Here’s the thread, but even at the time I posted it, I had forgotten the crucial details of the thing. The whole thing was a total embarrassment and I was probably mistaken anyway.
You make some interesting points, although I think if you could find a correlation between z and x and y where x and y are bigger than 1, that would be more substantial.
Nitpick: The problem with using this is not the size of (n-1)! – you only need *modular *arithmetic so the numbers in calculation don’t exceed n.
The problem is just the size of n itself.
The loosely related Fermat’s Little Theorem also involves (n-1) conceptual multiplications, but because the factors are constant (a^(n-1) rather than (n-1)!), only about log n operations are required. This theorem is used for prime testing. Unfortunately, while most composite numbers flunk a Fermat test, some (“pseudoprimes”) do not. Indeed certain composite numbers of which the smallest is 561 pass the Fermat primality test for all a.
I find that hard to believe.
Nitpick acknowledged. The size of the primes that we are interested in trumps most ordinary means of manipulating those numbers. Smaller primes are well known and we need only look them up for most purposes. Predictable patterns are as far as I know unknown (apart form the trivial such as all all primes>2 are odd etc.)
I did not know that.
I was trying to recall Fermat’s Little theorem and primality test when I wrote that post. Memory and time failed me. Thanks for looking it up.
So, obviously, I have talked previously in length about the equation:
z=y**2+xy
Which calculates all possible numbers on the table- given the knowledge of x and y- and thus all the possible non-primes?
Is there another way in doing this?
Because, in theory, with the knowledge of two equations- I could use simulataneous equations to reverse the system- where z is given and x and y is supplied- deciding whether number is prime or not.
No, you couldn’t. Imagine you’re asked: Is the number 14573621 prime or not?
What x or y are you going to use in your hypothetical two equations?
As has been pointed out before in this thread z = y^2 + xy (y^2 is a more common way to write y squared by the way), can be rewritten z = y(x + y), so y and (x+y) are two factors of z. But if you know y, you know a factor of z so you already know that z is not prime, so there’s no need for a second equation or any solving of simultaneous equations.
I leave as an exercise for the reader why x (which doesn’t have to be a factor in z) is unlikely to be something you can find either.
Reminds me of the way various professions show that all odd numbers are prime:
Mathematician -
“1 is prime. 3=1+2 is prime. 5=3+2 is prime. 7=5+2 is prime. Therefore, by induction, all odd numbers are prime!”
Physicist -
“1 is prime, 3 is prime, 5 is prime, 7 is prime, 9? - experimental error, 11 is prime… I conclude all odd numbers are prime!”
Engineer -
“1 is prime, 3 is prime, 5 is prime, 7 is prime, 9 is prime, 11 is prime… I conclude all odd numbers are prime!”
Except that 1 isn’t prime.
What I thought, was that if there was another way of expressing x and y in order to build z, then by using simulataneous equations you could reverse the equation in order to prise x and y out pf z.
It still seems reasonable enough to me, I just do not have the time or ability to begin to think of what the second equation might look like.
I understand what you mean about y, but if you had another form of making z then it seems obvious to me that you could reverse the equation.
Maybe I am not seeing that, but it seems relevant to me.
Sheesh. I need to apologise for my previous post, which actually makes no sense at all. I started on one thought, got derailed and wrote a post based on the wrong destination I was at at the time. My actual original intention wasn’t to write that you need to reduce the equation to only one unknown, which of course you don’t when you have a system of two equations.
The problem with looking for a second equation is that your y and x are respectively a factor of z, and the difference between the remainder of z divided by y, and y itself.
I don’t have the time or ability to prove it, but I find it likely/“naively obvious” there’s no other way of combining the x and y of your equation to make z that applies to all z.
Now part of that is knowing that the world’s mathematicians have looked for ways of quickly determining primeness for literally millennia, and part of it is gut feeling.
Aren’t you just talking about the integer factorization problem?
It was the interference pattern that got me thinking about another equation:
obviously the modulus difficulties the situation- but it seems to me taht there is some form of a consistent pattern- which in theory could be translated into a second equation.
But you don’t need just another equation, you need another equation relating x and y to z.
I can’t comment on how your patterns arise, since I can’t figure out what they’re representing. Your descriptions are just too confusing.
As noted earlier, the equation z = y^2 + xy is the same as z = y(x + y). So you have two factors, y and x+y. The unique prime factorisation theorem tells us that there is only one set of numbers which you can use to construct y and x. There may well be more than one way of building y and x from them.
So the problem is exactly this:
[ol]
[li]Find the set of prime factors of z. [/li][li]All of the factors, and all products of factors that are less than or equal to the square root of z are valid solutions for y. You can choose any.[/li][li]Once you have y, x is trivially found ( x = product of the remaining prime factors less y)[/li][/ol]
This isn’t expressible as an equation. Worse, step one is actually the solution to your original problem. We know y must be a factor of z. The simple existence of y or not (for y ≠ z) is actually the entire problem. In that sense the use of z = y^2 + xy as a way of building up a mechanism for finding primes is actually circular. In order to find y in the first place you must solve a harder problem than you are setting out to use this as a solution to. That is rather than just decide if y:y≠z exits (and thus z is prime or not), you actually have to find y. Finding any factor of z is enough to show it is not prime, going on to find x and y is just more work.
In general a serious problem is that what you envisage isn’t a simple simultaneous equation. You require that the solutions be integers, and that makes it a whole new ball game in difficulty. There are no equivalent ways of manipulating a set of equations (as you would use to solve a set of simultaneous equations that can have non-integer solutions) if the solutions can only be integers (or are otherwise constrained to a set of discrete or non-continuous solutions). It is an important problem, as there are many things that require discrete solutions.
My theory is (it is too dificult for me to computer at the current time) to create a structure which is built only out of the first (smallest) factors of z.
You could do this by looping z through a set of numbers and indexing it in the list in order to get its first x and y position.
If this then created a structure capable of an equation, then the second equation would only concern the smallest factors- and thus any solution that came out as continuous- would be a prime.
This is assumnig that this structure ommits the 1s line.
Well, yes, the smallest factor is probably going to be the easiest one to find. That’s already taken into account in all methods of factoring.
Looking at the geometric pattern of the graph got me thinking:
There are geometric “vergences” (convergence, divergence) at the corners of greater and lesser boxes, greater and lesser triangles, and the center of meeting circles.
What values do these vergences have?
I think I understand what you mean.
Sorry, but about the “value” of these “verging points”, I think you might have to be a little be more specific for my incompetence.
On the ratio of the triangles, I believe they all perfectyl reflect each other:
So the ratio would be 1:1
The angle of the triangle would be interesting, but it would be consistent with the angle of the plane (seemingly, 45 degrees).