Addendum to “A stronger Bertrand’s postulate with an application to partitions”

Abstract

In [2] we proved, using only elementary techniques, that every positive integer, except 1, 2, 4, 6 and 9, is the sum of distinct odd primes. The purpose of this note is to bring to light some closely related results of which the author was unaware when [2] was published. These results were brought to the author's attention by Professor A. Makowski. They are as follows: H. E. Richert [5], using elementary methods, proved that every integer greater than 6 is the sum of distinct primes (not necessarily odd). R. Breusch [1], using intricate analytic methods, proved that if x_7 then between x and 2.x there is at least one prime of each of the following forms: 4k-1, 4k+ 1, 6k-1, 6k+ 1. A. Makowski [3], using these deep analytic results of Breusch and an elementary result of Richert [4], proved that every integer greater than 55, 121, 161, 205 is the sum of distinct primes of the form 4k-I, 4k+ 1, 6k-1, 6k+ 1 and that these lower bounds are the best possible. BIBLIOGRAPHY