i. Almost every positive integer is composite.
ii. Almost every prime has more than 1000 digits.
Are the above statements true?
Please comment.
1) Mostly composite: True: The Prime Counting Function approximates the number of primes less than n as n/log(n). As a fraction of n, this is 1/(number of digits in n). So, for example, of the first Googleplex of integers, only a fraction 1/Google are prime, and the rest are composite. QED
2) Mostly longer than 1000 digits: True. The fraction f, that are shorter than 1000 digits is computed taking the limit as m=10^n goes to infinity, or therefore as n goes to infinity:
f = lim {as n-> inf} (primes with less than 1000 digits/ primes total}
= lim{n->inf} (10^1000/3) / (10^n/n)
= lim{n->inf} n/3 10^(1000-n)
= lim{n->inf} n/(3 10^n) (then, using L'Hôpital's Rule)
= lim(n->inf) 1/(3 n 10^n-1)
= 1/inf = 0
QED
Edited on September 25, 2018, 10:53 am
attempted soln.s
