and also between 
and n(n+1). By modifying Paul Erdős’ proof of the Sylvester-Schur Theorem, we show that one or more primes lie within [ Φ(n-1), Φ(n) ) where Φ(n) returns an Oppermann boundary point (which we define as a semipronic number.)
Oppermann’s Theorem
Fred Daniel Kline
2 August 2011 --- Copyright © 2011
L. H. F. Oppermann conjectured that when n is a whole number >1, at least one prime number lies between n(n-1) and and also between and n(n+1). By modifying Paul Erdős’ proof of the Sylvester-Schur Theorem, we show that one or more primes lie within [ Φ(n-1), Φ(n) ) where Φ(n) returns an Oppermann boundary point (which we define as a semipronic number.)
Preliminaries
Landau’s Problems
From Wikipedia, the free encyclopedia [1]
At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about primes. These problems were characterised in his speech as “unattackable at the present state of science” and are now known as Landau’s problems. They are as follows:
1. Goldbach’s conjecture: Can every even integer greater than 2 be written as the sum of two primes?
2. Twin prime conjecture: Are there infinitely many primes p such that p + 2 is prime?
3. Legendre’s conjecture: Does there always exist at least one prime between consecutive perfect squares?
4. Are there infinitely many primes p such that p − 1 is a perfect square? In other words: Are there infinitely many primes of the form n^2 + 1?
We suggest that 3. Legendre’s conjecture would have been 3. Oppermann’s conjecture had Edmund Landau been aware of it at the time. Oppermann’s conjecture is the stronger of the two.
Oppermann’s Conjecture
From the Overview of the Royal Danish Sciences Institution’s work and its members’ work in the year 1882. [2] [3]
Oppermann, L. H. F., Professor, Lecturer in German at the University of Copenhagen; Knight of Dannebrog (Denmark) April 16th 1875.
In the notes from a meeting on March 9th 1877, after discussing papers by Legendre, J. W. L. Glaisher, and Meissel, Oppermann stated:
At the same occasion, I made people aware of the not yet proven conjecture, that when n is a whole number >1, at least one prime number lies between n(n-1) and 
and also between 
and n(n+1).
Danish-to-English translation by Jesper Madsen.
A Useful Function
The capital Phi function returns the semipronic number for the n th step, for all n > 0
Definitions
pronic numbers [4]
Pronic (or oblong) numbers are the products of two consecutive numbers, a × (a+1) for all a > 0
It is easy to see that 
and 
semipronic numbers
Semipronic is the term we use to describe the Oppermann boundary points, which are the numbers in the set of perfect squares interleaved with the pronic numbers.[5]
For the perfect squares, it is easy to see that 
and 
which leads us to these four equalities:
which we simplify by using the capital Phi function:
Theorem
