As an undergrad I had a hypothesis about the prime numbers that goes as follows:
For the nth prime P(n) and the mth prime P(m) Then P(n) * P(m) > P(n+m)
The base case for prime 2 is well known: 2 * P(n) > P(n+1)
At the time I wrote a computer simulation that verified the relation up to some low number like 10,000,000. I later took it to one of my professors who did work in differential equations who in turn pointed me towards somebody at the university doing number theory.
This professor claimed to have proven my hypothesis using something called Chebyshev constants, but the proof went over my head. The original professor, who was not a number theorist, encouraged me to consider publishing on this, the number theorist professor was more interested in having me join him on some of his other research. I did not.
I ended up dropping the topic as I was busy with other things. But I still sometimes think about it.
The hypothesis has several implications I've never seen mentioned in relation to prime number theory:
3*P(n) > P(n+2)
and
P(n)*P(n) > P(2n)
These things seem notable to me, granted I am not a mathematician.
Is this fact well known? Is it in fact easy to prove? Was my professor mistaken? Could it be worthy of publication?
As an outsider I just do not know. Appreciate the insight. Thanks!
