1. Introduction
In 1845, Bertrand conjectured what became known as Bertrand’s postulate: twice any prime strictly exceeds the next prime [1] . Tchebichef presented his proof of Bertrand’s postulate in 1850 and published it in 1852 [2] . It is now sometimes called the Bertrand-Chebyshev theorem. Surprisingly, a stronger statement seems not to be well known, but is elementary to prove: The sum of any two consecutive primes strictly exceeds the next prime, except for the only equality 2 + 3 = 5. After I conjectured and proved this statement independently, a very helpful referee pointed out that Ishikawa published this result in 1934 (with a different proof) [3] . This observation is a special case of a much more general result, Theorem 2, that is also elementary to prove (given the prime number theorem), and perhaps not previously noticed: If
denotes the nth prime,
with
and if
are natural numbers (not necessarily distinct), and
are positive integers (not necessarily distinct), and then there exists a positive integer N such that
for ll
. We also have another result: If
and j are nonnegative integers, then there exists a large enough positive integer N such that, for all
,
. We give some numerical results.
2. Main Result
Theorem 1. If
and j are nonnegative integers, then there exists a large enough positive integer N such that, for all
,
.
Applying Rosser’s theorem for all
, we have
For all
, we have
Consider the expression
We consider the following limit
Taking the ln of the numerator and denominator and applying L’Hospital Rule gives
Then we see
When
then
(Because
, for
)
Or, for
, N is a large enough positive integer, then
,
It turns out,
.
Theorem 2. If
are j nonnegative integers (not necessarily distinct), and
are i positive integers (not necessarily distinct), with
, then there exists a large enough positive integer N such that, for all
,
.
Applying Rosser’s theorem for all
, we have
For all
, we have
Consider the expression
We consider the following limit
Taking the ln of the numerator and denominator and applying L’Hospital Rule gives
Then we see
When
then
(Because
, for
and
)
Or, for
, N is a large enough positive integer, then
,
It turns out,
.
3. Concluding Remark
In this short note we have provided the prime number inequality via Rosser and Schoenfeld bounds [4] .
Acknowledgements
I thank VNU University of Science for accompanying me.