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The following elementary proof was published by Paul Erdős in 1932, as one of his earliest mathematical publications. [3] The basic idea is to show that the central binomial coefficients must have a prime factor within the interval ( n , 2 n ) {\displaystyle (n,2n)} in order to be large enough.
The book is dedicated to the mathematician Paul Erdős, who often referred to "The Book" in which God keeps the most elegant proof of each mathematical theorem. During a lecture in 1985, Erdős said, "You don't have to believe in God, but you should believe in The Book." [1]
Erdős found a proof for Bertrand's postulate which proved to be far neater than Chebyshev's original one. He also discovered the first elementary proof for the prime number theorem, along with Atle Selberg. However, the circumstances leading up to the proofs, as well as publication disagreements, led to a bitter dispute between Erdős and Selberg.
Copeland and Erdős's proof that their constant is normal relies only on the fact that is strictly increasing and = + (), where is the n th prime number. More generally, if is any strictly increasing sequence of natural numbers such that = + and is any natural number greater than or equal to 2, then the constant obtained by concatenating "0."
The Erdős–Faber–Lovász conjecture on coloring unions of cliques, proved (for all large n) by Dong Yeap Kang, Tom Kelly, Daniela Kühn, Abhishek Methuku, and Deryk Osthus. [4] The Erdős sumset conjecture on sets, proven by Joel Moreira, Florian Karl Richter, Donald Robertson in 2018. The proof has appeared in "Annals of Mathematics" in ...
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The theorem is named after Paul Erdős and Norman H. Anning, who published a proof of it in 1945. [1] Erdős later supplied a simpler proof, which can also be used to check whether a point set forms an Erdős–Diophantine graph, an inextensible system of integer points with integer distances. The Erdős–Anning theorem inspired the Erdős ...
El Bachraoui's proof is an extension of Erdős's arguments for the primes between n and 2n. Shevelev, Greathouse, and Moses (2013) discuss related results for similar intervals. [7] Bertrand’s postulate over the Gaussian integers is an extension of the idea of the distribution of primes, but in this case on the complex plane.