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 Showing results 1 to 25 of 56 Search took 0.01 seconds. Search: Posts Made By: mickfrancis
 2020-07-20, 15:55 Replies: 4 Views: 3,437 Posted By mickfrancis Thanks for this. I shall cogitate upon it! Thanks for this. I shall cogitate upon it!
 2020-07-20, 08:04 Replies: 4 Views: 3,437 Posted By mickfrancis Thanks for the reply, Robert. I don't think this... Thanks for the reply, Robert. I don't think this is quite what I'm saying, though I may just be misunderstanding - forgive me if so. What I have noticed is that \phi\left(p^{n} \pm q^{n}\right) is...
 2020-07-18, 19:17 Replies: 4 Views: 3,437 Posted By mickfrancis Factors of Euler Totient Function of sum/difference of prime powers It appears that for prime p and q, p>= q, \phi\left(p^{n} \pm q^{n}\right) is divisible by (often a high power of) n. The power of n seems to increase with the number of distinct prime factors of n....
 Forum: Math 2017-03-01, 07:17 Replies: 16 Views: 2,109 Posted By mickfrancis I'd never come across this before - thanks! ... I'd never come across this before - thanks! Mick.
 Forum: Math 2017-02-28, 13:44 Replies: 16 Views: 2,109 Posted By mickfrancis Thanks for your input on this, Mick. Thanks for your input on this, Mick.
 Forum: Math 2017-02-27, 21:42 Replies: 16 Views: 2,109 Posted By mickfrancis Changing the question... Having given some thought to this, I realise that what I really want is a way to find integer values for m such that (ceil(mr))^2- (mr)^2 < (mr)^s where s <= 1. I'm guessing this is a...
 Forum: Math 2017-02-27, 15:51 Replies: 16 Views: 2,109 Posted By mickfrancis Really helpful - thank you - plenty for me to get... Really helpful - thank you - plenty for me to get my teeth into there! Regards, Mick.
 Forum: Math 2017-02-27, 15:20 Replies: 16 Views: 2,109 Posted By mickfrancis Thanks for the response. I'm afraid you'll have... Thanks for the response. I'm afraid you'll have to forgive my ignorance, but I can't see at the moment how continued fractions help me here - any hints appreciated. Mick.
 Forum: Math 2017-02-27, 14:39 Replies: 16 Views: 2,109 Posted By mickfrancis Finding multiples of a real number that are close to a whole number Given a real number r, and a small positive value e arbitrarily close to 0, does anyone know of a fast way to find integer multipliers m such that either: {mr} < e or 1 - {mr} < e (where {mr}...
 Forum: Factoring 2016-11-09, 12:19 Replies: 2 Views: 1,010 Posted By mickfrancis Interesting - thanks Jason. Interesting - thanks Jason.
 Forum: Factoring 2016-11-09, 10:41 Replies: 2 Views: 1,010 Posted By mickfrancis Zhang's Special Quadratic Sieve Has anyone had any experience with implementing Zhang's Quadratic Sieve? I can't seem to find the implementation referred to (as being included with the paper) in Eric Landquist's paper "An...
 Forum: Math 2016-08-15, 13:08 Replies: 4 Views: 1,097 Posted By mickfrancis It can be rewritten as... It can be rewritten as \displaystyle\prod_{i=1}^{p} {(1 + \frac{n-p}{i+p})} , I think, but I'm not sure this helps...
 Forum: Math 2016-08-15, 12:23 Replies: 4 Views: 1,097 Posted By mickfrancis Good thought. I think the p and 2p are transposed... Good thought. I think the p and 2p are transposed in that ratio? I get: ((n+p)! * p!)/((2p)! * n!), which can be written as \frac{\displaystyle\prod_{i=1}^{p} {(n+i)}} {\displaystyle\prod_{i=1}^{p}...
 Forum: Math 2016-08-15, 09:31 Replies: 4 Views: 1,097 Posted By mickfrancis Fast calculation of binomial coefficients If I know {n} \choose {p} (mod m), is there a way of calculating {n+p} \choose {2p} (mod m) with time complexity better than $\mathcal{O}(p)$? m <= n p <= n/2
 Forum: Factoring 2016-05-06, 08:13 Replies: 2 Views: 3,965 Posted By mickfrancis Makes sense - thanks Jason. Makes sense - thanks Jason.
 Forum: Factoring 2016-05-04, 14:39 Replies: 2 Views: 3,965 Posted By mickfrancis Sieving with powers of small primes in the Small Prime variation of the Quadratic Sieve As I understand it, in the Small Prime Variation of the Quadratic Sieve, primes less than a threshold (Pmin, say) are not used for sieving, as the cost of sieving is disproportionately high given the...
 Forum: Factoring 2016-05-03, 08:50 Replies: 3 Views: 1,057 Posted By mickfrancis Thanks Jason. Thanks Jason.
 Forum: Factoring 2016-05-02, 17:54 Replies: 3 Views: 1,057 Posted By mickfrancis Special-q method for Quadratic Sieve Does anyone have a reference to / copy of a good description of the special q method as used with the Quadratic Sieve? All I can find is a brief reference in a paper entitled "On the Amount of...
 Forum: Factoring 2016-03-31, 06:21 Replies: 5 Views: 1,135 Posted By mickfrancis Ah, well, this is my point you see :) I have a... Ah, well, this is my point you see :) I have a non-sieving implementation, in Haskell, that does generate enough relations to factor a C80 in 14 hours. As far as I can tell, this is currently the...
 Forum: Factoring 2016-03-30, 20:57 Replies: 5 Views: 1,135 Posted By mickfrancis Can CFRAC really factor a C80 (in less than 14... Can CFRAC really factor a C80 (in less than 14 hours)?
 Forum: Factoring 2016-03-30, 15:24 Replies: 5 Views: 1,135 Posted By mickfrancis Thanks Jason. Interestingly, because of the way... Thanks Jason. Interestingly, because of the way the candidate x values are chosen, subsequent factoring is on values less than sqrt(n), sometimes by a factor of more than 10. Also, the values being...
 Forum: Factoring 2016-03-30, 13:47 Replies: 5 Views: 1,135 Posted By mickfrancis Non-sieving version of Quadratic Sieve I've been experimenting with a non-sieving version of the Quadratic Sieve algorithm. To date the performance is nowhere near as good as the basic Quadratic Sieve, let alone SIQS; gathering relations...
 Forum: Math 2016-03-29, 18:49 Replies: 9 Views: 1,320 Posted By mickfrancis I knew I should have invested in those volumes! I knew I should have invested in those volumes!
 Forum: Math 2016-03-29, 14:52 Replies: 9 Views: 1,320 Posted By mickfrancis I'm not sure why it would be a problem - it's... I'm not sure why it would be a problem - it's really just modular subtraction on the left and subtraction in a Residue Number System on the right isn't it? (I'm probably missing something here...
 2016-03-29, 14:26 Replies: 9 Views: 1,607 Posted By mickfrancis Hi Jason, Do you use Bernstein's Scaled... Hi Jason, Do you use Bernstein's Scaled Remainder Trees? If not, have you experimented with them at all? Mick.
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