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We know the limit is not 100 because of Euclid's proof of the infinitude of primes.
by idlewords 10y ago
We know the limit is not 100 because of Euclid's proof of the infinitude of primes.
- drdeca 10y agoThe primes become more sparse in the integers as you go farther. For most reasonable definitions of the density of composite integers in the integers, the density would be 100%. Iirc, it is believed that the number of primes less than n is O(n/log(n)), so primes/integers among the first n integers should go as O(1/log(n)), so goes to 0 as n goes to infinity.
- CUViper 10y agoI don't think that tells you anything without comparing to the infinitude of composites too. The fraction of numbers less than n that are prime is roughly 1/logn, which goes to zero in the limit. So I think the limit of numbers with factors less than n as n goes to infinity must indeed be 100%. That doesn't mean there's an n with 100%, just that there's no smaller limit that some n doesn't exceed.
- recursive 10y agoThere are infinite perfect squares also, but only 0% of all integers are perfect squares.
- idlewords 10y agoYet there are just as many perfect squares as integers.