October 1st 2005
# Admissible Prime Sets

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## 1. Introduction

This topic is based on 2 conjectures
made by Hardy-Littlewood in the early 20th century. The first one is the
k-Tuple Conjecture. The second one claims that the
prime counting function is convex.
These conjectures are incompatible with each other, which was proved in 1972
(see counter-example in this paper on page 3).
It is possible that both conjectures are wrong, but both cannot be true at the same time.

An Admissible Set is a set which fullfills the requirements for the
k-Tuple Conjecture, i.e. a set
of natural numbers 0 = *a*_{1} < *a*_{2} < ... < *a*_{k} so
that (*m* + *a*_{1})(*m* + *a*_{2})...(*m* + *a*_{k}) does not
have a permanent divisor independant from *m*.
The construction of such a set is easy:
Take the set of natural numbers [0, *n*[. For each prime number *p* < *n* eliminate a residue class
which does not hold 0 or *n*-1. If 0 and *n*-1 are not in the same residue class modulo *p*, there
are *p*-2 possibilities, otherwise *p*-1 possibilities. The remaining set obviously satisfies
the condition of the k-Tuple Conjecture.

*k* is the size of such a set, *n* the length. While finding any set for a given length *n* is trivial, finding
a set with maximal size *k* requires exhaustive search. Let r(*n*) = max *k* over all possible sets.
A necessary requirement for a counter-example for
the second Hardy-Littlewood conjecture is
that r(*n*) > p(*n*), where p(*n*) is
the prime counting function.

I had two goals:
Find the smallest number *N* with r(*n*) > p(*n*)
Find r(*n*) for all *n* < *N*

In September 2004, I found that 447 = r(3159) > p(3159) = 446. I have been calculating
r(*n*) for 2048 < *n* < 3159 since then (*n* < 2049 has already been
calculated exhaustively).

Without proof, here are my results.

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