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:

• https://www.quora.com/Why-do-integers-mod-a-prime-number-create-a-finite-field
• MIT lecture notes - commutative algebra
• List of supplementary commutative algebra texts - mathoverflow list

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:

• Published sequence A325914 [ oeis.org ] Proth primes
• sal683onetime (onetime encryption scheme) 
• oeis.org - author search