By using Prime numbers less than $\sqrt{N}$, the number of primes less than $'N'$ can be found out. Is this true or false?
I have verified it for upto from $10^1$ to $10^3$ for all the tens. So this means Number of prime numbers inside $10^{24}$ can be found out by using prime numbers less than $10^{12}$?
Whether there is more advanced of this theorem is there anywhere?
Example
For example, $N = 100 \rightarrow \sqrt{N} = 10$
So odd Prime numbers inside $\sqrt{N}$ are $2$, $3$, $5$, $7$
Total numbers = $100$
Even numbers = $50$
Odd numbers = $49$ (Excluding 1 as we already know 1 is not a prime)
So No. of Prime numbers inside $100$ is
= $49-[(49/3)]-[((49 - (49/3))/5)]-[((49 - (49/3)-(49/5))/7)]+1 = 25$
Same can be done for $200$, $300$ ... $10^3$ ... and so on (I have verified upto $10^3$)
Note: I have not rounded off the values in the above equation!! You may get a result within a range of $5$ from the actual value($+5$ or $-5$) but I need to work more on fine tuning this.