On Some New Questions on the Distribution of Prime Numbers
نویسنده
چکیده
have infinitely many solutions. If the well known hypothesis about prime twins is true, that is, if the equation pn+i—pn — 2 has infinitely many solutions, (1) and (2) of course are trivially satisfied. The first inequality of (2) is inserted only for the sake of completeness. I t follows from the well known fact that lim sup (pn+i—pn) = °° (since n\+2, n\+3, • • • , » ! + » are all composite). The proof of the other inequalities will be simple, but less trivial. Clearly pn-1pn+x>p 2 n implies (pn-i+pn+i)/2>pn and pm>(pm-i +pm+i)/2 implies pm>pm-ipm+i. The well known relations between the various mean values suggest the following questions: Is it true that for every / the inequalities
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