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 Showing results 1 to 17 of 17 Search took 0.07 seconds. Search: Posts Made By: jshort
 Forum: Math 2020-10-06, 18:20 Replies: 12 Question about number of the form 2^n + 1 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 Question about number of the form 2^n + 1 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... 2020-10-01, 22:15 Replies: 16 Repeating residues in LL tests of composite Mersenne numbers 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} =... 2020-10-01, 20:15 Replies: 3 alternative 2nd stage of p-1 factoring algorithm 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... 2020-09-30, 20:05 Replies: 3 alternative 2nd stage of p-1 factoring algorithm 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); ... 2020-09-30, 02:11 Replies: 3 alternative 2nd stage of p-1 factoring algorithm 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 Proths of the form 2^{k+1}(2^{k} - 1) + 1, 2k+1 is prime 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... 2019-12-19, 17:31 Replies: 5 How much smaller can "primitive" factors be? 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... 2019-12-15, 22:48 Replies: 4 How are composite Mersenne's sieved (weeded) out? 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. 2019-12-15, 17:34 Replies: 4 How are composite Mersenne's sieved (weeded) out? 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... 2019-11-25, 00:41 Replies: 10 Other ways of finding WR primes 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... 2019-11-24, 22:32 Replies: 10 Other ways of finding WR primes Views: 1,716 Posted By jshort Thank you for this. Thank you for this. 2019-11-24, 22:24 Replies: 10 Other ways of finding WR primes 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... 2019-11-24, 20:30 Replies: 10 Other ways of finding WR primes 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 Factoring composite Mersenne numbers using Pollard Rho 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 Factoring composite Mersenne numbers using Pollard Rho 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 Factoring composite Mersenne numbers using Pollard Rho 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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