- #1

MevsEinstein

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- TL;DR Summary
- So I was creating a function ##R(x)## such that gives you the gap between the largest two primes less than or equal to ##x##. I was trying to define it using the prime counting function, but I ran into problems

Hi PF!

I created a function ##R(x)## that gives the gap between the largest two primes less than or equal to ##x##. To define it, I used this property: $$\pi(x+R(x))=\pi(x)+1$$ Which is true since the ##x## distance between ##\pi(x)## and ##\pi(x)+1## is ##R(x)##. If we solve for ##R(x)## we get $$R(x) = \pi^{-1}(\pi(x)+1)-x$$ But ##\pi^{-1}(x)## isn't defined since ##\pi(x)## is a step function. Is there any other non-problematic way I can define ##R(x)##?

I created a function ##R(x)## that gives the gap between the largest two primes less than or equal to ##x##. To define it, I used this property: $$\pi(x+R(x))=\pi(x)+1$$ Which is true since the ##x## distance between ##\pi(x)## and ##\pi(x)+1## is ##R(x)##. If we solve for ##R(x)## we get $$R(x) = \pi^{-1}(\pi(x)+1)-x$$ But ##\pi^{-1}(x)## isn't defined since ##\pi(x)## is a step function. Is there any other non-problematic way I can define ##R(x)##?