Problems & Puzzles: Problems

Problem 47 . Holes and Crowds-II

Now we'll focus on the opposite concept to prime-holes, 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 k-tuples(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 8-tuplets (p1, p2, ...p8)?

Here are some basic things:

1. What is the minimal distance between the extreme primes, p1 & p8, namely, s(8)= p8-p1?.
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 8-tuple 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 p24-p1 = 100 and that the 24-tuplets 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 p1 such that p24 - p1 =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 k-tuples:
A prime k-tuples 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 5-tuplet , but {3, 5, 7, 11, 13} doesn't because all three residues modulo 3 are represented

(2) Table that summarizes the valid patterns of k-tuples 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.

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