematics, Bertrand's postulate (actually now a theorem) states that for each [math]\displaystyle{ n \ge 2 }[/math] there is a prime [math]\displaystyle{ p }[/math] such that [math]\displaystyle{ n\lt p\lt 2n }[/math]. First conjectured in 1845 by Joseph Bertrand, it was first proven by Chebyshev, and a shorter but also advanced proof was given by Ramanujan.

The following elementary proof was published by Paul Erdős in 1932, as one of his earliest mathematical publications. The basic idea is to show that the central binomial coefficients need to have a prime factor within the interval [math]\displaystyle{ (n, 2n) }[/math] in order to be large enough. This is achieved through analysis of the factorization of the central binomial coefficients.

The main steps of the proof are as follows. First, show that the contribution of every prime power factor [math]\displaystyle{ p^r }[/math] in the prime decomposition of the central binomial coefficient [math]\displaystyle{ \textstyle\binom{2n}{n}=\frac{(2n)!}{(n!)^2} }[/math] is at most [math]\displaystyle{ 2n }[/math]. Then show that every prime larger than [math]\displaystyle{ \sqrt{2n} }[/math] appears at most once.

The next step is to prove that [math]\displaystyle{ \tbinom{2n}{n} }[/math] has no prime factors in the interval [math]\displaystyle{ (\tfrac{2n}{3}, n) }[/math]. As a consequence of these bounds, the contribution to the size of [math]\displaystyle{ \tbinom{2n}{n} }[/math] coming from the prime factors that are at most [math]\displaystyle{ n }[/math] grows asymptotically as [math]\displaystyle{ \theta^{\!\;n} }[/math] for some [math]\displaystyle{ \theta\lt 4 }[/math]. Since the asymptotic growth of the central binomial coefficient is at least [math]\displaystyle{ 4^n\!/2n }[/math], the conclusion is that, by contradiction and for large enough [math]\displaystyle{ n }[/math], the binomial coefficient must have another prime factor, which can only lie between [math]\displaystyle{ n }[/math] and [math]\displaystyle{ 2n }[/math].

The argument given is valid for all [math]\displaystyle{ n \ge 468 }[/math]. The remaining values of [math]\displaystyle{ n }[/math] are by direct inspection, which completes the proof.

Lemmas in the proof

The proof uses the following four lemmas to establish facts about the primes present in the central binomial coefficients.

Lemma 1

For any integer [math]\displaystyle{ n\gt 0 }[/math], we have

[math]\displaystyle{ \frac{4^n}{2n} \le \binom{2n}{n}. }[/math]

Proof: Applying the binomial theorem,

[math]\displaystyle{ 4^n = (1 + 1)^{2n} = \sum_{k = 0}^{2n} \binom{2n}{k}=2+\sum_{k = 1}^{2n-1} \binom{2n}{k} \le 2n\binom{2n}{n}, }[/math]

since [math]\displaystyle{ \tbinom{2n}{n} }[/math] is the largest term in the sum in the right-hand side, and the sum has [math]\displaystyle{ 2n }[/math] terms (including the initial [math]\displaystyle{ 2 }[/math] outside the summation).

Lemma 2

For a fixed prime [math]\displaystyle{ p }[/math], define [math]\displaystyle{ R = R(n,p) }[/math] to be the p-adic order of [math]\displaystyle{ \tbinom{2n}{n} }[/math], that is, the largest natural number [math]\displaystyle{ r }[/math] such that [math]\displaystyle{ p^r }[/math] divides [math]\displaystyle{ \tbinom{2n}{n} }[/math].

For any prime [math]\displaystyle{ p }[/math], [math]\displaystyle{ p^{R }\le 2n }[/math].

Proof: The exponent of [math]\displaystyle{ p }[/math] in [math]\displaystyle{ n! }[/math] is given by Legendre's formula

[math]\displaystyle{ \sum_{j = 1}^\infty \left\lfloor \frac{n}{p^j} \right\rfloor\!, }[/math]

so

[math]\displaystyle{ R =\sum_{j = 1}^\infty \left\lfloor \frac{2n}{p^j} \right\rfloor - 2\sum_{j = 1}^\infty \left\lfloor \frac{n}{p^j} \right\rfloor =\sum_{j = 1}^\infty \left(\left\lfloor \frac{2n}{p^j} \right\rfloor - 2\!\left\lfloor \frac{n}{p^j} \right\rfloor\right)\!. }[/math]

But each term of the last summation must be either zero (if [math]\displaystyle{ n/p^j \bmod 1\lt 1/2 }[/math]) or one (if [math]\displaystyle{ n/p^j \bmod 1\ge 1/2 }[/math]), and all terms with [math]\displaystyle{ j\gt \log_p(2n) }[/math] are zero. Therefore,

[math]\displaystyle{ R \leq \log_p(2n), }[/math]

and

[math]\displaystyle{ p^R \leq p^{\log_p{2n}} = 2n. }[/math]

Lemma 3

If [math]\displaystyle{ p }[/math] is odd and [math]\displaystyle{ \frac{2n}{3} \lt p \leq n }[/math], then [math]\displaystyle{ R(n,p) = 0. }[/math]

Proof: There are exactly two factors of [math]\displaystyle{ p }[/math] in the numerator of the expression [math]\displaystyle{ \tbinom{2n}{n}=\tfrac{(2n)!}{(n!)^2} }[/math], coming from the two terms [math]\displaystyle{ p }[/math] and [math]\displaystyle{ 2p }[/math] in [math]\displaystyle{ (2n)! }[/math], and also two factors of [math]\displaystyle{ p }[/math] in the denominator from one copy of the term [math]\displaystyle{ p }[/math] in each of the two factors of [math]\displaystyle{ n! }[/math]. These factors all cancel, leaving no factors of [math]\displaystyle{ p }[/math] in [math]\displaystyle{ \tbinom{2n}{n} }[/math]. (The bound on [math]\displaystyle{ p }[/math] in the preconditions of the lemma ensures that [math]\displaystyle{ 3p }[/math] is too large to be a term of the numerator, and the assumption that [math]\displaystyle{ p }[/math] is odd is needed to ensure that [math]\displaystyle{ 2p }[/math] contributes only one factor of [math]\displaystyle{ p }[/math] to the numerator.)

Lemma 4

An upper bound is supplied for the primorial function,

[math]\displaystyle{ x\# = \prod_{p \leq x} p, }[/math]

where the product is taken over all prime numbers [math]\displaystyle{ p }[/math] less than or equal to the real number [math]\displaystyle{ x }[/math].

For all real numbers [math]\displaystyle{ x\ge 3 }[/math], [math]\displaystyle{ x\#\lt 2^{2x-3} }[/math].

Proof: Since [math]\displaystyle{ x \# = \lfloor x \rfloor \# }[/math] and [math]\displaystyle{ 2^{2\lfloor x\rfloor-3}\le2^{2x-3} }[/math], it suffices to prove the result under the assumption that [math]\displaystyle{ x = m }[/math] is an integer, [math]\displaystyle{ m\ge3. }[/math] Since [math]\displaystyle{ \binom{2m-1}{m} }[/math] is an integer and all the primes [math]\displaystyle{ m+1 \le p \le 2m-1 }[/math] appear in its numerator but not in its denominator, we have

[math]\displaystyle{ \frac{(2m-1)\#}{m\#} \le \binom{2m-1}{m} = \frac{1}{2}\!\left(\binom{2m-1}{m-1} + \binom{2m-1}{m}\right) \lt \frac{1}{2}(1+1)^{2m-1} = 2^{2m-2}. }[/math]

The proof proceeds by complete induction on [math]\displaystyle{ n. }[/math]

• If [math]\displaystyle{ n = 3 }[/math], then [math]\displaystyle{ n\# = 6 \lt 8 = 2^{2n-3}. }[/math]
• If [math]\displaystyle{ n = 4 }[/math], then [math]\displaystyle{ n\# = 6 \lt 32 = 2^{2n-3}. }[/math]
• If [math]\displaystyle{ n \ge 5 }[/math] is odd, [math]\displaystyle{ n=2m-1 }[/math], then by the above estimate and the induction assumption, since [math]\displaystyle{ m\ge3 }[/math] and [math]\displaystyle{ m\lt n }[/math] it is

[math]\displaystyle{ n\# = (2m-1)\# \lt m\#\cdot2^{2m-2} \lt 2^{2m-3}2^{2m-2} = 2^{4m-5} = 2^{2n-3}. }[/math]

• If [math]\displaystyle{ n=2m }[/math] is even and [math]\displaystyle{ n \ge 6, }[/math] then by the above estimate and the induction assumption, since [math]\displaystyle{ m\ge3 }[/math] and [math]\displaystyle{ n-1\lt n }[/math] it is

[math]\displaystyle{ n\# = (n-1)\# \lt 2^{2(n-1)-3} \lt 2^{2n-3} }[/math].

Only [math]\displaystyle{ x\# \lt 4^x }[/math] is used in the proof.

Proof of Bertrand's Postulate

Assume that there is a counterexample: an integer n ≥ 2 such that there is no prime p with n < p < 2n.

If 2 ≤ n < 468, then p can be chosen from among the prime numbers 3, 5, 7, 13, 23, 43, 83, 163, 317, 631 (each being the largest prime less than twice its predecessor) such that n < p < 2n. Therefore, n ≥ 468.

There are no prime factors p of [math]\displaystyle{ \textstyle\binom{2n}{n} }[/math] such that:

• 2n < p, because every factor must divide (2n)!;
• p = 2n, because 2n is not prime;
• n < p < 2n, because we assumed there is no such prime number;
• 2n / 3 < p ≤ n: by Lemma 3.

Therefore, every prime factor p satisfies p ≤ 2n / 3.

When [math]\displaystyle{ p \gt \sqrt{2n}, }[/math] the number [math]\displaystyle{ \textstyle {2n \choose n} }[/math] has at most one factor of p. By Lemma 2, for any prime p we have p^R(p,n) ≤ 2n, so the product of the p^R(p,n) over the primes less than or equal to [math]\displaystyle{ \sqrt{2n} }[/math] is at most [math]\displaystyle{ (2n)^{\sqrt{2n}} }[/math]. Then, starting with Lemma 1 and decomposing the right-hand side into its prime factorization, and finally using Lemma 4, these bounds give:

[math]\displaystyle{ \frac{4^n}{2n} \le \binom{2n}{n} = \left(\,\prod_{p\!\; \le \sqrt{2n}} p^{R(p,n)}\right) \!\! \left(\prod_{\sqrt{2n} \!\;\lt \!\; p \!\;\le 2n/3} p^{R(p,n)}\right) \lt (2n)^{\sqrt{2n}} \prod_{1 \lt \!\; p \!\;\leq 2n/3 } p = (2n)^{\sqrt{2n}} \left( \frac{2n}{3}\right)\# \le (2n)^{\sqrt{2n}} 4^{2n/3}. }[/math]

Taking logarithms yields to

[math]\displaystyle{ {\frac{\log 4}{3}}n \le (\sqrt{2n}+1)\log 2n\; . }[/math]

By concavity of the right-hand side as a function of n, the last inequality is necessarily verified on an interval. Since it holds true for [math]\displaystyle{ n=467 }[/math] and it does not for [math]\displaystyle{ n=468 }[/math], we obtain

[math]\displaystyle{ n \lt 468. }[/math]

But these cases have already been settled, and we conclude that no counterexample to the postulate is possible.

Addendum to proof

It is possible to reduce the bound for n to [math]\displaystyle{ n=82 }[/math].

Lemma 1 can be expressed as

[math]\displaystyle{ \frac{2\cdot4^n}{n} \le \binom{2n}{n} }[/math]

for [math]\displaystyle{ n \ge 13 }[/math], and because [math]\displaystyle{ \frac{\pi(n)}{n}\lt \frac{1}{2} }[/math] for [math]\displaystyle{ n \ge 13 }[/math], we can say that the product [math]\displaystyle{ p^R }[/math] is at most [math]\displaystyle{ (2n)^{\sqrt{\frac{n}{2}}} }[/math], which gives

[math]\displaystyle{ {\frac{\log 4}{3}}n \le\! \left(\sqrt{\frac{n}{2}}+1\right)\log 2n \; }[/math]

which is true for [math]\displaystyle{ n=81 }[/math] and false for [math]\displaystyle{ n=82 }[/math].

External links

• Chebyshev's Theorem and Bertrand's Postulate (Leo Goldmakher): https://web.williams.edu/Mathematics/lg5/Chebyshev.pdf
• Proof of Bertrand’s Postulate (UW Math Circle): https://sites.math.washington.edu/~mathcircle/circle/2013-14/advanced/mc-13a-w10.pdf
• Proof in the Mizar system: http://mizar.org/version/current/html/nat_4.html#T56
• Weisstein, Eric W.. "Bertrand's Postulate". http://mathworld.wolfram.com/BertrandsPostulate.html.