How to find primes between given range

I saw movie "the man who knew infinity"

In which ramanujan had formulated a formula to calculate number of prime numbers between a range of numbers

My teacher told me that there exist no program which can generate primes

My friend told me that someone had found a counter example of ramanujan'a formula which was billions of billion

So I want to know that did he really formulated the formula, if yes then what is it and how it works (proof required).

• various computer programs can generate prime numbers - some of them very simple. – Cato Dec 22 '16 at 13:23
• Your teacher told you that "there exist no program which can generate primes"??? You can tell him or her, pardon my French, that's a bunch of baloney. – barak manos Dec 22 '16 at 13:23
• As with regards to "number of prime numbers within a given range": You can approximate the number of primes $\in[x,y]$ as $\frac{y}{\ln(y)}-\frac{x}{\ln(x)}$. There are also tighter bounds. Though I'm not familiar with the historical details, I would guess that Ramanujan had conjectured such tighter bound, which was later refuted explicitly with a counterexample (or possibly more interestingly - refuted by proving that a counterexample existed)... – barak manos Dec 22 '16 at 13:27
• @barakmanos That really big number too' big for me to realistically care about much? :D – Simply Beautiful Art Dec 22 '16 at 13:29
• @SimpleArt: I understand each word in your comment, but I am unable to connect them into a meaningful statement. Would you mind rephrasing it? – barak manos Dec 22 '16 at 13:35

Now, take out all the numbers divisible by $2$. Then all the numbers divisible by $3$. Then $5,7,\dots,$ all the way till you reach the square root number. The numbers you have left are prime numbers, and this is easily doable with a computer.
If you are interested in approximations though, consider the prime-counting function. One simple approximation to finding the amount of prime numbers between $x$ and $y$:
$$\frac y{\ln y}-\frac x{\ln x}$$