Question

Let EE be the set of all positive even integers. We call a number e∈Ee∈E "eprime" if ee cannot be expressed as a product of two other members of EE.

a) Give an example of an eprime that is greater than 100 and has two different eprime factorizations.

b) Are the eprimes dense in E?

c) What is the proportion of eprimes in EE? (meaning that if I were to take a random, contiguous, really large set of values in E, how many of them would be eprime relative to the size of the set)

Answer 1

(a) 102 is an example of such eprimes as 102=2.3.17 it is never been a product of two positive even integers. And 102 is divisible by 6 and 34 where both of them are eprime.

(b) The numbers which are divisible 4 are not eprimes. Only eprimes are 2(mod 4) i.e. of the form 2+4k. Hence eprimes are not dense in E.

(c) two consecutive two eprimes having difference 4 and in between them there is one and only non eprime. So that ever large interval or large set in E you consider there is half of them are eprime and half of them are not. Hence density of eprimes in E is exacly 1/2 i.e. half of them are eprimes in E.