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Forum: Math 2020-10-06, 18:20
Replies: 12
Views: 715
Posted By jshort
Thanks Dr Sardonicus. I didn't think a...

Thanks Dr Sardonicus.

I didn't think a counter example could be found for such small n. My mistake was in assuming that the non-primitive part of 2^{n} + 1 must contain many factors from smaller...
Forum: Math 2020-10-03, 23:11
Replies: 12
Views: 715
Posted By jshort
Question about number of the form 2^n + 1

Let a_{n} be the nth Highly Composite Integer* and b_{n} = lcm(2^{a_{1}} + 1, 2^{a_{2}} + 1, ...,2^{a_{n}} + 1). If c_{n} = \frac{2^{a_{n}} + 1}{gcd(2^{a_{n}} + 1,b_{n-1})}, will c_{n} always be...
Forum: Number Theory Discussion Group 2020-10-01, 22:15
Replies: 16
Views: 2,756
Posted By jshort
Let S_{1} = 4 = (2 + \sqrt{3}) + (2 - \sqrt{3})...

Let S_{1} = 4 = (2 + \sqrt{3}) + (2 - \sqrt{3}) and S_{n+1} = S_{n}^{2} - 2.

We'll first prove via induction that S_{n} = (2 + \sqrt{3})^{2^{n-1}} + (2 - \sqrt{3})^{2^{n-1}}.

Clearly S_{1} =...
Forum: Number Theory Discussion Group 2020-10-01, 20:15
Replies: 3
Views: 455
Posted By jshort
I know its bad form to write answers to your own...

I know its bad form to write answers to your own question. To be honest I don't have a straightforward answer as to whether or not this way of conducting a 2nd-stage to the p-1 method is faster than...
Forum: Number Theory Discussion Group 2020-09-30, 20:05
Replies: 3
Views: 455
Posted By jshort
What your describing is something completely...

What your describing is something completely different altogether and it isn't the Pollard rho factoring algorithm;

This is the Pollard-rho factoring algorithm;

rho(n)=
{
local(x,y);

...
Forum: Number Theory Discussion Group 2020-09-30, 02:11
Replies: 3
Views: 455
Posted By jshort
alternative 2nd stage of p-1 factoring algorithm

Suppose we're factoring an integer via the p-1 method and we've already completed the first stage ie. L = a^{B!} mod(n) where n is the composite we wish to factor.

In the 2nd stage, we assume...
Forum: Proth Prime Search 2020-06-11, 23:46
Replies: 5
Views: 2,098
Posted By jshort
Proths of the form 2^{k+1}(2^{k} - 1) + 1, 2k+1 is prime

(I guess am sort of answering my own question here......but here goes)

Are Proth Primes of the form 2^{k+1} \cdot (2^{k} - 1) + 1 worthwhile to search for?

Fyi - I'm specifically referring to...
Forum: Other Mathematical Topics 2019-12-19, 17:31
Replies: 5
Views: 1,213
Posted By jshort
How much smaller can "primitive" factors be?

When looking for "easy" to find integers N that are forced to be congruent to 1 + k \cdot p where the congruence factor p is "as large as possible" compared to N, are the primitive factors of...
Forum: Number Theory Discussion Group 2019-12-15, 22:48
Replies: 4
Views: 1,625
Posted By jshort
Thank you sir. Interesting that the p-1 test...

Thank you sir.

Interesting that the p-1 test isn't used straight away, but rather a trivial Sieve of Eratosthenes.

Anyways, I'd imagine that the bounds B and B' must be awfully small.
Forum: Number Theory Discussion Group 2019-12-15, 17:34
Replies: 4
Views: 1,625
Posted By jshort
How are composite Mersenne's sieved (weeded) out?

Is there a sieving process that is done before the Lucas-Lehmer primality test is run to test if a Mersenne is prime?

Normally (that is for random integers), one simply uses a primordial number...
Forum: Number Theory Discussion Group 2019-11-25, 00:41
Replies: 10
Views: 1,716
Posted By jshort
Looking at these results F(n,p) for n = 1 - 6 it...

Looking at these results F(n,p) for n = 1 - 6 it looks rather discouraging to say the least!

I knew that as n increase the number of primes drops drastically, however for n > 3, the frequency of...
Forum: Number Theory Discussion Group 2019-11-24, 22:32
Replies: 10
Views: 1,716
Posted By jshort
Thank you for this.

Thank you for this.
Forum: Number Theory Discussion Group 2019-11-24, 22:24
Replies: 10
Views: 1,716
Posted By jshort
Thank you for the references. I wasn't aware of...

Thank you for the references. I wasn't aware of Mike Oakes's work, but will definitely read up on him :)

I haven't considered [$]b^{2^n} +1[/$], with b > 2 because their bit-size is even bigger...
Forum: Number Theory Discussion Group 2019-11-24, 20:30
Replies: 10
Views: 1,716
Posted By jshort
Other ways of finding WR primes

I've been thinking about this for some time as to why Mersenne integers are the #1 candidate for finding WR prime numbers, and have basically come to these two reasons:

1) There is a fast...
Forum: Factoring 2019-03-20, 22:52
Replies: 9
Views: 1,186
Posted By jshort
Re - storing residues You generally only...

Re - storing residues

You generally only need to store one residue. It's a "reference" point that we assume is already inside the periodic cycle for which we compare future residues with. If our...
Forum: Factoring 2019-03-17, 23:45
Replies: 9
Views: 1,186
Posted By jshort
Thanks for the reference ThilaHarich…. Yeah...

Thanks for the reference ThilaHarich….

Yeah the factorization of F_{8} was the most successful application of the Pollard Rho method to date for sure. Interestingly Brent mentions that there...
Forum: Factoring 2019-03-14, 20:23
Replies: 9
Views: 1,186
Posted By jshort
Factoring composite Mersenne numbers using Pollard Rho

First post here - sorry if this is either an obvious or mundane question.

The Pollard Rho factoring method is suppose to have a running time proportional to n^{\frac{1}{4}}, however if you're...
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