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Page history last edited by Bill McEachen 2 years, 4 months ago

Math Stuff

 

 

 
 
 
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caveat: please credit the author for anything used from this page (Bill R McEachen, CCCSD/POD)

 

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this wiki is now organized with this main page treating my main primorial conjecture and other topics via links:

 

= = = =

Feb 2015  general cleanup.

Mar 2014  general cleanup.  Doing a bit of work w/ bitmaps

May 29 2012  I just saw the work of the Generalized Fermat factorization ( Devaraj conjectures ), interesting

 

Apr 6 2012

I should post a few links here to a few screencasts on the Dynac ES SCADA system;  Here is the first on intro&basic navigation apps (should play in QuickTime player, VLC Player or similar):  (typ.<100Mb,mp4 format).  I have no ability to post-edit, and rudimentary tools, so go easy on me

 

               see this page:  Dynac-related (videos etc)

 

Jul 15 2011 I added comment and formula to OEIS A084639  (relating to bit strings).  Here is some backgnd xplanation:  OEIS A084639

 

Apr 6 2009 I must thank Sephan Fenn of Germany who has been must helpful with working up a java pgm for working with large bases.  Mr Fenn is connected with the program Digi][Trans for conversions between unicode, binary,hex, etc.  He has been most helpful with my questions.

 

= = = =

 

*Primorials

In May 2007 I built upon my earlier 2006 HCN conjecture as follows, which is much more sweeping:

MacEachen's Conjecture (note old country spelling to honor my ancestors!)

A required (but not sufficient) condition for primeness is associated with Sloane's OEIS entry A129912 submitted June 4th 2007 by me. It turns out this appears to be connected in a convoluted way to another sequence, A071562 (see later).  (1/20/2017 Note that this does not just apply to every odd#, the first failure being 51).

 

Every prime number >2 must have an absolute distance to a sequence entry (primorials, primorial products) that is itself prime, aside from the special cases prime=2 and those primes immediately adjacent to a sequence entry (primorials, primorial products). The property is required but not sufficient ...it considers distances no larger than the candidate

Rephrasing, this merely means every odd prime number must either be adjacent to, or a prime distance away from a primorial or primorial product.

(the distance will be a prime smaller than the candidate)

I believe I can show that if Goldbach's Conjecture is true, the above conjecture is true ...or versa visa....it is obviously related to prime gaps

I think Sokol's fails early applying the condition that the distance must be < the candidate prime (required to be useful for as a primality condition).  An example fail is 331, where (331-2),(331-6),(331-30) and (331-210) are all composite.

Note that my conjecture now appears (with a nominal prize offered) independently at:

Simon Fraser Univ Open Problem Garden

I make note of basic work done with primorials by Sokol and Potter independently:

http://johnsokol.blogspot.com/ Sokol blog/email

from Sokol, it reads (sic)

"Sokol's conjecture:

A primes can only exists + or - a prime from a primorial.

Where 1 is considered a prime and 2 is not."

[http://primorialconjecture.org/default.aspx] Bob Potter note website link recently moved! (as did Graeme McRae's http://2000clicks.com/MathHelp/ )

from Potter, it reads:

"A contiguous sequence of prime numbers when in the vicinity of a primorial (or primorial multiple) will combine with other (probably non contiguous) primes to make a Goldbach pair for the primorial. The length of sequence for which this effect holds increases as the value of the primorial or primorial multiple increases, in the limit the sequence will tend to an infinite contiguous sequence".

In particular, Potter's Goldbach partition work is reinforced by my thinking that the use of MC provides a solution subset to Goldbach's for every non-adjacent prime (not every even).

A reviewer, Jens K Andersen (aka PrimeHunter) of Wikipedia, confirms the conjecture holds to 10^9, which is approx the first 51 million primes...I hope someone xtends into 64-bit region.

 

As an aside, Bertrand's Postulate (theorem) falls out as part of the conjecture (at least one prime between N and 2N-2.

 

Here is a simple demo exe to show how things work. It will requires the file c:\\prime2\\prevprime2.txt exist (I include it). Run the exe out of the DOS console, it will pre-fill a prime array. Then follow directions - if the output shows a list of sequence entry offsets, some not marked composite, the # is VERY likely (probable) prime if one of the distances is prime. File output is generated to c:primespdist.txt.

prime list

 

demo exe

For example, entering 2807 reports it as composite. Entering 8191 reports the applicable sequence entry offsets. Several offsets are potentially non-composite (81

79, 8161, 8011, 5881, 5669, 4409 and 3517. We can confirm at least 1 is prime, so 8191 is probable prime (and it is). The pgm does this with 3 prime lookups OR 76 factor divisions.

Here are my brute force source and exe (C++). Please don't criticize my code, I just banged it out to verify the conjecture and is very quick&dirty. It will NOT work above specifying a value of 2.2E9. Entering 1000001 beginning with no previous primes took approx 56 minutes on my XP 3.2 Ghz box (I must switch to sieve of Atkin!). The program produces screen output as well as (2) files, c:primesmylist.txt and c:primesprimality.txt. The former is the primes identified by the program and the latter reports highest prime target and the % numbers classed as composite via the conjecture. Various limits and array sizing could be upsized if using 64-bit. Note this version (cpp source) is missing 1 of the 69 terms in the array (415800), thanks to Vladeta Jovovic. In fact, replace the source array with Vladeta's which is here in both Word and Open Office, thru 80 terms, though my code uses array beginning at 2, not 1

Word

 

OpenOffice

 

G++ source

 

G++ exe

 

  • can someone provide a counter example?? or indicate if this conjecture is already known to math circles? To properly report a conjecture failure for my checked range, one would need to have a verified prime number, and then show that none of the up to 80 offsets were prime themselves.

*

One would then have to provide any sequence entries used beyond the 80th. To check a number like 1E8-1, the largest offset that must be checked is of the same size, ie 1E8-3, which could involve verifying against a sequence entry of approx. 2E8, which is beyond entries I have computed - not a single sequence entry can be missed or one may likely get such false breakdowns.

The true usefulness manifests when one is working with very large candidate primes.

One basic method for primality (deterministic if conjecture is true) involves the aforementioned as the first step of 2 steps.

Obviously it is a tremendous advantage to involve differences rather than division.

My commentary is on the Wikipedia Primorial discussion page, with less than enthusiastic response. Scrutinize and/or run the program above and hopefully you'll see it works as advertised...its drawback of course it that one must have all previous primes available, otherwise one is merely substituting testing a series of smaller numbers for primality in lieu of the larger original. I have another version that does an initial read in of primes off file but anyone can add that code to what I'm sure will be their improved coded version to mine.

Now, looking at the prime factor breakout of the sequence terms, it appears strangely related to another OEIS sequence A071562, in a bit of a convoluted manner I will explain later. Suffice to say, the connection makes it quite a bit easier to generate sequence terms. In fact, an email from the author (Vladeta Jovovic) of that sequence claims, and I insert his words here:

"

They are of the form:

2^k(1)*3^k(2)*5^k(3)*...*p(s)^k(s),

where p(s) is s-th prime

k(i)>0, i = 1, 2, ... , s

.

k(i) - k(i-1) = 0 or 1 for i = 2, 3, ... , s

 

  1. { k(1), k(2), ... , k(s) } = k(1)"

where #A ( or A ) denotes the cardinality of set A ...

as an aside, here is a plot (that I consider interesting) of the min. distance of primes from sequence primes, thru 10^5 and normalized.  The image is now at WikiCommons:  commons.wikimedia.org/wiki/File:OEIS_A129912_spin1.svg

It is fairly easy to recognize the lower part of curve is being sucked up to the 0 line...as "+" indicates where sequence entry >N, this tells one for larger primes, one should look to prime offsets near sequence entries > the candidate. Here the plotted value is (dmin/N)^2, where dmin is the minimum prime distance to a sequence entry

 

    • my prelim HCN conjecture is really just Fortune's Conjecture, which is a very small part of my new conjecture...see this link for details:

http://www-lipn.univ-paris13.fr/~banderier/Computations/prime_factorial.html

Jul 23rd: ok, done with Pari-Gp implementation. here is the pari output for A129912 (primorials within ULL), 82 entries. $PEND pari script to generate the array

pari list

I must again emphasize how helpful Vladeta Jovovic has been re: my work ...I am rechecking the TP found via using 100th primorial...here is my crude pari script for this specific search: http://billymac00.pbwiki.com/f/chek.gp

circa Dec '07 I had a fortunate meeting with a Canadian mathematician on my trip, appreciate his feedback (Irwin Pressman). 

 

Nov 2009  I came across A068372 which treats numbers that are a prime distance from a specific primorial...I adde a comment concerning many of the entries P being a prime offset from all primorials <2P-1

 

**END

 

billymac00.myopenid.com  (http://www.sagenb.com/login uses newest pswd, NOT via openid)

alternately, for Octave, http://www.verbosus.com/octavus.html  std older login

 

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