Yet a conjecture about the prime counting function.

Question

This is a conjecture emanating from random tests with my set routines written in Forth.

$a>b \;\wedge \pi(a+b)=\pi(a)+\pi(b)\implies b<11$

Furthermore, if $b\in\mathbb N$, then $b\in\{0,1,2,3,4,9,10 \}$

I would like (partly) proofs or counterexamples.

Tested for $b<a<100,000$.

Testing the numbers of hits for the non trivial values of $b$ and for some upper limits:

               1       2       3       4       9      10
    100       74      48       8      14       2       2
   1000      832     332      34      66       4       6
  10000     8771    2454     204     406      11      20
 100000    90408   19180    1223    2444      37      72
1000000   921502  156992    8168   16334     165     328

Answer 1

Your claim is a consequence of the Secondly Hardy Littlewood conjecture which has been open since 1923. We don't even know if $\pi(a+b) \le \pi(a) + \pi(b)$ holds in general so proving $\pi(a+b) = \pi(a) + \pi(b)$ implies $b < 11$ is beyond the current state of art.