Dear Tom, Thank you very much for your crackajack work.

About your “An admissible k-tuple of 447 primes can be created in an interval of 3159 integers,

while p(3159) = 446.”,

can I realize:

An admissible 447-tuple has been created in an interval of 3159 integers, while p(3159) = 446?

It is not:

An admissible prime 447-tuple has been created in an interval of 3159 integers,while p(3159) = 446.

Where we call a k-tuple is admissible, if it does not cover all congruence classes modulo any

prime p, we call the k-tuple a prime k-tuple when all of its components are primes by Daniel M. Gordon and Gene Rodemich.

So that: if the original k-tuple conjecture is true, then Hardy-Littlewood conjecture

p(x+y) - p(x) <= p (y)

is fails with a value of y = 3159.

In my paper I try prove that admissible prime k-tuples are infinite rather then admissible k-tuples will infinitely often be simultaneously primes. The original k-tuple conjecture may not true, example the prime of the form n^2-1.

Could you read my paper please and we will discuss some interesting problem.

Fengsui Liu.

Tom <

tom@...> 写道： On the k-tuples, have just finished an update of the permissible

patterns site www.opertech.com/primes/k-tuples.html

At the site the trophies (contradiction patterns/super dense

constellations) are listed up to packing 100 additional primes in an

interval. In fact 17 additional prime can be packed in an interval of

length of just 8509 integers.

The chart at www.opertech.com/primes/kpiwchart.html shows the crossover

(s) and the growth.

It is estimated that all intervals of more than 5980 integers can

demonstrate a super-dense condition.

Enjoy

Thomas J Engelsma

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