proof of the prime number formula(It's real)

Question

Proof that this formula always generates primes.Also it generates all primes(grate prime numbers formula)

I have tried this formula it also generates primes orderd except the prime 2.

I am really interested in this formula because it generates all primes there are such formulas for example this one$\left\lfloor\text{A}^{3^n}\right\rfloor$.But it don't generates all primes and $A$ is a dangerous number and that becomes calculating hard.Also there are some formula that sometimes don't generates primes but they are easy to use.By the way if anyone can proof this formula more people can interest.I also don't belive that it's real but I tried it to 50 numbers and it worked.

score 5 · Accepted Answer · answered Jun 04 '16 at 16:13

5

Wilson's theorem states that $(p-1)!+1\equiv 0\mod{p}$ if and only if $p$ is prime. Thus $2m+1\mid (2m)!+1$ if and only if $(2m)!+1\equiv 0\mod{2m+1}$ if and only if $2m+1$ is prime. Therefore $\frac{(2m)!+1}{2m+1}$ is an integer if and only if $2m+1$ is prime.

So if $2m+1=p$ is prime, the formula reduces to $$H(m) = 2\left(\frac{2m+1}{2}\right)^{\lfloor \frac{2m+1}{(2m)!+1}\cdot \frac{(2m)!+1}{2m+1}\rfloor} = 2\cdot \frac{2m+1}{2} = 2m+1,$$ which is prime.

If $2m+1$ is not prime, then since $\frac{(2m)!+1}{2m+1}$ is not an integer, its floor is less than $\frac{(2m)!+1}{2m+1}$, so the exponent is equal to zero and we get $H(m) = 2$.

answered Jun 04 '16 at 16:13

rogerl

22,399

1

And how to proof it generates primes ordered – Taha Akbari Jun 04 '16 at 16:17
I think this proof shows that. As $m$ increases, $2m+1$ ranges over all odd numbers, so the odd primes are generated in order, with $2$'s interspersed for odd nonprimes. – rogerl Jun 04 '16 at 16:18
And how to proof it generates all primes?????? – Taha Akbari Jun 04 '16 at 16:19
1

That should really be obvious if you examine the argument. As I just said, the odd primes are generated in order, since $H(m) = 2m+1$ for $2m+1$ a prime. – rogerl Jun 04 '16 at 16:29
oh, yes sorry . – Taha Akbari Jun 04 '16 at 16:32

proof of the prime number formula(It's real)

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