Let $n$ be a positive integer and let $f(n)$ be the counting function for non-composite numbers.
So $$f(0)=0,f(1)=1,f(2)=2,f(3)=3,f(4)=3,f(5)=4$$
Now my mentor noticed as a kid that apparantly
$$f(x+y) \leq f(x)+f(y) + 1$$
and he considered it intuitive.
Does this look familiar ?
Yes, it is a weaker version of the famous second Hardy- Littlewood conjecture :
Let $\pi(x)$ be the prime counting function. The second Hardy- Littlewood conjecture states that $\pi(x+y) \leq \pi(x) + \pi(y)$, $x,y \geq 2$.
So we get the variation "conjecture T":
$\pi(x+y) \leq \pi(x) + \pi(y) + 2$
Now you are maybe thinking, so what ? Just a random conjecture ? Nothing special ?
But now comes the question.
It is known that :
The statement of the second Hardy–Littlewood conjecture is equivalent to the statement that the number of primes from x + 1 to x + y is always less than or equal to the number of primes from 1 to y. This was proved to be inconsistent with the first Hardy–Littlewood conjecture on prime k-tuples, and the first violation is expected to likely occur for very large values of x.[2][3] For example, an admissible k-tuple (or prime constellation) of 447 primes can be found in an interval of y = 3159 integers, while π(3159) = 446.
See : https://en.wikipedia.org/wiki/Second_Hardy%E2%80%93Littlewood_conjecture
So I wonder :
Is conjecture T still consistant with the first Hardy–Littlewood conjecture ?
How much do we need to make the second Hardy–Littlewood conjecture weaker to be consistant with the first Hardy–Littlewood conjecture ?