i. Almost every positive integer is composite.
ii. Almost every prime has more than 1000 digits.
Are the above statements true?
The second one is easier than the first. There are finitely many primes with 1000 or fewer digits, but infinitely many with more than 1000 digits.
The first statement is more nuanced. The set of primes and the set of composites are both infinite; both have cardinality Aleph-null. However, for any given size limit there are more composite numbers than primes. As the limit of consideration gets larger and larger, the fraction of primes approaches 1/ln(n), which approaches zero.
Consider what would be the case if the universe were discovered to be infinite. Would it still be true that there are many more red-dwarf stars than stars like the sun? ... even though they are both countably infinite in number.
But again, there are Aleph-null primes and Aleph-null composites altogether.
Posted by Charlie
on 2018-09-25 09:48:16