- Definitions:

"

An ACF conjecture is a conjecture which is *Almost Certainly

False* due to rigorous heuristical argumentation, but which

is not proven to be false.

An ACT conjecture is a conjecture which is *Almost Certainly

True* due to rigorous heuristical argumentation, but which

is not proven to be true.

"

Both types of conjectures, ACF and ACT, exist in their own right.

Both types of conjectures have no (easy) proof or disproof (while

a 'not yet found' counterexample would always suffice for both).

We saw examples of both these types of conjectures on this list

in the last couple of days. They were closely related to prime

numbers. And we know of some old and famous conjectures (about

primes) which also fit in one of the ACF or ACT classes.

I believe that it might be useful to introduce a classification

of ACF and ACT conjectures into the field of prime numbers. If

you have a conjecture about primes, then the *very* first thing

you should try is to classify it as ACF or ACT (by use of good

heuristics or *many* numbers).

We should try to collect as many of very hard to prove or disprove

conjectures about primes as ever possible of any of the types ACF

or ACT to convince Chris Caldwell that such an ACF/ACT conjectures

page would be a page of it's own right among his prime pages.

Hans

PS: Which ACF/ACT conjectures about (or including) primes are

known to you? > We should try to collect as many of very hard to prove or disprove

Hans, here's my ACT conjecture, of you like :

> conjectures about primes as ever possible of any of the types ACF

> or ACT to convince Chris Caldwell that such an ACF/ACT conjectures

> page would be a page of it's own right among his prime pages.

>

> Hans

>

> PS:

> Which ACF/ACT conjectures about (or including) primes are

> known to you?

For fixed, relatively prime naturals a,b, every large enough number

of the form ax+b is the average of two primes of the form ax+b.

This, apparently, is a GC-like conjecture for the infintely many

primes on the series ax+b. Goldbach's conjecture is actually a

special case of the above conjecture as the set of naturals, in which

GC is defined, is a special case of ax+b, with a=1.

Kaveh