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Definition - Denizen

A sequence $\lbrace a_k \rbrace$ is a denizen if all of it's members are prime numbers, i.e $a_0, a_1, ... a_n \in \mathbb{P} $; and it satisfies the following condition; if "$a_{x_1} =y_1 $", "$a_{x_2} =y_2$", "$x_1 \pm m_1y_1 \neq x_2 \pm m_2 y_2 $ when $y_2<y_1 $" and "$m_3$ isn't divisible by $y_1$"; then "$a_{x_1 \pm m_1y_1}=y_1$" and $"a_{x_1 \pm m_3} \neq y_1$" (where $m \in\mathbb{N}$ where $y \in\mathbb{P}$ and where $x \in\mathbb{Z}$).

Let a denizen consisting of prime numbers up to and including $p_\alpha$ be denoted $\lbrace a_k \rbrace ^{p_\alpha} $. For example; a denizens that can be denoted as $\lbrace a_k \rbrace^7 $ is {2,7,2,3,2,5,2}.

Question

What is the maximum length $\lbrace a_k \rbrace ^{p_\alpha} $ can take?

Attempt

In order to find the maximum length a denizen $\lbrace a_k \rbrace ^{p_\alpha} $ can take I considered denizens of two different types.

I first considered a denizen $\lbrace a_k \rbrace ^{p_\alpha} $ of a type which corresponds to the sequence of natural numbers $ \lbrace 2,3,4,...,p_{\alpha +1} -1 \rbrace $. The corresponding denizen is $\lbrace 2,3,2,...,2 \rbrace $. This type of denizen had been created such that $a_i=d_i|i$, where $d_i$ is some divisor of $i$, implies $a_i \in $$\lbrace a_k \rbrace$. The consequence of this property is that this type of denizen can be considered a sequence of the lowest prime divisors of the natural numbers from $2$ to $p_{\alpha +1} -1$ respectively. For example, of this type; $\lbrace a_k \rbrace ^{11} = \lbrace 2,3,2,5,2,7,2,3,2,11,2 \rbrace $ and corresponds to $\lbrace 2,3,4,5,6,7,8,9,10,11,12 \rbrace$.

A denizen $\lbrace a_k \rbrace ^{p_\alpha} $ of this type which has a length of $ p_{\alpha +1} -2 $.

However I found a larger type of denizen corresponding to the sequence of integers $\lbrace -(p_{\alpha -1} -1), ..., -4,-3,-2,-1,0,1,2,3,4, ..., p_{\alpha -1} -1 \rbrace $. The corresponding denizen is $\lbrace 2, ..., 2,3,2,p_\alpha,2,p_{\alpha -1},2,3,2, ..., 2 \rbrace $. This type of denizen can also be considered a sequence of lowest prime factors of the integer sequence above, however it also requires the replacement of $-1$ and $1$ with the two primes $p_{\alpha}$ and $p_{\alpha -1}$ respectively and involves replacing $0$ with $2$. For example, of this type $\lbrace a_k \rbrace ^{11} = \lbrace 2,5,2,3,2,7,2,11,2,3,2,5,2 \rbrace $ and corresponds to $\lbrace -6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6 \rbrace$.

This type of denizen has a length of $ 2p_{\alpha -1} -1 $ and also has an apparent symmetry as defined below. this type of denizen has a length greater or equal to the length of the previous type because of the identity $2p_{x-2} \geq p_{x}-1$ proved here.

So my next question is; is the second type of denizen the largest lengthed $\lbrace a_k \rbrace ^{p_\alpha} $ denizen possible? How could you prove it was?

I have tried to prove this, using the concept of symmetry. I defined the symmetric depth as the largest prime number $p_N$ such that $a_{x+ p_N}=a_{x- p_N} = p_N$ and $a_{x+ p_{i}}=a_{x- p_{i}} =p_i$ for all prime numbers $p_i$ less than $p_N$, centred on some $a_x\in \lbrace a_k \rbrace$. I let $\lbrace a_k \rbrace ^{p_\alpha} _ {p_N} $ be a denizen consisting prime numbers upto and including $p_\alpha$ with symmetric depth $p_N$, and hoped to prove that the length of a denizen $\lbrace a_k \rbrace ^{p_\alpha} _ {p_N} $ is always greater than or equal to the length of a denizen $\lbrace a_k \rbrace ^{p_\alpha} _ {p_{N-1}} $. However I found a counter example to this, due to the fact that the gaps between large prime numbers tend to be greater than the gaps between smaller prime numbers.

However I have yet to produce a denizen $\lbrace a_k \rbrace ^{p_\alpha} $ greater in length than $\lbrace a_k \rbrace ^{p_\alpha} _ {p_{\alpha -2}} $.

So any ideas to further this?

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  • $\begingroup$ Question also posted, without notice to either site, to m.se: math.stackexchange.com/questions/1404429/… $\endgroup$ – Gerry Myerson Aug 24 '15 at 6:45
  • $\begingroup$ Distantly related: some years ago, my co-authors and I defined the concept of an IRDCS of length $n$. The smallest example, of length 11, is $6,9,3,4,5,3,6,4,3,5,9$. Note that for each $m$ appearing in the list, $m$ appears at every $m$th position, and only at every $m$th position; also, every $m$ that appears appears at least twice. $\endgroup$ – Gerry Myerson Aug 24 '15 at 23:19
  • $\begingroup$ I'm curious about what the purpose for creating the concept of an IRDCS was and why the numbers $6,9,3,4,5$ were chosen? Essentially a placement condition is one of the requirements of a denizen; a prime number will appear in the $m$th position if it is the lowest prime factor of $m$. I note that an IRDCS allows two consecutive terms to be odd so it doesn't look like it was created to represent a section of the natural number line. Denizens were created to represent the sequence of composite numbers between two consecutive primes. $\endgroup$ – Brad Graham Aug 24 '15 at 23:46
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    $\begingroup$ There's a theorem that says you can't partition the integers into finitely many arithmetic progressions with different common differences. An IRDCS is a partition of a finite segment of the integers into arithmetic progressions with different common differences, so it shows you can't weaken the theorem hypotheses too much. The numbers were chosen because they work: there is no IRDCS shorter than length 11, and the only one of length 11 is the one I gave, and its reverse. Try it! $\endgroup$ – Gerry Myerson Aug 25 '15 at 2:41
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    $\begingroup$ 23252a232b2_23272_232_25232a2723252b232 . Length 39. a and b represent the primes 11 and 13. Fill in the three blanks with 17, 19, and 23. Use Chinese Remainder Theorem to calculate the numbers corresponding to the digits, or lookup Westzynthius. Gerhard "And There Are Larger Examples" Paseman, 2015.08.30 $\endgroup$ – Gerhard Paseman Aug 31 '15 at 2:49

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