Number Theory Problem
So, the following "fact" came up in the topology class I'm helping teach. Lots of people seem pretty sure it's true but I've stared at it for five minutes and can't prove it. Call a rational a "p2" rational if it can be written in the form p/2^n for some prime, p, and natural, n. Given ε >0, can we find a p2 rational in the interval [1/3, 1/3+ ε ) ?
The route of least resistance towards a proof seems to be looking at the asymptotic density of primes but I can't really make any headway.
[Edit (9/4): Um.. I kind of misread it. p doesn't have to be prime, so it really is obvious. Calling non-prime numbers p confuses Adam.]
The route of least resistance towards a proof seems to be looking at the asymptotic density of primes but I can't really make any headway.
[Edit (9/4): Um.. I kind of misread it. p doesn't have to be prime, so it really is obvious. Calling non-prime numbers p confuses Adam.]