A simple elementary proof of $M(x)=\sum_{n≤x} μ(n)=o(x)$

A simple proof of Zariski's Lemma

‎Our aim in this very short note is to show that the proof of the‎ ‎following well-known fundamental lemma of Zariski follows from an‎ ‎argument similar to the proof of the fact that the rational field‎ ‎$mathbb{Q}$ is not a finitely generated $mathbb{Z}$-algebra.

A truly elementary proof of Bertrand’s theorem

Bertrand’s theorem is a surprising conclusion of Newtonian mechanics, which states that only the attractive linear and inverse-square forces can yield closed, if bounded, orbits. (For an English translation of Bertrand’s paper, see Santos et al.; for a near-transcription of Bertrand’s proof using modern notation, see Greenberg.) Since the harmonic potential cannot be physically extended over as...

Elementary Proof of MacMahon’s Conjecture

Major Percy A. MacMahon’s first paper on plane partitions [4] included a conjectured generating function for symmetric plane partitions. This conjecture was proven almost simultaneously by George Andrews and Ian Macdonald, Andrews using the machinery of basic hypergeometric series [1] and Macdonald employing his knowledge of symmetric functions [3]. The purpose of this paper is to simplify Macd...

A Simple Combinatorial Proof of

A Simple Proof of The

We give elementary derivations of some classical summation formulae for bilateral (basic) hypergeometric series. In particular, we apply Gauß’ 2F1 summation and elementary series manipulations to give a simple proof of Dougall’s 2H2 summation. Similarly, we apply Rogers’ nonterminating 6φ5 summation and elementary series manipulations to give a simple proof of Bailey’s very-well-poised 6ψ6 summ...