MathDB

Show that every set

\{p_1,p_2,\dots,p_k\}

of prime numbers fulfils the following: The sum of all unit fractions (that are fractions of the type

\frac{1}{n}

), whose denominators are exactly the

k

given prime factors (but in arbitrary powers with exponents unequal zero), is an unit fraction again. How big is this sum if

\frac{1}{2004}

is among this summands? Show that for every set

\{p_1,p_2,\dots,p_k\}

containing

k

prime numbers (

k>2

) is the sum smaller than

\frac{1}{N}

with

N=2\cdot 3^{k-2}(k-2)!

2