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Search took 0.03 seconds. Search: Posts Made By: mickfrancis

Forum: Number Theory Discussion Group

2020-07-20, 15:55

Replies: 4

Factors of Euler Totient Function of sum/difference of prime powers

Views: 5,372

Posted By mickfrancis

Thanks for this. I shall cogitate upon it!

Thanks for this. I shall cogitate upon it!

Forum: Number Theory Discussion Group

2020-07-20, 08:04

Replies: 4

Factors of Euler Totient Function of sum/difference of prime powers

Views: 5,372

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...

Forum: Number Theory Discussion Group

2020-07-18, 19:17

Replies: 4

Factors of Euler Totient Function of sum/difference of prime powers

Views: 5,372

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

Finding multiples of a real number that are close to a whole number

Views: 3,196

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

Finding multiples of a real number that are close to a whole number

Views: 3,196

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

Finding multiples of a real number that are close to a whole number

Views: 3,196

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

Finding multiples of a real number that are close to a whole number

Views: 3,196

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

Finding multiples of a real number that are close to a whole number

Views: 3,196

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

Finding multiples of a real number that are close to a whole number

Views: 3,196

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

Zhang's Special Quadratic Sieve

Views: 1,417

Posted By mickfrancis

Interesting - thanks Jason.

Interesting - thanks Jason.

Forum: Factoring

2016-11-09, 10:41

Replies: 2

Zhang's Special Quadratic Sieve

Views: 1,417

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

Fast calculation of binomial coefficients

Views: 1,543

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

Fast calculation of binomial coefficients

Views: 1,543

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

Fast calculation of binomial coefficients

Views: 1,543

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

Sieving with powers of small primes in the Small Prime variation of the Quadratic Sieve

Views: 6,039

Posted By mickfrancis

Makes sense - thanks Jason.

Makes sense - thanks Jason.

Forum: Factoring

2016-05-04, 14:39

Replies: 2

Sieving with powers of small primes in the Small Prime variation of the Quadratic Sieve

Views: 6,039

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

Special-q method for Quadratic Sieve

Views: 1,488

Posted By mickfrancis

Thanks Jason.

Thanks Jason.

Forum: Factoring

2016-05-02, 17:54

Replies: 3

Special-q method for Quadratic Sieve

Views: 1,488

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

Non-sieving version of Quadratic Sieve

Views: 1,607

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

Non-sieving version of Quadratic Sieve

Views: 1,607

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

Non-sieving version of Quadratic Sieve

Views: 1,607

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

Non-sieving version of Quadratic Sieve

Views: 1,607

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

CRT optimisation?

Views: 1,775

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

CRT optimisation?

Views: 1,775

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...

Forum: Computer Science & Computational Number Theory

2016-03-29, 14:26

Replies: 9

Speeding up (n mod m) for very large fixed n and varying smaller m

Views: 2,203

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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