Power, Simplicity and Beauty
In 1949, Leo Moser gave a proof of Bertrand’s postulate in the paper “A theorem of the distribution of primes”(American Mathematical Monthly,624-624,1949,vol56). Maybe this proof is the simplest one among all proofs of Bertrand’s postulate. …
Leo Moser’s proof of Bertrand’s postulate Read MoreSylvester’s theorem the binomial coefficient \begin{equation}{n\choose k}=\frac{n(n-1)(n-2)\cdots(n-k+1)}{k!}\end{equation} has a prime divisor \(p\) greater than \(k\). In other words, if \(n\geq 2k\), then the product of \(k\) consecutive integers greater than \(k\) is divisible by a …
Generalizations of Bertrand’s postulate Read MoreBertrand’s postulate states that if \(x\geq4\), then there always exists at least one prime \(p\) with\(x<p<2x-2\). A weaker but more elegant formulation is: for every \(x>1\) there is always at least one prime p such that \(x<p<2x\). In …
Ramanujan’s proof of Bertrand’s postulate Read More