Main page ( http://www.opertech.com/primes/k-tuples.html
The latest updates have been posted, with improvements to more than
2200 widths. Currently there are 29 million plus patterns.
Of these 2200 improvements about 20 are new first occurence densities.
The graph is still getting stepper.
As this graph continues to climb, some questions need to be asked and
In the following,
k(w) is the maximum number primes permissible in a width of 'w'
pi(w) is the standard number of primes from 1 to w.
As you look at the graph, the curve 'appears' to be concave up, as
evidenced by the 'hockey stick' start. The best fit line on the
graph is for tracking purposes and not to imply linearity.
But this line also shows this concave up tendency. Also, the 'flat'
spots at 19000 and 32000 are mainly due to sequence of programming.
The upper bound of the searching was raised from 5200s, to 12000s, to
19000s, to 32000s, and finally to 41741 - meaning the smaller widths
have seen more searching.
Hensley and Richards showed that the two HL conjectures were
incompatible, and left the impression that the k-tuples conjecture
was the true conjecture.
Montgomery and Vaughn showed that k(w) <= 2 * pi(w) using the 'large
If this graph is truly concave up or linear, then the Montgomery and
Vaughn proof would be contradicted, so one of two things must be true,
A) the graph has an inflection point, or
B) the k-tuples conjecture is false.
Concerning A) this inflection point appears highly unlikely as these
quantities can be described as combinatorial objects.
What is the concensus of math world on the HL conjectures?
1: k-tuples true and 2nd HL false
2: k-tuples false and 2nd HL true
3: k-tuples false and 2nd HL false
Looking for all the input I can get.
Thomas J Engelsma