I’m a 31 year old engineer, and I’ve recently came up with a way to exactly predict the probability of the number of prime numbers between two different integers.

For example using my way, the number of prime numbers between $0$ and $100$ is between $0$ and $50$. And it turns out that it is correct, since there are $25$ primes between $0$ and $100$.

But is this of any real importance that would lead me to publish a paper? Also my way is purely elementary and so I suspect that mathematicians would even bother to give it a look.

**Answer**

The method in question is noting that no primes other than 2 and 5 are divisible by 2 or 5, and thus any ten numbers $10n,10n+1,\ldots10n+9$ (really, any ten consecutive numbers) contain at most 4 primes, assuming the first term is larger than 5.

This can be improved by removing multiples of 3, so that no 30 consecutive numbers can have more than 8 primes (as long as the smallest is more than 5). It’s an improvement since the basic method would give only a bound of 12 for such an interval. Of course this can be increased by throwing in new primes like 7.

Indeed, with more subtlety you can prove things about intervals smaller than the least common multiple of the primes used, and this has been done for intervals of lengths up to perhaps 3000. For larger intervals analytic methods have been used which restrict the number of primes which can appear.

This is not a new result, but the family of results is important! In particular, they were an ingredient in improving Zhang’s theorem, a major step toward the twin prime conjecture. So what you have is just a piece of the puzzle, but it’s a very large, intricate puzzle. Be glad you have seen part of it!

**Attribution***Source : Link , Question Author : George Belcher , Answer Author : Charles*