- --- On Mon, 3/11/13, WarrenS <warren.wds@...> wrote:
> If it is your goal to choose X so

How many yield square roots of -1 that aren't the same (or additive inverses) from the two SPRP tests?

> there are few spsp{2, X},

> then X=735 does pretty well.

> There are 57 spsp{2, 735} below 2^32

> and 1149508 spsp{2, 735} below 2^64.

> The latter is substantially better than Broadhurst's

Once you've got a SPRP, on average each new SPRP test removes only a quarter of the fakes. Culling 140/140 with one test seems unlikely. I'm not sure if that quarter includes the square (and higher) root hack.

> X=858945; there are 1311216 spsp{2,858945} below 2^64.

> I make no claim 735 is best, it is merely comparatively

> good.

>

> It would probably be possible to devise an infallible

> isprime test for

> <=64-bit integers by performing spsp(2) and spsp(735)

> tests, and then hashing

> the 1149508 failures into 8192 bins (each bin representing

> about 140; need to

> devise the hash function to get good equidistribution)

> and then the content of that table entry tells you which

> base B to use for a final spsp(B)

> test. (B chosen to kill all fakes in its hash-table

> bin.)

> This primality test, in net, would involve three spsp tests

The $620 test (not $640) can already be performed with a cost of 3 SPRP tests, so this wouldn't be an improvement over what's already done.

> and one hash-table lookup in a table with 8192 entries; you

> could probably compute the table in a few days.

> I am in fact working on an infallible 64-bit isprime test,

In the kind of science I like, reports of failure are useful too. Knowing what alleyways are blind is useful, but also someone else might have a final insight to patch things up.

> but not using the method I just outlined,instead based on

> combining an spsp(2) test with a Lehmer test.

> I'll report on that later, if I succeed.

Phil

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[stolen with permission from Daniel B. Cristofani] - On 3/13/2013 5:47 AM, Phil Carmody wrote:
> No I couldn't. That's so overly verbose and redundant it makes me twitch, I can

Apologies, my Pari/GP script-fu is definitely not on par with DJB's. I just

> barely bring myself to repeat it!

> "lift(Mod(p*s, lift(znorder(Mod(2,s))))) == 1" is just

> "p*s % znorder(Mod(2,s)) == 1"

> Having 3 exit conditions to the loop is overkill too.

wanted to provide a script so that people could plug and chug an r,p pair to see

what psp's would be generated. Also, I didn't know that the % (mod) operator

still worked in Pari. I thought everything had to be done with Mod(). Thanks

for that insight.

> The latter worries me a bit, as it might imply wasted effort. I'm trying to

And apologies here too, I mis-remembered a statement from his Category S page

> picture how these duplicates arise. Given a n, the maximal prime factor p|s is

> uniquely defined, and r as order_2(p) is uniquely defined. Therefore n can only

> appear with pair (r,p)?

and mis-spoke by applying it to the Category E psp's.

On 3/14/2013 11:02 AM, WarrenS wrote:

> 2. Consulting the Cunningham project pages,

> http://homes.cerias.purdue.edu/~ssw/cun/index.html

> every Mersenne-form number 2^r - 1 now is fully factored if

> r<929. Apparently the first two open cases are r=929 and 947

> yielding 214 and 217 digit numbers to factor.

An update here: M929 has been factored, and a group of people have already

started factoring M947. You can find the factor for M929 here:

http://homes.cerias.purdue.edu/~ssw/cun/page125

And, you can see who is factoring which Cunningham number here:

http://homes.cerias.purdue.edu/~ssw/cun/who

And more importantly, you can find all known*1 factors for all important*2

numbers in the online factor database here:

factordb.com

Once there, you can type in 2^929-1, and it will show you all the factors of

that number and that it is Fully Factored (FF). Currently, you can type in

2^947-1 and see that it is a CF, composite number, with factors known, but not

yet fully factored.

*1 = All known factors that have been stored into the factordb.

*2 = All numbers that people are interested in and store in the factordb.

Also, the factordb stores prime numbers too. Below 300 digits it will just

prove the number prime, and above that it will accept Primo certificates and

verify them locally.

-David C.