MHB Showcase of 2016 Consecutive Numbers w/ 100 Primes

[kaliprasad]

Show that there exists 2016 consecutive numbers that contains exactly 100 primes.

 

[kaliprasad]

Show that there exists 2016 consecutive numbers that contains exactly 100 primes.

My solution

Spoiler

we know that number of primes less than 1000 is $= 168$
Now let f(x) be number of primes in a sequence of 2016 primes starting at x.
$f(1) > 100$.
now when we move to next number the number of primes increases/decreases by 1 or remains unchanged
$f(2017!+2) = 0$ as 2016 numbers starting from this number all are composite
So from 1 going upto 2017!+2 the starting number ( $>100$) remains unchanged or increases by 1 or decreases by 1
going to 0.
Hence at some point it is 100.