# On the Number of Primes between n^2 and (n+1)^2

### On the Number of Primes between n^2 and (n+1)^2

For any positive integer, n ≥ 2, there are at least two prime numbers between $$n^{2}$$ and$$(n+1)^{2}$$ .

Guest

### Re: On the Number of Primes between n^2 and (n+1)^2

The question is stronger than the related Legendre's Conjecture (https://en.wikipedia.org/wiki/Legendre%27s_conjecture).

And the key result is:

$$\pi((n+1)^{2})$$ - $$\pi(n^{2})$$ $$\rightarrow$$ $$\pi(n)$$ = n/log(n) as $$n \rightarrow \infty.$$

Example: $$\pi(75,001^{2})$$ - $$\pi(75,000^{2})$$ - $$\pi(75,000)$$

= 262,872,577 - 262,865,922 - 7493 = -738 (almost 10% error).

That excellent result corresponds very well with our theory!

Dave the antworker
Guest

### Re: On the Number of Primes between n^2 and (n+1)^2

Guest wrote:The question is stronger than the related Legendre's Conjecture (https://en.wikipedia.org/wiki/Legendre%27s_conjecture).

And the key result is:

$$\pi((n+1)^{2})$$ - $$\pi(n^{2})$$ $$\rightarrow$$ $$\pi(n)$$ = n/log(n) as $$n \rightarrow \infty.$$

Example: $$\pi(75,001^{2})$$ - $$\pi(75,000^{2})$$ - $$\pi(75,000)$$

= (262,872,577 - 262,865,922) - 7493

= 6755 - 7493 = -738 (almost 11% error);

6755 - 75,000/log(75,000) = 6755 - 6681

= 74 (almost 1% error).

Those excellent results correspond very well with our theory!

Dave the antworker
Guest