Will Prime Numbers (@_primes_) tweet the final prime number with 140 digits before the heat death of the universe?
Prime Numbers (@_primes_) is a Twitter feed that tweets successive prime numbers on an hourly basis. As we know, tweets are limited to 140 characters and the number of prime numbers extend infinitely beyond 140 digits. So how long will it take @_primes_ to reach the final prime number with 140 digits, thus ending its awesome yet tedious mathematical mission? More importantly, will it reach that final tweet before the heat death of the universe in roughly 10^100 years? As it says on the homepage of @_primes_:
Every prime number, eventually. (Or the heat death of the universe; whichever happens first.)
Let’s start by finding out what the final prime number with 140 digits is. A quick search on WolframAlpha reveals that this is a number of 139 consecutive 9s and a 7, which is just 3 shy of 10^140. Next, we want to figure out the approximate number of prime numbers between 0 and 10^140 or π*(10^140) which is equal to the number of hours needed. Using the prime number theorem, which is:
π (N) ~ N/ln N
or, our number (10^140) divided by the natural logarithm of our number (322.36191302) is equal to the approximate number of prime numbers between 0 and 10^140. We plug in our numbers and get:
π (10^140) ~ 10^140/322.36191302
π (10^140) ~ 3.10210… × 10^137
Therefore, when we factor in the primes that have already been tweeted (@_primes_ is currently on 5 digit primes, which barely scratches the surface) that final tweet will take a little less than 3.10210… × 10^137 hours. So what’s that in years? Well, there are 8760 hours in a year and when we divide our number of hours (3.10210… × 10^137) by 8760 we get 3.54121… × 10^133 years. This is significantly longer than the 10^100 years until the end of everything. As long as @_primes_ is able to tweet on an hourly basis until the heat death of the universe, it will still be a long ways away from reaching its final tweet.
* Note that π here is not referring to the mathematical constant of 3.14… rather it signifies the number of prime numbers below or at the number given.