Let’s say we have 10 primes at or below x, and 6 primes at or below x/2. That means that there are at least 4 primes (10 − 6) on our magic line. The lower point can include one of the primes “at or below” it. So one of the 6 primes at or below the lower endpoint of the magic line (as I originally defined it as “half the starting point, rounded up”—it’s changed now) could be located right on the endpoint. If that was included as one of the primes on the magic line, then there would have to be 5 primes on the magic line—a contradiction. So no, I think the lower endpoint must not be included. I fixed the post by altering the definition of the lower endpoint of the magic line and credited you at the end.
That’s a great question! Let’s think about it.
Let’s say we have 10 primes at or below x, and 6 primes at or below x/2. That means that there are at least 4 primes (10 − 6) on our magic line. The lower point can include one of the primes “at or below” it. So one of the 6 primes at or below the lower endpoint of the magic line (as I originally defined it as “half the starting point, rounded up”—it’s changed now) could be located right on the endpoint. If that was included as one of the primes on the magic line, then there would have to be 5 primes on the magic line—a contradiction. So no, I think the lower endpoint must not be included. I fixed the post by altering the definition of the lower endpoint of the magic line and credited you at the end.
This example doesn’t fit the updated definition:
Good read, I don’t think I’d heard of Ramanujan primes before.
Thanks :D I’ll update that soon.