Primordinal Soup

Time and again here, we will be returning to THAT man, Kurt Godel, the guy who proved that not everything is provable within mathematics.

A simple example follows, it is called "The Goldbach Conjecture", and involves prime numbers.

Now, a prime number is a number which has no factors, that is, can only be divided by itself and 1.

Obviously, no even number (apart from 2) can be a prime number, since all even numbers are divisible by 2.

Which of these are prime numbers? 3, 11, 21, 31, 41, 51.

The answers are 3, 11, 31, and 41, because 21 is 7x3, 51 is 17x3. These are called "composites".

The Goldbach Conjecture says this:

"Every even number is the sum of two (odd) prime numbers" (the word "odd" is there because 2 is excluded).

Let's call this the "GC" property of a number

Examples:

12 = 5 + 7
22 = 19 + 3 = 17 + 5
32 = 19 + 13 = 29 + 3
42 = 19 + 23 = 29 = 13 = 31 + 11 = 37 + 5

are all "GC".

The question arises: Is the Goldbach Conjecture true?

That is, is EVERY even number like this? Actually, it has never been proved or disproved. We just know that there are lots of them. No highest even number has ever been found, yet there is no proof that that there are infinitely many of them.

It has been speculated that this is one of those truths which can never be proved, the existence of which was showed by Godel in 1931.

There IS stuff out there that is true, but nevertheless cannot be proved.

Let's introduce a similar concept - the "Reverse Goldbach Conjecture". This says:

"Every even number is the difference of two (odd) prime numbers".

Let's call this the "RGC" property.

This has ramifications for computers.

Let's pick a number, say a trillion.

Is It GC or RGC or both, or neither?

You can write a program to test all primes up to a trillion to test if it is GC, and the program will finish at 1 trillion with an answer, YES, or NO. This called a "bounded" test.

However, testing for RGC is a bit more complicated, simply because a subtraction is involved. Testing for RGC has to start at 1 trillion and go UP, trying to find two odd primes which, when the smaller is subtracted from the larger, a trillion emerges.

It is not obvious how far up the number ladder we will have to go to get such a pair, if we ever find one.

For being GC, the testing process terminates in a finite amount of time, always.

For being RGC, the testing process may take us to numbers so high, it may take decades to find the pair, but there also exists the possibility that such a pair does not exist. Either way, we may never know. This kind of test is called "unbounded".

I mentioned computers because computers are particularly good at making loops to test for ever-increasing numbers.

The ramification is that the programmer must make sure his "loops are "bounded", and never "unbounded".

I have made that mistake myself with simple FORTRAN programs.