## Re: [PrimeNumbers] Largest Ordinary Prime?

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• Firstly, congratulations to Paul, Paul, David and Bouk. Secondly, DAMN. I had this great idea while I was on holiday about ten days ago. It appears to be much
Message 1 of 22 , Oct 3, 2002
Firstly, congratulations to Paul, Paul, David and Bouk.

Secondly, DAMN. I had this great idea while I was on holiday
about ten days ago. It appears to be much the the same idea that
you guys had a long while earlier ;-)

I used my experience in trying to prove Gigantic GRUs to model
how many digits of prime factors I would expect to extract from
a cyclotomic factor of a given size given the amount of ECM work
I have done on such factors to date (about 7-10 GHz hours on each
composite on average, though some I've just started and a few have

I then generated a list of promising composite exponents c such that
2^c-1 gave a decent chance of being useful in proving 2^a - 2^b +/- 1
where c=a-b. 64680 came out at 19.85%, so your 30% indicates how much
more work you did on it (and a bit of luck). I reckon the best few
exponents of around the same size are: 65520, 75600, 69300, 83160,
73920, 70560 then 64680.

Another thought I had was to use 2^c-1 where it's a Mersenne prime
... or c=2p, 3p etc where 2^p-1 is a Mersenne prime: diminishing
returns but worth a try for small multiples.

Ah well, I'll stick to the GRUs.

Good luck,
Andy
• ... Then recall that Paul U restricted his search of 2^a-2^b-1 to b
Message 2 of 22 , Oct 4, 2002
Andy:

> I reckon the best few exponents of around
> the same size are: 65520, 75600, 69300,
> 83160, 73920, 70560 then 64680.

Then recall that Paul U restricted his
search of 2^a-2^b-1 to b<=40.

So that gives 280 chances for those 7
values of c=a-b, whereas
ln(N) ~ ln(2^70000) = 50,000,
giving you odds of order 100:1 against.

It's neat that one of your 280 showed up,
with b=15, and even neater that Bouk's eagle
eyes spotted it in Henri's pages.

PS: How's the latest GRU cooking?

Remember that Phil has just offered
double the usual extra bits,
if/when you need them :-)

Also Paul L was holding NFS cycles
in reserve for 2^15*(2^64680-1)-1
which we didn't need. So maybe
you could do something really
impressive with another 150 digits
from him, in some strategic place :-?

Best wishes

David
• ... b
Message 3 of 22 , Oct 4, 2002
> > I reckon the best few exponents of around
> > the same size are: 65520, 75600, 69300,
> > 83160, 73920, 70560 then 64680.
>
> Then recall that Paul U restricted his
> search of 2^a-2^b-1 to b<=40.
>

b<44 actually. Also a<100000.

Bouk suggests having a look at 118080.

I will find minimal PrPs for all the suggestions at the end of the
month; also for 2^a-2^b+1. ;-)

Paul
• ... Did you mean 110880? I get an expected factorisation of 18.4% for that one, but only 9.7% for 118080. By my model, 110880 is the most promising exponent in
Message 4 of 22 , Oct 5, 2002
Paul Underwood wrote:

> Bouk suggests having a look at 118080.

Did you mean 110880? I get an expected factorisation of
18.4% for that one, but only 9.7% for 118080. By my model,
110880 is the most promising exponent in (85680,750000]

Here's a list of exponents where each entry has a higher
expected factorisation yield than any larger:

Exponent, factors, expectation
1680,40,100.00%
1740,24,96.33%
1890,32,95.48%
2100,36,94.98%
2520,48,91.05%
2640,40,88.53%
2940,36,88.41%
3360,48,86.53%
3780,48,82.88%
3960,48,80.44%
4200,48,79.05%
4620,48,78.67%
5040,60,78.06%
5460,48,71.49%
5880,48,69.92%
6120,48,69.12%
7560,64,68.14%
7920,60,63.67%
9240,64,62.61%
10080,72,61.54%
10920,64,56.45%
12600,72,53.33%
13860,72,52.73%
15120,80,51.39%
16380,72,46.94%
18480,80,46.48%
20160,84,43.47%
21840,80,41.16%
27720,96,39.70%
30240,96,36.29%
32760,96,35.11%
36960,96,31.72%
37800,96,30.69%
40320,96,28.96%
41580,96,28.95%
42840,96,28.73%
55440,120,28.00%
65520,120,24.39%
75600,120,21.10%
83160,128,21.05%
85680,120,19.43%
110880,144,18.40%
131040,144,15.89%
138600,144,15.06%
166320,160,14.15%
171360,144,12.41%
196560,160,12.14%
221760,168,11.37%
240240,160,10.17%
277200,180,9.96%
332640,192,8.95%
360360,192,8.35%
393120,192,7.64%
415800,192,7.26%
443520,192,6.75%
471240,192,6.43%
480480,192,6.31%
498960,200,6.29%
554400,216,6.21%
556920,192,5.47%
720720,240,5.39%
739200,192,4.09%
742560,192,4.08%
748440,192,3.93%
749700,162,3.33%

HTH,
Andy
• ... 110880 it should be indeed. Moreover because phi(36960,2) is a prp. 36960 * 2 = 73920 2^73920-1 is already BLS 36960 * 3 = 110880 2^110880-1 is about 1300
Message 5 of 22 , Oct 5, 2002
> Paul Underwood wrote:
>
> > Bouk suggests having a look at 118080.
>
> Did you mean 110880? I get an expected factorisation of
> 18.4% for that one, but only 9.7% for 118080. By my model,
> 110880 is the most promising exponent in (85680,750000]

110880 it should be indeed. Moreover because phi(36960,2) is a prp.

36960 * 2 = 73920

36960 * 3 = 110880

2^110880-1 is about 1300 digits short for Konyagin-Pomarance.

Bouk.

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• ... Not yet... ;-) Phil ===== First rule of Factor Club - you do not talk about Factor Club. Second rule of Factor Club - you DO NOT talk about Factor Club.
Message 6 of 22 , Oct 5, 2002
--- Bouk de Water <bdewater@...> wrote:
> Moreover because phi(36960,2) is a prp.
^^^
> 36960 * 2 = 73920
>

Not yet... ;-)

Phil

=====
First rule of Factor Club - you do not talk about Factor Club.
Second rule of Factor Club - you DO NOT talk about Factor Club.
Third rule of Factor Club - when the cofactor is prime, or you've trial-
divided up to the square root of the number, the factoring is over.

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• ... True, oh sage. But 2312 digits would be a mere bagatelle for primo. More significantly what a would make 2^a*(2^73920-1)-1 PrP? On average you have to
Message 7 of 22 , Oct 5, 2002
Phil:

> Not yet... ;-)

True, oh sage.

But 2312 digits would be
a mere bagatelle for primo.

More significantly what "a" would make
2^a*(2^73920-1)-1
PrP?

On average you have to wait a long time
and by then 2^a gives you most of BLS anyway,
thus eroding you factorization work.

It was Paul U's a=15 that was really notable in

2^15*(2^64680-1)-1

David
• ... [snip] ... To be precise, 239 phi-factors and 5.895144%: http://groups.yahoo.com/group/primenumbers/files/Factors/m720720.zip Best, Andrey [Non-text
Message 8 of 22 , Oct 13, 2002
Andy Steward wrote:

> Here's a list of exponents where each entry has a higher
> expected factorisation yield than any larger:
>
> Exponent, factors, expectation
[snip]
> 720720,240,5.39%

To be precise, 239 phi-factors and 5.895144%: