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This is an idea that I had about 3 months ago. I tried some college professors, they didn't care. I tried to solve, but with no luck. I ask for help to find the closed form of the following product series. I know it would be safer only to put the product, but I just would like to see this solved, and if I put why I think this is important, the chance of someone helping would be higher.

My idea is:

If $\prod_{k=2}^{X-1}{\sin(\pi\frac{X}k)}≠0$, than the number $X$ is prime. Else, $X$ is a non-prime.

Why:

  • $\sin(\pi\frac{X}k)$ tests if a number k divides X. If the value returned is $0$, than k divides X. Because for a number to divide another, the result must be an integer, and $\sin(\pi\alpha)$ is equal to $0$ if and only if $\alpha$ is an integer. So, to test if a number, for example, k=3 divides 9, we could try that like $\sin(\pi\frac{9}3)=\sin(3\pi)=0$. Now let's try to see if 2 divides 11 this way: $\sin(\pi\frac{11}2)=\sin(5.5\pi)=-1≠0$
  • Using logic, we could extend this to the product above. We only need one sine being $0$ for the product to be $0$. That means that with only one divisor from range $2$ to $X-1$, the number X is a non-prime. And the only way for the product to be $0$ is if one or more of the sines is $0$: You can't multiply numbers $≠0$ and get $0$.

For example, 7 is prime if: $$\sin(\pi\frac{7}2)\times \sin(\pi\frac{7}3)\times \sin(\pi\frac{7}4)\times \sin(\pi\frac{7}5)\times \sin(\pi\frac{7}6)\ne 0$$ and that's the case.

I know I could test only until $\sqrt{7}$ rounded up, but if someone can get the closed form of the product, that wouldn't matter, and we wouldn't have to calculate the square root.

Thank you for reading this.

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  • $\begingroup$ it will be very tough if not impossible , as there are no results for sin(a/b) as compared to those for sin(a+b) or even sin(a*b). Nice to see someone thinking new stuff $\endgroup$ – VigneshM Nov 9 '14 at 18:46
  • $\begingroup$ It's good that you're trying things out for yourself, but unfortunately this sort of primality test is ultimately useless. There's not going to be a closed form for this or any similar product. Take a look at this thread, this thread, or this thread for ideas similar to yours. Someone named "C.P. Willans" seems to be the originator of such methods. $\endgroup$ – curious Nov 9 '14 at 19:11
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Assuming $X$ is an integer $\geq3$ your test is fine from a logical point of view. But it is completely useless if you want to find out whether $X:=63\,176\,591$ is prime. You would need millions of terms of the sine series in order to compute a single factor of your product, say $\sin\bigl(\pi{X\over 31}\bigr)$. But even if you computed all these factors to $10^6$ places, in a numerically unlucky situation you would not be able to decide definitively that the product is $\ne0$.

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  • $\begingroup$ Yes, this product series is useless unless someone could find its closed form. There are closed forms really close to this, like $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$. But it seems this one is impossible, like the others have said. $\endgroup$ – Paulo Felipe Nov 10 '14 at 0:03
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This test is useless because this product tends to zero for big numbers. Even the result for 1273 (prime) is $0$ for the computer, because it gets too small. The sine values varies between 0 and 1, so the product gets smaller and smaller... I even did an approximation for that product (possibly wrong), which gave me a complex answer but also returned $0$ for big numbers.

I'm trying to think now of a function that returns $f(integer)=0$, but gives $|f(non-integer)|\ge1$

Since I already wasted some time trying to do this approximation I'll share it. But again, I don't know if this is correct:

$P(X)=\prod_{k=2}^{X-1}{\sin(\pi\frac{X}k)}$

$P(X)=(2i)^{2-x}\prod_{k=2}^{X-1}{(e^{i\pi X/k}-e^{-i\pi X/k})}$

$P(X)=(2i)^{2-x}\prod_{k=2}^{X-1}{e^{i\pi X/k}}\times\prod_{k=2}^{X-1}{(1-e^{-2i\pi X/k})}$

$$P(X)=(2i)^{2-x}\prod_{k=2}^{X-1}{e^{i\pi X/k}}\times(-1)^{X-2}\times\prod_{k=2}^{X-1}{(e^{-2i\pi X/k}-1)}$$

$\prod_{k=2}^{X-1}{e^{i\pi X/k}}=exp(\sum_{2}^{X-1}{\frac{i\pi X}{k}})=exp(i\pi X\times\sum_{2}^{X-1}{\frac{1}{k}})$

$\prod_{k=2}^{X-1}{e^{i\pi X/k}}\approx exp[i\pi X(lnX+0.57722+\frac{1}{2X}-1)]$

$$\prod_{k=2}^{X-1}{e^{i\pi X/k}}\approx exp[i\pi X(lnX-0.4228+\frac{1}{2X})]$$

Now using this: Evaluating the infinite product $\prod\limits_{k=2}^\infty \left ( 1-\frac1{k^2}\right)$

$\prod_{k=2}^{X-1}{(e^{-2i\pi X/k}-1)}=(-1)^{X-2}\times\prod_{k=2}^{X-1}{(1-e^{-2i\pi X/k})}$

$=(-1)^{X-2}\times\prod_{k=2}^{X-1}{(1-\frac{1}{exp(2i\pi X/k)}})=$

$=(-1)^{X-2}\times\prod_{k=2}^{X-1}{(1-\frac{1}{exp(i\pi X/k)})(1+\frac{1}{exp(i\pi X/k)}})=$

$=(-1)^{X-2}\times\prod_{k=2}^{X-1}{(\frac{exp(i\pi X/k)-1}{exp(i\pi X/k)}\times\frac{exp(i\pi X/k)+1}{exp(i\pi X/k)})}$

Where $\frac{exp(i\pi X/k)+1}{exp(i\pi X/k)}=a_k$ and $\frac{exp(i\pi X/k)-1}{exp(i\pi X/k)}=\frac{1}{a_{k-1}}$

$$\prod_{k=2}^{X-1}{(e^{-2i\pi X/k}-1)}=(-1)^{X-2}\times(\frac{1+exp(\frac{i\pi X}{X-1})}{exp(\frac{i\pi X}{X-1})})\times(\frac{exp(i\pi X)}{1+exp(i\pi X)})$$

So, the answer would be:

$$P(X)\approx(2i)^{2-X}\times exp[i\pi X(lnX-0.4228+\frac{1}{2X})] \times (\frac{1+exp(\frac{i\pi X}{X-1})}{exp(\frac{i\pi X}{X-1})})\times(\frac{exp(i\pi X)}{1+exp(i\pi X)})$$

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