it is
:.
Only is used in the proof.
Proof of Bertrand's Postulate
Assume that there is a counterexample
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a ...
: an integer ''n'' ≥ 2 such that there is no prime ''p'' with ''n'' < ''p'' < 2''n''.
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'' < 2''n''. Therefore, ''n'' ≥ 468.
There are no prime factors ''p'' of such that:
* 2''n'' < ''p'', because every factor must divide (2''n'')!;
* ''p'' = 2''n'', because 2''n'' is not prime;
* ''n'' < ''p'' < 2''n'', because we assumed there is no such prime number;
* 2''n'' / 3 < ''p'' ≤ ''n'': by Lemma 3
Lemma may refer to:
Language and linguistics
* Lemma (morphology), the canonical, dictionary or citation form of a word
* Lemma (psycholinguistics), a mental abstraction of a word about to be uttered
Science and mathematics
* Lemma (botany), a ...
.
Therefore, every prime factor ''p'' satisfies ''p'' ≤ 2''n'' / 3.
When the number has at most one factor of ''p''. By Lemma 2
Lemma may refer to:
Language and linguistics
* Lemma (morphology), the canonical, dictionary or citation form of a word
* Lemma (psycholinguistics), a mental abstraction of a word about to be uttered
Science and mathematics
* Lemma (botany), a ...
, for any prime ''p'' we have ''p''''R''(''p'',''n'') ≤ 2''n'', so the product of the ''p''''R''(''p'',''n'') over the primes less than or equal to is at most . Then, starting with Lemma 1
Lemma may refer to:
Language and linguistics
* Lemma (morphology), the canonical, dictionary or citation form of a word
* Lemma (psycholinguistics), a mental abstraction of a word about to be uttered
Science and mathematics
* Lemma (botany), a ...
and decomposing the side into its prime factorization, and finally using Lemma 4
Lemma may refer to:
Language and linguistics
* Lemma (morphology), the canonical, dictionary or citation form of a word
* Lemma (psycholinguistics), a mental abstraction of a word about to be uttered
Science and mathematics
* Lemma (botany), a ...
, these bounds give:
:
Taking logarithm
In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
s yields to
:
By concavity
In calculus, the second derivative, or the second order derivative, of a function is the derivative of the derivative of . Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, ...
of the right-hand side as a function of ''n'', the last inequality is necessarily verified on an interval. Since it holds true for and it does not for , we obtain
:
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 .
Lemma 1 can be expressed as
for , and because for , we can say that the product is at most , which gives
:
which is true for and false for .
References
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
The Mizar system consists of a formal language for writing mathematical definitions and proofs, a proof assistant, which is able to mechanically check proofs written in this language, and a library of formalized mathematics, which can be used i ...
: http://mizar.org/version/current/html/nat_4.html#T56
*
{{DEFAULTSORT:Bertrands postulate, proof of
Prime numbers
Factorial and binomial topics
Article proofs
Theorems about prime numbers