Treffer: Some problems of combinatorial number theory related to Bertrand's postulate

Summary: The well-known number theory text of \textit{G.Hardy} and \textit{E. M. Wright} [ An introduction to the theory of numbers, 5th ed., Oxford University Press (1978; Zbl 0423.10001)] contains the following remark (p.373): `Bertrand's Postulate' is that, for every \(n > 3\), there is a prime \(p\) satisfying \(n < p < 2n-2\). Bertrand verified this for \(n < 3, 000, 000\) and Chebyshev proved it for all \(n>3\) in 1850. Bertrand's Postulate is essentially equivalent to the statement that the first \(2k\) integers can always be arranged in \(k\) pairs so that the sum of the entries in each pair is a prime. We give the simple proof of this statement, and discuss some generalizations whose proofs seem to be quite intractable, even though they can be supported by numerical exploration and simple probabilistic analysis.