Brainstorms: Question About Prime Numbers

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Topic: Question About Prime Numbers

chimp
Member
Member # 333

posted 28. September 2003 16:05

Twin primes are prime numbers such as 5 and 7, 11 and 13, 17 and 19,
etc.

There are strings of prime numbers that are n-units apart:

3, 5, 7, [3 prime numbers, 2 units apart]

5, 11, 17, 23, 29, [5, 6 units]

7, 157, 307, 457, 607, 757, 907, [7, 150 units]

11... ? ...? ...? ...

The question becomes: For all odd prime numbers P, are there P number of
primes that are the same numerical[equal] distance apart?

[ 28. September 2003, 17:25: Message edited by: Russell E. Rierson ]

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chimp
Member
Member # 333

posted 30. September 2003 03:19

Zakir Zeidov sent me this interesting message:

quote:

Interesting problem, Russell! For p=11, minimal d = 4911773580 (OEIS A088430), and AP contains maximal number, 11, primes. For p=13, d should be a factor of 2310. Who first find it (and then try 17,19,...)? Zak BTW I guess that found d is indeed minimal not unique- there is no reason of absense of other larger d's.

So, for 11, minimal d = 4911773580 ...very interesting!

11
4911773591
9823547171
14735320751
19647094331
24558867911
29470641491
34382415071
39294188651
44205962231
49117735811

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chimp
Member
Member # 333

posted 01. October 2003 02:00

Interesting...

http://listserv.nodak.edu/scripts/wa.exe?A2=ind0111&L=nmbrthry&P=R233

quote:

Date: Wed, 7 Nov 2001 09:17:30 -0500 Reply-To: Phil Carmody Sender: Number Theory List From: Phil Carmody Subject: Prime-producing Linear Polynomials
The linear polynomial f(X) = dX+q can have at most q successive terms f(0)...f(q-1) prime, (and q must be prime for f(0) to be prime, evidently). It remains an open question, and one with few data-points, whether such maximal q-based q-length Arithmetic Progressions exist for every q.
q=3, d=2 yield the primes 3,5,7; q=5, d=6 yield the primes 5,11,17,23,29; q=7, d=150 yield the primes 7,157,307,457,607,757,907.
In 1986, Löh discovered for q=11 d=1536160080, and for q=13 d=9918821194590.
[The above was a synthesis of what Paulo Ribenboim has in The New Book of Prime Number Records]
At the start of 2001 I started tackling the q=17 problem, and I wrote some brute force code to attack it. The code was peppered with bugs, and despite finding a record Cunningham Chain with the broken code http://listserv.nodak.edu/scripts/wa.exe?A2=ind0103&L=nmbrthry&P=R423 I gave up both on the project and the code.
However, I recently encountered other tasks which seemed like a good target for the code, and easier than the Arithmetic Progression problem, and resolved to de-mothball the code. Within a day of thinking this, Tom Hadley posted the very result I thought I might search for - a minimal 15-tuple ( http://www.ltkz.demon.co.uk/kt15.txt ). Rather than kill the idea, this encouraged me!
So I (think I) fixed the bugs, and started the search again. After roughly 5 days on a single 533MHz Alpha (21164), I found the following result, finally toppling the 15-year-old record.
The record is now q=17 d=341976204789992332560
And the primes are 17, 341976204789992332577, 683952409579984665137, 1025928614369976997697, 1367904819159969330257, 1709881023949961662817, 2051857228739953995377, 2393833433529946327937, 2735809638319938660497, 3077785843109930993057, 3419762047899923325617, 3761738252689915658177, 4103714457479907990737, 4445690662269900323297, 4787666867059892655857, 5129643071849884988417, 5471619276639877320977
q=19 anyone?
(The scaling factors indicate that it might be possible with a collaborative effort, and my code parallelises)

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Claire
Member
Member # 725

posted 03. October 2003 22:03

Is logic a form of persuasion?

I think that logic is a vector reducing element. It is also very persuasive and leading on a descriptive level and closely related to a type of hypnotic state that half the brain naturally creates. I think that even cognitive scientific understanding, in one sense, deflates a component of reality, it is an attractive force/system. I think irreducible complexity is one example of a deflation principle, also I relate this t o a type of reduced and concentrated attention by the principle used for quantifying the quantum world by reduction to a packet of quanta or packet of energy, or a real number and an imaginary number to complex ones, unlike real numbers used in classical thinking as an example. I think the objective world is closer to us than we think, but something is not enabling us to create a completely new portal to it, it could be a type of clever lecherous mechanism inside the brain or noise reduction part. Maybe we shouldn't know too much! Logic is a form of hypnotic influence on the right irrational side of the mind. This thought came about by thinking about the left and right brain systems. The left-brain rationalises mostly, we might even rationalise what needn't be rational as common sense would tell, you know what they say about common sense*.

I haven’t yet watched my recorded TV programme on Chaitin and Penrose but I think a new element in logic could be conceived by an increase in too much rationality/logic, not a balance that introduces a stopping point or stops at a threshold. The mind tricks us accordingly by the lead of the rational. It cannot be turned off very easily when we are awake (allowing for only illogical right brained thoughts when dreaming) unless we are very tired. The hypnotism comes to mind during the awake phases mainly. The mind makes sense, order and logic of non-sense, disorder. Does that mean all objective reality is logical?

The type of logic used is based on common thinking and works well but that might not signify that the entire universe that is thought about also works to that extent. We make sense of it until we hit a wall. When riding up to that wall, our own rational mind takes us for the ride!

*"It should make more sense than common sense".

However on the other hand it could be noted that in order to make less sense than common sense, the lesser the sense it makes (compared to the more common sense that it doesn't) the more it does make more sense and the more common sense makes much less sense.

Claire

[ 08. November 2003, 23:43: Message edited by: Claire ]

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Stuart Harris
Member
Member # 152

posted 06. October 2003 19:15

Russell,

Does this follow from Dirichlet's result on primes in arithmetic progression?

See: Primes in Arithmetic Progression

Stu

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Claire
Member
Member # 725

posted 16. October 2003 23:15

I was right.

"Post-formal operational" thinking as a cognitive technique used for understanding in science, that gets us away (at intervals) from the persuasion of preliminary logic thinking. This requires a different kind of mind. I used my own post-formal operational thinking for this problem for instance to find this out. This closely relates to IC and SC. Its implications for information theory and mathematics are enormous in this sense.

Claire

[ 17. October 2003, 22:38: Message edited by: Claire ]

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chimp
Member
Member # 333

posted 17. October 2003 03:18

Thank you Stuart and Claire.

Yes, for any specifically chosen prime number P, there exist P number primes with a common distance "d".

For the number "2", first set of d = 1

2,3

For the number 3, the first set of d = 2

3, 5, 7

For the number 5, the first set of d = 6

etc...

Thanks for the arithmetic progression link [Smile]

So, could it be possible to construct a machine[function] that takes nonlinear data[e.g prime distributions, or random bits] and transforms it into a linear relation through associated invariance[duality] principles?

[ 17. October 2003, 03:24: Message edited by: Russell E. Rierson ]

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