I had read Prof.Chris.Caldwell's article "An amazing prime heuristic". I recommend this

article for all friends, especially for some amateur like me.

From this article you will know where does the Hardy's prime k-tuple conjecture come

from, you will know that Prof.Chris made a persistent unrelenting application of one simple idea

to a wide variety of problems, you will know that the value of heuristic argument and its fail.

You will see the living idea of mathematician.

I see never any paper, which claimed "Even after great care and testing you should not stake to much on their predications".

What does fail?

Where is the flaw?

If a k-tuple < b1,...,bi,...,bn> is inadmissible, then it include a complete set of residues of a prime pn,

< b1,...,bi,...,bn> = <0,1,...,i,...,pn - 1> mod pn.

w(p) = p.

The adjustment factor

c =( (1 - W(p))/p)/(1 - 1/p)^k = 0.

The probability that they would be simultaneously prime would be 0. we shall not discuss the inadmissible k-tuple. e.g., x^3 - x + 9, <5,7,11,13,29>.

The key is the admissible k-tuple.

By the heuristic, Hardy had proved that each admissible k-tuple takes on simultaneous prime values infinitely often. But x^2-1, 3^n-1 is admissible. Obviously, there is not any prime of this form except 2. So that Hardy doubted the probability, " Probability is not a notion of pure mathematics". Of course , he might doubt the error term in prime theorem, doubt the Riemann hypothesis, and so on also.

Prof.Chris wrote: "And to be blunt, sometimes heuristic fail! not only that they sometimes fail for even the most cautious of user."

Yes, the inadmissible k-tuple is only thing that can stop a prime k-tuple from yielding simultaneous prime infinitely often. Another counterexample all are reducible f(x) and

{x: P(f(x))} = empty. Where the predicate P(f(x)) denote the f(x) be a prime.

Our logical conclusions are all holding in all possible world, but {x: P(f(x))} = empty is not the possible world of the predicate P(f(x)). The predicate P(f(x)) has not any interpretation or model. In the impossible world {x: P(f(x))} = empty our logical conclusions may fail. Thus the counterexample x^2-1, 3^n-1 is not a flaw of the heuristic at least, perhaps, the fail is that Hardy applied heuristic conclusions to the impossible world {x: P(f(x))} = empty. Of course , it is another problem whether "Probability is not a notion of pure mathematics ".

In the Phil's example " demons, bucket and balls", the proof of infinity is very simple, there is no anything may be doubted, but, we had proved the bucket is empty, or from the contradiction between empty and infinity by Godel completness theorem: a theory is consistent iff it has a model, we proved again that the bucket is empty, we only accepted the logical conclusions infinity in the impossible world-------the empty bucket fail.

To get rid of above infinite paradox, we modify the prime k-tuple conjecture to be : every admissible pattern of prime k-tuple occurs infinitely often. Where the pattern of prime k-tuple is that a k-tuple <a+b1,...,a+bi,...,a+bk> they are simultaneous prime. Namely, if the admissible prime k-tuple R(k,a) existed {a:R(k,a)} =/= empty , then |{a:R(k,a)}| = infinite. Perhaps, it is easy to prove the modified conjecture.

I intrepidly say out my some idea, I hope enthusiastic Prof.Chris say out more interesting something. Perhaps, I am wrong, please any man point out.

China Liu Fengsui.