02-24-2002, 08:22 PM
Okay, let's take the even number 12, and cut off the ends (0 and 12), because our "AND hole" (where two primes sit opposite) must be in the middle to produce two separate primes (measured to zero in two opposite directions) to sum:
h f f f h f
11 10 9 8 7 6
1 2 3 4 5 6
h h h f h f
The reason why there cannot be a frequency (a number divisible by a prime) in the top line opposite every prime in the bottom line is because, if there were, the bottom line would be a superior "key" into the top line. You cannot produce such a magic key out of nothing, just by folding it.
In other words, suppose we take this line in line, not stacked, put on a blindfold, and pick a "slot" at random. Our chances of hitting a hole (a prime) are one in six. So if we hit a hole, we will "know" we have landed in 1 of six possible spots with equal probability.
With this much information, we will not be able to say with certainty what the contents of the slot immediately to our right or left will be. If we have landed on 2, to our right will be a hole. If we have landed on 5, to our right will be a frequency.
To say that we cannot say where we are, and we cannot say what number will be 2 to our right, but if we fold the line, we can now know what will be above us from a sample of one, is not possible. We cannot know that much from one spot, there must be a probability of hitting a frequency.
Or, on the flip side, if we could "know" we would hit a frequency vertically, then we would not gain any additional information by sampling it, which is also not possible. For instance, if we take a random two-slot sample from bottom line, we can know we will always have a better idea of our location than if we took one.
Now, it is quite a different case if we use an odd-numbered line, because the even numbers are not as complex. We know, other than 2, evens are all frequencies. So of course we can use a hole as a key into the contents of an even number, but that does not change by folding the line.
Even in a flat line, we can always say if we hit a hole, there will be a frequency 3 to our right. So, we could say the same thing that, if we fold an odd line, we cannot use the bottom line as a key into one above and one to the left, and if we fold an even line, we cannot use our hole as a key into one above.
But that's Goldbach's other conjecture. That any odd number is a sum of two primes and 1, hence the 1-above-and-1-to-the-left offset. Keep in mind, when you fold it, you cannot knwo form how far away you are selecting, only whether that distance is odd or even. So it is a random number, known to be odd or even, to the right.
Anyway, you get the idea, in a flat line we know we cannot predict what will be a random even number to our right from a random sample of one. If we hit a hole - and we know exactly what the line sampled from looks like - we will know there is an exact probaility we have hit 5, and that there is an exact equal chance our random number will be 2 or 4, and an exact probability the number mapped to will be 7 or 9, and hence an exact uncertainty if it will be a hole or a frequency.
So by folding the line, we now cannot know what will be above. In other words, our knowledge of what number is a random even number to the right of any randomly selected hole, even if we know exactly what the line looks like, cannot change. So, inevitably, if we fold the line and examine what is above what, we must discover that some holes have holes above them!
So we can predict, with certainty, that our knowledge must not change. We can prove that if we fold it, and look at it again - just as we looked at it flat before we took our random sample - we will have learned nothing new.
Proved.
eLROY
h f f f h f
11 10 9 8 7 6
1 2 3 4 5 6
h h h f h f
The reason why there cannot be a frequency (a number divisible by a prime) in the top line opposite every prime in the bottom line is because, if there were, the bottom line would be a superior "key" into the top line. You cannot produce such a magic key out of nothing, just by folding it.
In other words, suppose we take this line in line, not stacked, put on a blindfold, and pick a "slot" at random. Our chances of hitting a hole (a prime) are one in six. So if we hit a hole, we will "know" we have landed in 1 of six possible spots with equal probability.
With this much information, we will not be able to say with certainty what the contents of the slot immediately to our right or left will be. If we have landed on 2, to our right will be a hole. If we have landed on 5, to our right will be a frequency.
To say that we cannot say where we are, and we cannot say what number will be 2 to our right, but if we fold the line, we can now know what will be above us from a sample of one, is not possible. We cannot know that much from one spot, there must be a probability of hitting a frequency.
Or, on the flip side, if we could "know" we would hit a frequency vertically, then we would not gain any additional information by sampling it, which is also not possible. For instance, if we take a random two-slot sample from bottom line, we can know we will always have a better idea of our location than if we took one.
Now, it is quite a different case if we use an odd-numbered line, because the even numbers are not as complex. We know, other than 2, evens are all frequencies. So of course we can use a hole as a key into the contents of an even number, but that does not change by folding the line.
Even in a flat line, we can always say if we hit a hole, there will be a frequency 3 to our right. So, we could say the same thing that, if we fold an odd line, we cannot use the bottom line as a key into one above and one to the left, and if we fold an even line, we cannot use our hole as a key into one above.
But that's Goldbach's other conjecture. That any odd number is a sum of two primes and 1, hence the 1-above-and-1-to-the-left offset. Keep in mind, when you fold it, you cannot knwo form how far away you are selecting, only whether that distance is odd or even. So it is a random number, known to be odd or even, to the right.
Anyway, you get the idea, in a flat line we know we cannot predict what will be a random even number to our right from a random sample of one. If we hit a hole - and we know exactly what the line sampled from looks like - we will know there is an exact probaility we have hit 5, and that there is an exact equal chance our random number will be 2 or 4, and an exact probability the number mapped to will be 7 or 9, and hence an exact uncertainty if it will be a hole or a frequency.
So by folding the line, we now cannot know what will be above. In other words, our knowledge of what number is a random even number to the right of any randomly selected hole, even if we know exactly what the line looks like, cannot change. So, inevitably, if we fold the line and examine what is above what, we must discover that some holes have holes above them!
So we can predict, with certainty, that our knowledge must not change. We can prove that if we fold it, and look at it again - just as we looked at it flat before we took our random sample - we will have learned nothing new.
Proved.
eLROY