Bertrand’s conjecture, proof of

This is a version of Erdős’s proof as it appears in Hardy and Wright.

We begin with the following lemma.

Lemma.

Let $n$ be a positive integer and $p$ be a prime. The highest power of $p$ dividing $n!$ is $\displaystyle\sum_{j}\left\lfloor\frac{n}{p^{j}}\right\rfloor$, where $\lfloor x\rfloor$ is the floor of $x$.

Proof.

Let $p^{k}$ divide $n!$ with $k$ as large as possible. Every $p$th term of the sequence  $1,\dots,n$ is divisible by $p$, so contributes a factor to $p^{k}$. There are $\displaystyle\left\lfloor\frac{n}{p}\right\rfloor$ such factors. Every $p^{2}$th term contributes an extra factor above that, providing $\displaystyle\left\lfloor\frac{n}{p^{2}}\right\rfloor$ new factors. In general, the $p^{j}$th terms contribute $\displaystyle\left\lfloor\frac{n}{p^{j}}\right\rfloor$ extra factors to $p^{k}$. So the highest power of $p$ dividing $n!$ is $\displaystyle\sum_{j}\left\lfloor\frac{n}{p^{j}}\right\rfloor$. ∎

Bertrand’s conjecture.

Let $n>2$ be a minimal counterexample to the claim. Thus there is no prime $p$ such that $n.

In the sequence of primes

 $2,3,5,7,13,23,43,83,163,317,631,1259,2503,$

each succeeding term is smaller than the double of its predecessor. This shows that $n\geq 2503$.

The binomial expansion (http://planetmath.org/BinomialTheorem) of $(1+1)^{2n}$ has $2n+1$ terms and has largest term $\binom{2n}{n}$. Hence

 $\binom{2n}{n}\geq\frac{4^{n}}{2n+1}\geq\frac{4^{n}}{(2n)^{2}}.$

For a prime $p$ define $r(p,n)$ to be the highest power of $p$ dividing $\binom{2n}{n}$. To compute $r(p,n)$, we apply the lemma to $(2n)!$ and $n!$. We get that

 $r(p,n)=\sum_{j\geq 1}\left\lfloor\frac{2n}{p^{j}}\right\rfloor-2\sum_{j\geq 1}% \left\lfloor\frac{n}{p^{j}}\right\rfloor=\sum_{j\geq 1}\left(\left\lfloor\frac% {2n}{p^{j}}\right\rfloor-2\left\lfloor\frac{n}{p^{j}}\right\rfloor\right).$

The terms of the sum are all 0 or 1 and vanish when $\displaystyle j>\left\lfloor\frac{\log(2n)}{\log p}\right\rfloor$, so $\displaystyle r(p,n)\leq\left\lfloor\frac{\log(2n)}{\log p}\right\rfloor$, that is, $p^{r(p,n)}\leq 2n$.

Now $\displaystyle\binom{2n}{n}=\prod_{p}p^{r(p,n)}$. By the inequality  just proved, primes larger than $2n$ do not contribute to this product  , and by assumption  there are no primes between $n$ and $2n$. So

 $\binom{2n}{n}=\prod_{\begin{subarray}{c}1\leq p\leq n\\ p\text{\ prime}\end{subarray}}p^{r(p,n)}.$

For $n>p>\frac{2n}{3}$, $\frac{3}{2}>\frac{n}{p}>1$ and so for $p>\frac{2n}{3}>\sqrt{2n}$ we can apply the previous formula   for $r(p,n)$ and find that it is zero. So for all $n>4$, the contribution of the primes larger than $\frac{2n}{3}$ is zero.

If $p>\sqrt{2n}$, all the terms for higher powers of $p$ vanish and $\displaystyle r(p,n)=\left\lfloor\frac{2n}{p}\right\rfloor-2\left\lfloor\frac{% n}{p}\right\rfloor$. Since $r(p,n)$ is at most 1, an upper bound  for the contribution for the primes between $\sqrt{2n}$ and $\frac{2n}{3}$ is the product of all primes smaller than $\frac{2n}{3}$. This product is $\exp\left(\vartheta(\frac{2n}{3})\right)$, where $\vartheta(n)$ is the Chebyshev function  $\vartheta(n)=\sum_{\begin{subarray}{c}p\leq n\\ p\text{\ prime}\end{subarray}}\log p.$

There are at most $\sqrt{2n}$ primes smaller than $\sqrt{2n}$ and by the inequality $p^{r(p,n)}\leq 2n$ their product is less than $(2n)^{\sqrt{2n}}$. Combining this information, we get the inequality

 $\frac{4^{n}}{(2n)^{2}}\leq\binom{2n}{n}\leq(2n)^{\sqrt{2n}}\exp\left({% \textstyle\vartheta(\frac{2n}{3})}\right).$

Taking logarithms and applying the upper bound of $n\log 4$ for $\vartheta(n)$ (http://planetmath.org/UpperBoundOnVarthetan), we obtain the inequality $\frac{n}{3}\log 4\leq(\sqrt{2n}+2)\log(2n)$, which is false for sufficiently large $n$, say $n=2^{11}$. This shows that $n<2^{11}$.

Since the conditions $n\geq 2503$ and $n<2^{11}$ are incompatible, there are no counterexamples  to the claim. ∎

References

• 1 G.H. Hardy, E.M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, 1938.
Title Bertrand’s conjecture, proof of BertrandsConjectureProofOf 2013-03-22 13:18:55 2013-03-22 13:18:55 CWoo (3771) CWoo (3771) 15 CWoo (3771) Proof msc 11N05