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December 21st, 2015, 03:19 PM   #1
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Interesting property of the Primes

Hi. Recently when I was investigating integer factorization techniques which is a big interest of mine, I found something very interesting regarding the primes. I'm sure this is expounded somewhere in the literature, but if not and anyways, here goes:

I was examining numbers which can be represented in the form: z^2 - (2*z + 1), specifically prime numbers, i.e. those such that k*p = z^2 - (2*z +1) where k is an integer coefficient, and p is a prime.

Anyway, this equation is a nice quadratic, and rearranging for z we have that z is satisfied when the determinant, which is sqrt(k*p + 2), is a whole number. Not exactly earth shattering stuff so far, granted. But bear with me, because this is where it gets interesting. I started plugging in prime numbers, including 1 as a prime, which it isn't technically I know, but anyway.

I then checked to see which primes produced whole number determinants, i.e. satisifed the quadratic in integers. What I discovered surprised me. The primes divided neatly into two groups, based on whether they satisifed the equation or not. I now provide a list of the primes which satisifed the equation (including the number 1 for completeness) up to 200:
1, 2, 7, 17, 23, 31, 41, 47, 71, 73, 79, 89, 97, 103, 113, 127, 137, 151, 167, 191, 193, 199.

To be honest, I only know the prime numbers from 1 - 200 so I stopped there, but suffice to say a pattern emerges. All the other primes in the interval from 1 - 200 don't satisfy the equation. Okay, interesting, I thought, but still not great. Then I started examining composites, and what I found was even more interesting.

When I multiplied primes from the list above together, the resulting number always satisifed the original equation in integers. Moreover, the k values which satisified the qudaratic were themselves either prime numbers featured in the included list above (which I assume extends indefinitely to infinity) or composites whose factors were included in the list.

As far as I know, I haven't seen this mentioned anywhere in my admittedly shallow understanding of the literature around number theory. I should add that furthermore, when any composite number had at least one factor not included in the list above, i.e. one which didn't satisfy the quadratic equation in integers, then the overall composite number itself also didn't satisfy the equation.

Conversely, any composite numbers made up ONLY of prime numbers which are in the list above, and indeed those that extend beyond it, will always also satisfy the quadratic in integers, by empirical observation. I feel that this might have interesting and useful implications in integer factorization, because for certain numbers, particularly semi-primes which can be shown to satisfy the quadratic equation above in integers, we can effectively reduce the number of primes to be looked for in trying to find a factorization by half, to just those on the list above and that extend beyond it.

I'd like to hear your thoughts on whether this is interesting or not, in any way new information, and perhaps useful.

Last edited by Jopus; December 21st, 2015 at 03:21 PM.
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