Problems & Puzzles:
Problems
Problem
47 . Holes and
CrowdsII
Now we'll focus on the opposite concept to
primeholes, that is to say, we will discuss these regions where k primes
are as close (crowds) as they can
be,
according to divisibility rules.
This is exactly the core of an issue technically named
prime ktuples^{(1)}.
The best site in the Web to find all kind of theory and records about,
is the well known Tony Forbes's site (patterns,
minimal,
maximal)
Let's suppose we define a k value. Let's suppose, just
to illustrate, that this k value is 8. What things we would be interested to know
about these prime 8tuplets (p1, p2, ...p8)?
Here are some basic things:
1. What is the minimal distance between the extreme
primes, p1 & p8,
namely, s(8)= p8p1?.
2. Are there different valid arrangements ( 'patterns') that these 8 primes may
adopt, preserving the same minimal distance s(8)?
3. What is the earliest set of 8 primes, for each pattern?
4. What is the largest known set of 8 primes, for each pattern?
For the case of k=8 the answers are:
1. s(8) = p8 p1 = 26
2. There are 3 patterns that the eight primes in a 8tuple may adopt. These
patterns are described by the following set of numbers, representing the
distance between each prime to the first one:
a) pi p1 = {0 2 6 8 12 18 20 26}
b) pi p1 = {0 2 6 12 14 20 24 26}
c) pi p1 = {0 6 8 14 18 20 24 26}
3. The first prime for the earliest example for each pattern is:
p1 = 11, for the first pattern
p1 = 17, for the second pattern
p1 = 88793, for the third
pattern
4. The first prime for the largest known example (June 2003), for each pattern is:
p1= 15234072433401 * 375# + 43813839521, for the first pattern
p1= 243551752728*320# + 1277, for the second pattern
p1 = 22 * 10^38 + 2241278889512323, for the third pattern
Well, Frank Ellerman has noticed that we know
at least one example for at least one pattern, for every 2<=k<=25,
except for k=24 ^{
(2)}
For k=24, we certainly know that p24p1 = 100
and that the 24tuplets may adopt one of four (4) patterns:
a) pi  p1 = {0 4 6 10 12
16 24 30 34 40 42 46 52 60
66 70 72 76 82 84 90 94 96
100}
b) pi  p1 = {0 4 6 12 16 24 30 34
40 42 46 52 54 60 66 70 72
76 82 84 90 94 96 100}
c) pi  p1 = {0 4 6 10 16 18 24 28
30 34 40 46 48 54 58 60 66
70 76 84 88 94 96 100}
d) pi  p1 = {0 4 6 10 16 18 24 28
30 34 40 48 54 58 60 66 70
76 84 88 90 94 96 100}
But the first specific case for k=24, is still waiting its
discoverer. So, the obvious question (also proposed by Ellerman) is:
Q1. Find the earliest p_{1}
such that p_{2}_{4}  p_{1}
=100?
I would like to add one more question:
Q2. Find s(k) and the
quantity of patterns for k=200
_________
^{(1)} Tony Forbes provides the
following rigorous definition of prime ktuples:
A prime ktuples is then defined as
a sequence of consecutive primes {p1, p2, ..., pk} such
that for every prime q, not all the residues modulo q are
represented by p1, p2, ..., pk, and pk  p1
= s(k). Observe that the definition excludes a finite number
(for each k) of dense clusters at the beginning of the prime number
sequence  for example, {97, 101, 103, 107, 109} satisfies the conditions of
the definition of a prime 5tuplet , but {3, 5, 7, 11, 13} doesn't because
all three residues modulo 3 are represented
^{(2) }
Table that summarizes
the valid patterns of ktuples for 2 to 50
(in red if
no one single example is known, Jun 2001).
s(k) = pk  p1
k 
s(k) 
# of patterns 
2 
2 
1 
3 
6 
2 
4 
8 
1 
5 
12 
2 
6 
16 
1 
7 
20 
2 
8 
26 
3 
9 
30 
4 
10 
32 
2 
11 
36 
2 
12 
42 
2 
13 
48 
6 
14 
50 
2 
15 
56 
4 
16 
60 
2 
17 
66 
4 
18 
70 
2 
19 
76 
4 
20 
80 
2 
21 
84 
2 
22 
90 
4 
23 
94 
2 
24 
100 
4 
25 
110 
18 
26 
114 
2 
27 
120 
8 
28 
126 
10 
29 
130 
2 
30 
136 
2 
31 
140 
2 
32 
146 
4 
33 
152 
14 
34 
156 
20 
35 
158 
2 
36 
162 
2 
37 
168 
2 
38 
176 
6 
39 
182 
26 
40 
186 
26 
41 
188 
8 
42 
196 
2 
43 
200 
6 
44 
210 
18 
45 
212 
4 
46 
216 
4 
47 
226 
4 
48 
236 
2 
49 
240 
2 
50 
246 
22 
Thomas
J. Engelsma has let me know (Set. 2003) that
he has a page
related to these issues and certainly a published
solution for Q2.
***
