Having used the example of Goldbach’s conjecture several times in my paper on metaphysics and conceivability, I’ve been giving the conjecture (we’ll call it GC from here on out) some extended thought. In plain english, GC states “For every even integer greater than 2, it can be expressed as the sum of two primes.” I found it helpful to express it in a symbolic logic, so here is where I started;
2n • n > 1 –> 2n = p + q • p,q ≠ mo
The latter part, where I state “p,q ≠ mo” is just to mean that p and q are primes, as primes cannot be expressed as the product of two integers. Hence the “not equal to the product of two integers,” i.e. the variables m and o.
We’re interested primarily in the latter half of the statement, basically just that
2n = p + q
where we remember that p and q are primes. Notice that simplifying the equation, we can state
n = (p + q)/2
which is, in other words, an average mean. Thus the difference between n and p and n and q must be the same. Thus I propose what I call Laliberte’s postulate:
For every integer there is are a pair of primes for which the distance between the integer and the primes is equal.
This can be expressed logically like this;
n –> n – d = p • n + d = q
However, this is where the problem for GC comes to the fore (Laliberte’s postulate is just a derivation of GC). Since primes are defined not in according with some positive formula, but by a negative formula (p ≠ mo), there cannot be a way for knowing what d (the equal distance) shall be in the case for every integer. Primes are, in other words, random.
This means there can be no method of determining whether, for some integer, where the primes will occur. There is no pattern in the distribution of primes, and therefore no formula.
Goldbach’s conjecture must be unprovable. Likewise for other mathematical theorems that are about primes.
Goldbach’s conjecture must be unprovable. Likewise for other mathematical theorems that are about primes.
Not quite. I suspect you may not have stated this narrowly enough. Examples of proven Mathematical theorems about primes:
There are infinitely many prime numbers. I do recall this and if I had some practice I could probably at least re-understand Euclid’s proof.
The Prime Number Theorem which deals with distribution of primes. I won’t even touch the proofs here, though. Either I slept through it, never understood it, or forgot it, or some combination of the three.
And Wilson’s Theorem which is essentially a primality test using some modulo stuff, is cool but kind of useless. But still a proven Theorem.
Was there something else you were driving at there?
Where the theorem is a statement of how primes relate to other numbers, that’s where it breaks down.