Thursday, April 13, 2023

sal1520 abundant relations make it ...

 A new [financial] year and a new book

First 6 chapter titles listed below:

  • Chapter 1: Metaverse - Sharing a power - asymmetric setup t=228 (silver)
  • Chapter 2: Metaverse - Object exchange using (small) data bandwidth t=329
  • Chapter 3: Metaverse - test key examples are 'fluted'
  • Chapter 4: Metaverse - Recap and source of asymmetric strength
  • Chapter 5: Metaverse - Bare bones draft spec for inter metaverse object exchange
  • Chapter 6: GP example and links

...


Next the downloadable pdf

( free copy just include a couple of termite images to distinguish from a paid copy )

Sunday, January 1, 2023

Agile Mathematics for Strong Hashing

 A new year and a new book

First 6 chapter titles listed below:

  • Chapter 1: Ingredients
  • Chapter 2: Large proth primes and computer testing / calculation
  • Chapter 3: Pivoting into other areas of Mathematical exploration
  • Chapter 4: 2014 to 2017/2018
  • Chapter 5: 2018 and 2019 starting to gel
  • Chapter 6: Review and claims

...


Next the downloadable pdf

( free copy just include a couple of termite images to distinguish from a paid copy )

 

Thursday, September 12, 2019

proth research - directories and more directories

After several years of working with Proth numbers, looking for interesting properties, a filesystem will get pretty packed!








How do I organise directories in a mass of say 500 or 1000 directories? Starting off with fixed so prothleadNNNNNN (Example: prothlead8388607)

If/when I find something interesting, and think its worth generalising, then I just replace the NNNNNN number with a short sequence of characters.
They have no meaning generally, just a way of making things stand out in a mass of directories.

What you can see from the final image 'prSAL' is how I labelled the prothlead8388607 directories after I generalised away from just the t=23 case. It helped to quickly think up something short - just a way of less typing at the time.

I could still be writing prothlead8388607 today but it just felt like too much and so i found a shorter thing that just occurred to me at the time. ( pretty arbitrary )

I could keep posting screenshots until I had 10 images on screen here to show the problem in hunting around in Proth numbers over a period of 7 years and keeping things organised in directories!


When 2*16 congruent to 5*29 do we still have Unique Factors?

Commutative algebra / abstract algebra was a joy to study, and coupled with Number theory is a superb grounding for second and third year undergraduate study.

Current lecture notes and set texts seem to suggest that whenever p is prime, that Z/pZ is an integral domain, a unique factors setup, ... and a Field.




Quoting from some MIT lecture notes:
"We call R a Unique Factorization Domain(UFD) if every nonzero element is a product of irreducible elements in a unique way up to order and units"

Take a look at the image above and see what you think?

The key thing that is easy to overlook is the part of the definition that says 'irreducible elements'

Here we would need to check each of the 4 elements 2,16,5,29 are irreducible mod 113.
We would not say that 16 is 'irreducible'

Links and References:

Tuesday, September 10, 2019

subset of proth with ord2 property that is interesting - submission progress for these new primes

Have submitted a list of primes in a new sequence to oeis and waiting on publication approval.

Sometime these things take a while, so in order to ensure provenance and dates, here are a few snapshots of the in progress submission.

The publishing process has been a little more involved than I thought. Keeping the PROG element below for future reference.

One aspect that did not make it into the published version was the Conjecture that I added in an effort to unstick the process. I repeat it here so that it is not lost in old history.

Conjecture: There are NO Proth numbers of form pr=1+(-1+2^n)*2^(n+1) ), for which, working modulo pr the congruence -1== 2^(2*(1+2*n)) modulo pr is FALSE




Links and references:

Sunday, June 23, 2019

pari operation time - Miller Rabin

Setting 1 in the call to ispseudoprime() in Pari/GP will get you a strong pseudoprime test (Miller-Rabin)

Pseudoprime tests are more practical first attempts when dealing with huge inputs ( In this example the number has 632202 decimal digits )

How long did the operation take? ## shows 13 hours 58 minutes




Link and further reading:


Thursday, June 13, 2019

ABC and ABCDN formats for input to LLR and PFGW - numeric construction

With the creation of organised projects for prime number searching, and the retirement of some of the leading C program creators, the knowledge about ABC and ABCDN format is harder to find.

Here I gather a few links (see references) and show a source code sample that will help point you in the right direction.



LLR is great for standard formats (commonly mined prime areas) and is able to cope with many different formats, however where you differ from some of the more common +1 and -1 forms, the following warning is given in the program readme:
- K*b^n+c numbers with |c| < > 1 or k > b^n can only be PRP tested.

If the number is found PRP, the % of factorization is then shown,
     but note that it is relevant only if c == +1 or -1...


References and links:
  • LLR implements "Lucas-Lehmer-Riesel" and Proth algorithms - source
  • OpenPFGW - source
  • OpenPFGW page about ABC and ABC2 and ABCD - text
  • Caldwell pages at University of Tennesse at Martin - program links