I find it fascinating watching math people lie on newsgroups about basic things.
Some of you may think you understand probability and statistics, but consider a question like, if someone flips a coin 10 times and gets 10 heads, and THEN flips the coin 10 times and get 10 tails, is the coin flawed?
If you think the answer is yes, you're wrong.
The correct answer is, not enough information given.
If you think there is no way the coin can do that given the odds, then you are wrong.
But no trained scientist (properly trained) would get the question wrong.
So then, can a chi square test or any other statistical test prove that p mod 3 with p a prime greater than 3 does NOT gives a random sequence?
Nope.
It can give you a degree of confidence and be very likely to be right, but then again, any such test would take 10 heads in a row as proof of non-randomness.
So what is the resolution? Is there none?
YES!!!
Mathematicians may get used to non-practical results to the extent that they forget that in the real world, rules mean PREDICTION!!!
Our understanding of the laws of physics allows us to do things, well, like type up missives on wonderful devices that many people don't fully understand, or fly around the world, or chat up all those minutes on cellphones BECAUSE WE KNOW THE RULES WELL ENOUGH.
So, the answer is not a fight over statistical tests, but prediction.
If mathematicians wish to maintain that p mod 3 with a prime greater than 3 is not random, then give me the freaking rules.
Give me the laws of the behavior.
Consider in contrast that with the prime distribution itself—the count of primes—it is well-known that the probability a number x is prime is roughly 1/(ln x).
So you have an equation. You can say, for instance, that the probability of primeness up to 100 is about 1/(ln 100) which is 0.217 to three digits, saying roughly 22 primes and there are 25, so you have prediction and a test showing the validity of that prediction.
So then, if p mod 3 is not random, then there would be rules.
Consider an unknown hypothesized function PMOD3RULE(x), which like 1/(ln x), would tell you that the probability at x=1098340 that p mod 3 in that area is 1 is 70%.
Then you could test the rule, and see if you got more 1's than 2's.
People, our technological world rests on the truth within the debate that non-random things are predictable, to some extent, or we would not have clothes, would not have computers, and would not have nice big houses to sit in and type up dumb messages to each other.
So the debate is at the foundations of our technology, at the foundations of our science, at the foundations of our intellectual pursuits about how we know what we know in this world.
Non-random means prediction. So posters arguing with me can't just trot out a test.
Unfortunately many modern mathematicians behave like they are not part of the real world, so the possibility of prediction as my counter to claims against what I know is random behavior might have escaped them.
They are used to playing word games and not being challenged.
They are used to bulling their way through with enough math-ese to win debates in a political arena.
But I SAY SHOW ME.
I don't care if these nincompoops think they win because they convince a lot of you who don't know any better. Sure they may think that's all that matters, but I know it's not.
Most of you may never get it, like you may never quite comprehend how in the hell that big box with the green light works, or care as long as you can download your music files.
Most of you will probably never understand much of anything in this world, so yes, mathematicians can convince you until the day you die, and you will die in ignorance.
But damn them, I am going to ask them once again to SHOW ME!!!
So it is a reality test people. Or better yet, a reality check and mate.
A reply to someone who said that lots of posters had already told James that to predict the opposite of the last residue would provide prediction at better than 50%.
Prediction in science is more than just about some rough rule but also about REASONS behind the rules.
As an example, in the past, people thought all kinds of common sense and reasonable things like that feathers fall slower than a brick because one is heavier than the other.
But the reality has to do with wind resistance, so remove the wind, and you can use a simple equation that tells you how fast EACH will fall to earth.
So the problem here isn't to look at some particular lists of something that may be random and think you see a pattern and declare a rule, but to give the laws of the supposed rule as then they can be tested.
The requirement then is a formula for p mod 3, where you plug something in, and get a probability that you will have 1 at a higher than 50% chance or 2.
What you suggest is not at all the way science is done, but it is the way many people commonly think, like they see something happening a lot and conclude that it will always tend to happen, which is a common problem in the way people think.
Science is not just about finding patterns, but finding rules that govern those patterns, and then using those rules—in this case formulas—to make those predictions.
The example I gave for an area where rules DO work with primes is with the prime distribution where 1/(ln x) is approximately the probability of primesness for a natural number x, so you can use it to roughly count primes.
So I have explained in detail and given you a concrete example.
I emphasize that and may sound a bit pedantic, but people need to understand that the problem here is a dedicated resistance to the truth from intelligent people willing to work to convince you of things that are false, so I need to step out basic things—over and over and over and over again—and still it usually does not work!!!
Human beings are troublingly weak against other human beings who are intelligent and actively working to lie, who are part of a group that is trusted.
The point here is to drop the automatic trust of mathematicians.
Modern mathematicians quite simply, cannot all be trusted.
You have to at least consider that these people are willfully lying to you.
And I have to state basics over and over and over again and watch it not matter, unfortunately because too many of you still are naive about our modern world.
So you are taken in, by people who wouldn't surprise me if they would
skin you alive if it would suit their own needs.