Monday, August 20, 2007
About random, primes and statistics
Knowing that statistics is an area of mathematics that physics students get rather familiar with I thought it'd be of interest to explain a simple area where mathematicians routinely lie—I think in order to keep research grants.
First you need to learn a bit of number theory, as 7 mod 3 = 1, where 1 is the residue modulo 3, so the bit of math is that shown x mod y, you take x-ky where k is the largest positive integer that will fit, and use what's left over, which is the residue.
I picked 3 because there is a fascinatingly boring thing about numbers modulo 3—perfect regularity:
Starting at 1 and counting up you have
1, 2, 3 followed by 4, 5, 6 followed by 7, 8, 9 on out to infinity
which modulo 3 gives
1, 2, 0 followed by 1, 2, 0 followed by 1, 2, 0 repeated on out to infinity.
(It's like a perfect waltz to infinity!!!)
That is an absolute and I'd say a trivial one at that but it will challenge everything you think you know about mathematicians as decent researchers as you have primes and you have composites and composites are products of primes—kind of like their children!—so if primes tended to pick a particular residue modulo 3, then composites would follow along!
But they don't. They split evenly between 0, 1 and 2.
Therefore, I strongly suggest to you, primes split evenly between having a residue of 1 and 2 modulo 3.
If that is true then the residue of a prime other than 3 modulo 3 will be random.
Here is what you get with the first 23 primes greater than 3:
5 mod 3 = 2, 7 mod 3 = 1, 11 mod 3 = 2, 13 mod 3 = 1, 17 mod 3 = 2, 19 mod 3 = 1, 23 mod 3 = 2, 29 mod 3 = 2, 31 mod 3 = 1, 37 mod 3 = 1, 41 mod 3 = 2, 43 mod 3 = 1, 47 mod 3 = 2, 53 mod 3 = 2, 59 mod 3 = 2, 61 mod 3 = 1, 67 mod 3 = 1, 71 mod 3 = 2, 73 mod 3 = 1, 79 mod 3 = 1, 83 mod 3 = 2, 89 mod 3 = 2, 97 mod 3 = 1
So the sequence is
2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1
and I suggest to you it is random, like flipping a coin, but better as it is absolutely random.
Now then what if it were NOT random so that there was some pattern in there?
Well, composites are products of primes, so where primes go, so do the composites, so if primes tended to have say, a residue of 1 modulo 3, then so would composites, like 7*13 = 91 and 91 mod 3 = 1.
If there were some more hidden pattern, for instance, when I've brought this subject up before sci.math'ers would claim that if 1 tends to be followed by 2 and 2 by 1 and they produced statistical tests claiming those proved that, then the sequence is not random!
But if 1 tends to be followed by 2 and 2 by 1, how would they push the composites? Into what pattern?
Now then, let's leap forward and say you accept that the rigidity of the ordering of counting numbers, where you have
1, 2, 3 followed by 4, 5, 6 followed by 7, 8, 9 on out to infinity
which modulo 3 gives
1, 2, 0 followed by 1, 2, 0 followed by 1, 2, 0 repeated on out to infinity
convinces you that primes other than 3 modulo 3 give a random distribution, why should anyone care?
Well, random is useful in physics but more intriguingly, if you accept that then you end a lot of research paths in modern mathematics!!!
Because if you push the argument to p_1 mod p_2, where the p's are differing primes, then it turns out some supposedly big questions in modern math, like the Twin Primes Conjecture are easily answered!
But there are mathematicians getting federal funds for research in those areas, if random is the call then those funds cease.
BUT if no one notices, those mathematicians can work endlessly in an area where they can never get an answer, at least not a correct one.
That is a highlight of how you can find controversy in the math field in a simple area where you can run your own statistical analysis to see.
Oh yeah, and remember a while back the math awards where one guy famously refused to accept his, while the others did accept?
Well, if primes are random then one of those people won a prize for supposedly innovative research into an area where no pattern actually lies.
Maybe you believe I'm wrong and there is some hidden pattern in primes modulo 3, or more technically
p mod 3
with p not equal to 3, as you go out to positive infinity.
I've given the beginning of the sequence:
2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1
where the start was the number 2 which has itself as a residue modulo 3.
In answering my bringing this issue up before posters on sci.math claimed that this area has been checked and that statistical analysis proved non-randomness but there is no why here given.
Why would primes care? Especially when their products, the composites can't swing one way with the primes swinging another?
I suggest to you that math people lie in this area, where you can do the statistical analysis yourself to see that they lie, or I'm wrong. I've been wrong before, but I don't think I'm wrong here.
I know people lie. And I know some of you in the physics field who like to play with statistics can crank up some machinery and play with primes to see for yourself.
Remember, BIG research grants are wrapped up in this where some mathematicians have worked in this area for decades, so their entire livelihoods are about it NOT being true that p mod 3 is random.
Their entire LIVES are about that not being true, so they are totally invested.
First you need to learn a bit of number theory, as 7 mod 3 = 1, where 1 is the residue modulo 3, so the bit of math is that shown x mod y, you take x-ky where k is the largest positive integer that will fit, and use what's left over, which is the residue.
I picked 3 because there is a fascinatingly boring thing about numbers modulo 3—perfect regularity:
Starting at 1 and counting up you have
1, 2, 3 followed by 4, 5, 6 followed by 7, 8, 9 on out to infinity
which modulo 3 gives
1, 2, 0 followed by 1, 2, 0 followed by 1, 2, 0 repeated on out to infinity.
(It's like a perfect waltz to infinity!!!)
That is an absolute and I'd say a trivial one at that but it will challenge everything you think you know about mathematicians as decent researchers as you have primes and you have composites and composites are products of primes—kind of like their children!—so if primes tended to pick a particular residue modulo 3, then composites would follow along!
But they don't. They split evenly between 0, 1 and 2.
Therefore, I strongly suggest to you, primes split evenly between having a residue of 1 and 2 modulo 3.
If that is true then the residue of a prime other than 3 modulo 3 will be random.
Here is what you get with the first 23 primes greater than 3:
5 mod 3 = 2, 7 mod 3 = 1, 11 mod 3 = 2, 13 mod 3 = 1, 17 mod 3 = 2, 19 mod 3 = 1, 23 mod 3 = 2, 29 mod 3 = 2, 31 mod 3 = 1, 37 mod 3 = 1, 41 mod 3 = 2, 43 mod 3 = 1, 47 mod 3 = 2, 53 mod 3 = 2, 59 mod 3 = 2, 61 mod 3 = 1, 67 mod 3 = 1, 71 mod 3 = 2, 73 mod 3 = 1, 79 mod 3 = 1, 83 mod 3 = 2, 89 mod 3 = 2, 97 mod 3 = 1
So the sequence is
2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1
and I suggest to you it is random, like flipping a coin, but better as it is absolutely random.
Now then what if it were NOT random so that there was some pattern in there?
Well, composites are products of primes, so where primes go, so do the composites, so if primes tended to have say, a residue of 1 modulo 3, then so would composites, like 7*13 = 91 and 91 mod 3 = 1.
If there were some more hidden pattern, for instance, when I've brought this subject up before sci.math'ers would claim that if 1 tends to be followed by 2 and 2 by 1 and they produced statistical tests claiming those proved that, then the sequence is not random!
But if 1 tends to be followed by 2 and 2 by 1, how would they push the composites? Into what pattern?
Now then, let's leap forward and say you accept that the rigidity of the ordering of counting numbers, where you have
1, 2, 3 followed by 4, 5, 6 followed by 7, 8, 9 on out to infinity
which modulo 3 gives
1, 2, 0 followed by 1, 2, 0 followed by 1, 2, 0 repeated on out to infinity
convinces you that primes other than 3 modulo 3 give a random distribution, why should anyone care?
Well, random is useful in physics but more intriguingly, if you accept that then you end a lot of research paths in modern mathematics!!!
Because if you push the argument to p_1 mod p_2, where the p's are differing primes, then it turns out some supposedly big questions in modern math, like the Twin Primes Conjecture are easily answered!
But there are mathematicians getting federal funds for research in those areas, if random is the call then those funds cease.
BUT if no one notices, those mathematicians can work endlessly in an area where they can never get an answer, at least not a correct one.
That is a highlight of how you can find controversy in the math field in a simple area where you can run your own statistical analysis to see.
Oh yeah, and remember a while back the math awards where one guy famously refused to accept his, while the others did accept?
Well, if primes are random then one of those people won a prize for supposedly innovative research into an area where no pattern actually lies.
Maybe you believe I'm wrong and there is some hidden pattern in primes modulo 3, or more technically
p mod 3
with p not equal to 3, as you go out to positive infinity.
I've given the beginning of the sequence:
2, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 2, 1
where the start was the number 2 which has itself as a residue modulo 3.
In answering my bringing this issue up before posters on sci.math claimed that this area has been checked and that statistical analysis proved non-randomness but there is no why here given.
Why would primes care? Especially when their products, the composites can't swing one way with the primes swinging another?
I suggest to you that math people lie in this area, where you can do the statistical analysis yourself to see that they lie, or I'm wrong. I've been wrong before, but I don't think I'm wrong here.
I know people lie. And I know some of you in the physics field who like to play with statistics can crank up some machinery and play with primes to see for yourself.
Remember, BIG research grants are wrapped up in this where some mathematicians have worked in this area for decades, so their entire livelihoods are about it NOT being true that p mod 3 is random.
Their entire LIVES are about that not being true, so they are totally invested.
# posted by James Harris @ 20:31 
