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## ACF and ACT conjectures

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• Definitions: An ACF conjecture is a conjecture which is *Almost Certainly False* due to rigorous heuristical argumentation, but which is not proven to be
Message 1 of 2 , Dec 1, 2001
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?
• ... Hans, here s my ACT conjecture, of you like : For fixed, relatively prime naturals a,b, every large enough number of the form ax+b is the average of two
Message 2 of 2 , Dec 2, 2001
> 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?

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

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
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