I’m working through William Leveque’s Fundamentals of Number Theory.
There is a problem put at the end of section 6.7 (Think it’s 5a, but I don’t have the text in front of me right now) which I’ve had some difficulty with (paraphrase; if anyone has the text please check that I’ve summarized correctly): “Prove that for any value e there are an infinite number of consecutive primes pn and pn+1 such that pn(1+e) > pn+1.”
Perhaps I’m misreading this, but usually I have little difficulty with the problems in this book (after a thorough reading of the chapter, of course), but this one has stumped me for several days. Thought I’d throw it out to the board and see if someone can help, or perhaps correct my mis-interpretation of the problem.
FYI: Section 6.7, “The true order of pi(x)”, details a proof of Chebyshev’s theorem:
along with a proof bounding the size of the nth prime and some improvements on the order of the summation of inverted primes (e.g. 1/2 + 1/3 + 1/5 + …).